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Test your knowledge of introductory Algebra with this Algebra practice exam. Whether you are studying for a school math test or looking to test your math skills, this free practice test will challenge your knowledge of algebra.

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Free Algebra Questions and Problems with Answers

Free intermediate and college algebra questions and problems are presented along with answers and explanations. Free worksheets to download are also included.

Intermediate Algebra Questions with Answers

• sample 1 .(student generated solutions). Also included are the solutions with full explanations .
• sample 2 .(True / False type). Also included are the solutions with full explanations .
• sample 3 .(student generated solutions). solutions with full explanations are included.
• sample 4 .(True / False type). solutions with full explanations included.
• sample 5 .(Multiple choice questions type). Also included are the solutions with full explanations .
• sample 6 .(True / False type). solutions with full explanations included.

Intermediate Algebra Problems with Detailed Solutions

• Algebra Problems .
• Intermediate Algebra Problems With Answers - sample 1 : equations, system of equations, percent problems, relations and functions.
• Intermediate Algebra Problems With Answers - sample 2 :Find equation of line, domain and range from graph, midpoint and distance of line segments, slopes of perpendicular and parallel lines.
• Intermediate Algebra Problems With Answers - sample 3 : equations and system of equations, quadratic equations, function given by a table, intersections of lines, problems.
• Intermediate Algebra Problems With Answers - sample 4 . Functions, domain, range, zeros.
• Intermediate Algebra Problems With Answers - sample 5 . Scientific Notation
• Intermediate Algebra Problems With Answers - sample 6 . Equations of Lines
• Intermediate Algebra Problems With Answers - sample 7 . Slopes of Lines
• Intermediate Algebra Problems With Answers - sample 8 . Absolute Value Expressions
• Intermediate Algebra Problems With Answers - sample 9 . Solve Absolute Value Equations
• Intermediate Algebra Problems With Answers - sample 10 . Solve Absolute Value Inequalities
• Intermediate Algebra Problems With Answers - sample 11 . Simplify Algebraic Expressions by Removing Brackets
• Intermediate Algebra Problems With Answers - sample 12 . Simplify Algebraic Expressions with Exponents

Intermediate Algebra Worksheets

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• Worksheet (2) .
• Worksheet (3) .
• Worksheet (4) .
• Worksheet (5) .
• Worksheet (6) .

College Algebra Questions with Answers

• sample 1 .(multiple choice questions). Also Detailed solutions with full explanations are included
• sample 2 . (multiple choice questions)

College Algebra Problems with Answers

• sample 1: Quadratic Functions .
• sample 2: Composite and Inverse Functions .
• sample 3: Exponential and Logarithmic Functions .
• sample 4: Graphs of Functions .
• sample 5: Find Domain and Range of Functions .
• sample 6: Problems on Polynomials : Graphs, Factoring, Finding, Multiplying, Dividing, Factor theorem, Zeros
• sample 7: Equation of Circle : Finding equations, center, radius of circles
• sample 8: Equation of Ellipse : Finding equations, foci, center, vertices of ellipses
• sample 9: Equation of Parabola : Finding equations, focus, vertex, axis, directrix of parabola.
• sample 10: Equation of Hyperbola : Finding equations, foci, center and vertices of hyperbola.

College Algebra Worksheets

• Worksheet (1): Graphs of Basic Functions .
• Worksheet (2): Exponential Growth and Decay Problems .
• Worksheet (3): Graphing Exponential Functions .
• Worksheet (4): Graphing Logarithmic Functions .
• Worksheet (5): Solve Exponential Equations .
• Worksheet (6): Solve Logarithmic Equations .
• Worksheet (7): Multiple Choice Questions on Polynomials and Solutions
• Worksheet (8): Multiple Choice Questions on Rational Functions and Solutions .
• Worksheet (9): Graphing Inverse Functions .

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• Number Theory
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• Everyday Math
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Take a guided, problem-solving based approach to learning Algebra. These compilations provide unique perspectives and applications you won't find anywhere else.

Algebra through Puzzles

What's inside.

• Introduction
• Simplifying Shortcuts
• Arithmetic Logic and Magic
• Balancing Scales
• Rates and Ratios
• Equations and Unknowns
• Manipulating Exponents
• Algebra in Motion
• Common Misconceptions
• Function Fundamentals
• Transformations
• Powers and Radicals
• Polynomials
• Factoring Polynomials

Rational Functions

Piecewise functions, community wiki.

Browse through thousands of Algebra wikis written by our community of experts.

Expressions and Variables

• What Makes A Good Problem?
• Simplifying Expressions
• Distributive Property
• Zero Product Property
• Contest Math I
• Solving Equations
• Setting Up Equations
• Simple Equations
• Verifying Solutions
• Multi-step Equations
• Isolating a Variable
• Balance Puzzles
• System of Linear Equations (Simultaneous Equations)
• How are exponent towers evaluated?
• Does a square root have two values?
• Do Square Roots Always Multiply?
• Is 0.999... = 1?
• What is 0 to the power of 0?
• If AB=AC, does B=C?
• Is (a/b)/c = a/(b/c)?
• Does \(\sqrt{x^2+y^2}=x+y?\)
• Does cross multiply always work for inequalities?
• How does addition in the denominator work?
• List of Common Misconceptions
• Linear Equations
• Forms of Linear Equations
• Equations of Parallel and Perpendicular Lines

Systems of Linear Equations

• Solving Linear Systems Using Matrices
• Absolute Value
• Absolute Value Equations
• Absolute Value Inequalities
• Linear Inequalities
• Absolute Value Linear Inequalities
• System of Inequalities
• Linear Programming

Non-Linear Inequalities

• Polynomial Inequalities
• Classical Inequalities
• Absolute Value Inequalities - 1 Quadratic Term
• Exponential Inequalities
• Logarithmic Inequalities

Quadratic Equations (Parabolas)

• Factoring Quadratics
• Quadratic Equations
• Completing The Square
• Applications of Completing the Square
• Quadratic Discriminant

Square Roots (Radicals)

• Simplifying Radicals
• Radical Equations
• Square Roots
• Rationalizing Denominators

Arithmetic and Geometric Progressions

• Arithmetic Progressions
• Arithmetic Mean
• Harmonic Progression
• Harmonic Mean
• Geometric Progressions
• Geometric Mean
• Arithmetic and Geometric Progressions Problem Solving
• Arithmetic-Geometric Progression
• Evaluating Functions
• Function Composition
• Inverse Functions
• Function Terminology
• Graphs of Functions
• Transforming Graphs of Functions
• Functional Equations
• Multiplying Polynomials
• Polynomial Division
• Synthetic Division
• Solving Identity Equations
• Difference Of Squares
• Applying the Perfect Square Identity
• Applying the Perfect Cube Identity
• Factoring by Substitution
• Rational Expressions
• Simplifying Rational Expressions
• Factoring Compound Quadratics: \(\, ax^4 + bx^2 + c\)
• Factoring Cubic Polynomials
• Descartes' Rule of Signs
• Fundamental Theorem of Algebra
• Method of Undetermined Coefficients
• Remainder Factor Theorem
• Transforming Roots of Polynomials
• Intercepts of Rational Functions
• Rational Equations
• Graphing Rational Equations
• Partial Fractions - Linear Factors
• Partial Fractions - Cover Up Rule
• Partial Fractions - Irreducible Quadratics
• Partial Fractions - Repeated Factors
• Telescoping Series - Sum
• Floor Function
• Ceiling Function
• Trailing Number of Zeros
• Fractional Part Function
• Hermite's Identity
• Rules of Exponents
• Simplifying Exponents

Exponential Functions

• Rules of Exponents - Algebraic
• Solving Exponential Equations
• Graphs of Exponential Functions
• Exponential Functions - Problem Solving
• Interest Rate
• Leonhard Euler
• Solving Logarithmic Equations
• Polar Coordinates
• Converting Polar Coordinates to Cartesian
• Parametric Equations
• Complex Numbers
• Complex Conjugates
• Complex Numbers - Absolute Values
• Complex Plane
• Complex Numbers in Geometry
• Gaussian Integers
• Discrete Fourier Transform
• Euler's Formula
• De Moivre's Theorem
• Roots of Unity

Advanced Linear Algebra (Matrices)

• Linear Algebra
• Determinants
• Eigenvalues and Eigenvectors
• Kernel (Nullspace)
• Vector Space
• Cayley-Hamilton Theorem
• Row And Column Spaces
• Spectral Theorem
• Fundamental Subspaces
• Change of Basis
• Rank-Nullity Theorem
• Linear Transformations
• Linear Independence
• Jordan Canonical Form
• Affine transformations
• Vieta Root Jumping

Advanced Polynomials

• Algebraic Manipulation
• Algebraic Manipulation Identities
• Sum of n, n², or n³
• Telescoping Series - Product
• Nested Functions
• Cardano's Method
• Cubic Discriminant
• Gauss: The Prince of Mathematics
• Srinivasa Ramanujan
• Binomial Coefficient
• Pascal's Triangle
• Binomial Theorem
• Negative Binomial Theorem
• Fractional Binomial Theorem
• Rational Root Theorem
• Vieta's Formula
• Newton's Identities
• Completing the Square - Multiple Variables
• Algebraic Identities
• Factorization of Polynomials
• Sophie Germain Identity
• Algebraic Number Theory
• Fermat's Last Theorem
• Eisenstein's Irreducibility Criterion
• Group Theory
• Lagrange's Theorem
• Schwartz-Zippel Lemma
• Polynomial Interpolation by Remainder Factor Theorem
• Lagrange Interpolation
• Method of Differences
• Primitive Roots of Unity
• Symmetric Polynomials
• The \(uvw\) Method
• Chebyshev Polynomials - Definition and Properties

Advanced Inequalities

• Cauchy-Schwarz Inequality
• Titu's Lemma
• Rearrangement Inequality
• Chebyshev's Inequality
• Jensen's Inequality
• Hölder's Inequality
• Young's Inequality
• Muirhead Inequality
• Reverse Rearrangement Inequality
• Arithmetic Mean - Geometric Mean
• Applying the Arithmetic Mean Geometric Mean Inequality
• Power Mean Inequality (QAGH)
• Quadratic Mean
• Triangle Inequality
• AIME Math Contest Preparation
• JEE Exam Preparation
• RMO Math Contest Preparation
• INMO Math Contest Preparation
• KVPY Exam Preparation
• BITSAT Exam Preparation

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9 Algebra Questions And Practice Problems To Do With Your Middle Schoolers

Beki christian.

Algebra questions involve using letters or symbols to represent unknown values or values that can change. Here you will find \bf{9} algebra questions to test your knowledge and show you the different ways that algebra can be used to solve a problem to find an unknown value or to make generalizations.

Algebra in elementary school

Algebra in middle school, how to solve algebraic equations, algebra questions for middle school: basic algebra, algebra questions for middle school: forming and solving equations, algebra questions for middle school: graphs.

Students as early as kindergarten begin to solve problems with algebraic thinking. Algebraic thinking starts with students understanding the properties of the operations, the relationships between them, and the composition of numbers.

They begin to relate concrete models to abstract expressions and equations, and continue this work throughout elementary school. They also look at patterns and the order of operations that will contribute to their understanding of algebra in later years.

This flexible understanding with the composition and representation of numbers, as well as the operations sets them up for more complex and abstract equations once they get to middle school. 

Algebra encompasses many skills and concepts to help us make sense of the world and solve problems. In middle school, we learn to write basic algebraic expressions, write and solve linear equations, write and solve a system of linear equations, and graph linear equations on the coordinate plane. Later, we further develop these skills which allow us to work with more complex equations such as quadratic equations, polynomial equations, and exponential equations.

9 Algebra Questions for Middle School

Use these 9 algebra questions to help your middle school students understand algebra.

When you are presented with an algebraic problem, it is important to make sense of the problem. Here are some of the key terms along with what they mean:

• Solve the equation – find out the value of the unknown.
• Substitute – put the values you have been given into the algebraic expression.
• Coefficients – the amount a term has been multiplied by. For example, in the expression 4c+2, \; 4 is the coefficient.
• Constants – values which are not variable and stay the same. For example, for 4c+2, \; 2 is the constant.
• Binomial – a binomial or binomial expression is one which has two parts, like our example 4c+2.
• Simplify – collect together like terms to make the expression or equation look simpler.
• Expand – multiply out the expressions inside parentheses.
• Factorize – put into parentheses.
• In terms of x – rewrite the equation in the form x = …

Remember, when working with algebra, we must still apply the order of operations, PEMDAS. i.e. Parenthesis, Exponents (powers, square roots), Multiplication, Division, Addition, Subtraction.

When working with algebraic expressions and equations we must consider carefully which operations to deal with first.

Algebra practice questions for middle school

1. A chocolate bar costs c cents and a drink costs d cents. Write down an expression for the cost of 2 chocolate bars and 2 drinks.

2 chocolate bars would cost 2 lots of c, or 2c, and 2 drinks would cost 2 lots of d, or 2d.

2. Simplify the expression 4m+5+2m-1.

We need to collect together like terms here so 4m + 2m = 6m and 5-1 = 4 (watch out for the negative).

In middle school, we learn a variety of different algebra techniques to answer algebra questions and to practice problem solving with algebra. These include:

• Simplifying algebraic expressions
• Expanding brackets and factoring
• Forming algebraic equations from word problems
• Solving algebraic equations and inequalities
• Substituting into expressions
• Changing the subject of an equation
• Working with real life graphs and straight line graphs

3. In this pyramid, you add two adjacent blocks to find the value of the block above.

What expression will be in the top box?

4. Brian is a window cleaner. He uses the following formula to calculate the amount to charge his customers:

Charge = \$20 + 4n

Where n is the number of windows a house has.

If a house has 7 windows, how much would Brian charge?

In this question, n is 7 so we can substitute 7 into the formula.

Charge = \$20 + 4 × 7

Charge = \$48

5. The area of a rectangle is 4x-6.

Which of the following pairs could be the length and width of the rectangle?

2x and 2x-3

There are two ways of attempting this question. We know that \text{area of a rectangle } = \text{ length} \times \text{width} so we could multiply each pair together to see which pair makes 4x-6.

Alternatively, if we factorise 4x-6 we get 2(2x-3) meaning the sides could be 2 and 2x-3.

6. The formula for changing degrees Celsius to degrees Fahrenheit is:

Rearrange this formula to make C the subject.

7. Work out the size of the smallest angle.

The angles in a triangle add up to 180^{\circ} therefore we can write

Now we have an equation we can solve.

The angles are :

The smallest angle is 34^{\circ} .

8. Jamie’s dad is 4 times older than Jamie. In 14 years time, Jamie’s dad will be twice the age of Jamie.

What is the sum of Jamie’s age now and Jamie’s dad’s age now?

To solve this we need to write an equation.

Let Jamie’s age now be x . Then Jamie’s dad’s age is 4x .

In 14 years time Jamie’s age will be x + 14 and Jamie’s dad’s age will be 4x + 14 .

Since we know Jamie’s dad’s age will be two times Jamie’s age, we can write

4x+14=2(x+14)

Jamie is currently 7 years old meaning his dad is 28 years old. The sum of their ages is 35 .

Note: when algebraic equations contain denominators on either side, we can use the cross multiplication method to help us work out the answer. For example with the following expressions:

This can then become, ad = bc and so on.

9. Which of the following lines passes through the point (2, 5)?

At the point (2, 5), \; x is 2 and y is 5. We can check which equation works when we substitute in these values:

Looking for more algebra math questions?

• Ratio questions
• Probability questions
• Trigonometry questions
• Venn diagram questions
• Long division questions
• Pythagorean theorem questions

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.

Pythagoras Theorem Questions [FREE]

Downloadable Pythagoras theorem worksheet containing 15 multiple choice questions with a mix of worded problems and deeper problem solving questions.

Includes an answer key and follows variation theory with plenty of opportunities for students to work independently at their own level.

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Solving Word Questions

With LOTS of examples!

In Algebra we often have word questions like:

Example: Sam and Alex play tennis.

On the weekend Sam played 4 more games than Alex did, and together they played 12 games.

How many games did Alex play?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

Turning English into Algebra

To turn the English into Algebra it helps to:

• Read the whole thing first
• Do a sketch if possible
• Assign letters for the values
• Find or work out formulas

You should also write down what is actually being asked for , so you know where you are going and when you have arrived!

Also look for key words:

Thinking Clearly

Some wording can be tricky, making it hard to think "the right way around", such as:

Example: Sam has 2 dollars less than Alex. How do we write this as an equation?

• Let S = dollars Sam has
• Let A = dollars Alex has

Now ... is that: S − 2 = A

or should it be: S = A − 2

or should it be: S = 2 − A

The correct answer is S = A − 2

( S − 2 = A is a common mistake, as the question is written "Sam ... 2 less ... Alex")

Example: on our street there are twice as many dogs as cats. How do we write this as an equation?

• Let D = number of dogs
• Let C = number of cats

Now ... is that: 2D = C

or should it be: D = 2C

Think carefully now!

The correct answer is D = 2C

( 2D = C is a common mistake, as the question is written "twice ... dogs ... cats")

Let's start with a really simple example so we see how it's done:

Example: A rectangular garden is 12m by 5m, what is its area ?

Turn the English into Algebra:

• Use w for width of rectangle: w = 12m
• Use h for height of rectangle: h = 5m

Formula for Area of a Rectangle : A = w × h

We are being asked for the Area.

A = w × h = 12 × 5 = 60 m 2

The area is 60 square meters .

Now let's try the example from the top of the page:

Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?

• Use S for how many games Sam played
• Use A for how many games Alex played

We know that Sam played 4 more games than Alex, so: S = A + 4

And we know that together they played 12 games: S + A = 12

We are being asked for how many games Alex played: A

Which means that Alex played 4 games of tennis.

Check: Sam played 4 more games than Alex, so Sam played 8 games. Together they played 8 + 4 = 12 games. Yes!

A slightly harder example:

Example: Alex and Sam also build tables. Together they make 10 tables in 12 days. Alex working alone can make 10 in 30 days. How long would it take Sam working alone to make 10 tables?

• Use a for Alex's work rate
• Use s for Sam's work rate

12 days of Alex and Sam is 10 tables, so: 12a + 12s = 10

30 days of Alex alone is also 10 tables: 30a = 10

We are being asked how long it would take Sam to make 10 tables.

30a = 10 , so Alex's rate (tables per day) is: a = 10/30 = 1/3

Which means that Sam's rate is half a table a day (faster than Alex!)

So 10 tables would take Sam just 20 days.

Should Sam be paid more I wonder?

And another "substitution" example:

Example: Jenna is training hard to qualify for the National Games. She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days. She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?

• The number of "5 hour" days: d
• The number of "3 hour" days: e

We know there are seven days in the week, so: d + e = 7

And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d + 3e = 27

We are being asked for how many days she trains for 5 hours: d

The number of "5 hour" days is 3

Check : She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours

Some examples from Geometry:

Example: A circle has an area of 12 mm 2 , what is its radius?

• Use A for Area: A = 12 mm 2
• Use r for radius

And the formula for Area is: A = π r 2

We are being asked for the radius.

We need to rearrange the formula to find the area

Example: A cube has a volume of 125 mm 3 , what is its surface area?

Make a quick sketch:

• Use V for Volume
• Use A for Area
• Use s for side length of cube
• Volume of a cube: V = s 3
• Surface area of a cube: A = 6s 2

We are being asked for the surface area.

First work out s using the volume formula:

Now we can calculate surface area:

An example about Money:

Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate. One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?

• Joel's normal rate of pay: $N per hour
• Joel works for 40 hours at $N per hour = $40N
• When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour
• Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N
• And together he earned $660, so:

$40N + $(12 × 1¼N) = $660

We are being asked for Joel's normal rate of pay $N.

So Joel’s normal rate of pay is $12 per hour

Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660

More about Money, with these two examples involving Compound Interest

Example: Alex puts $2000 in the bank at an annual compound interest of 11%. How much will it be worth in 3 years?

This is the compound interest formula:

So we will use these letters:

• Present Value PV = $2,000
• Interest Rate (as a decimal): r = 0.11
• Number of Periods: n = 3
• Future Value (the value we want): FV

We are being asked for the Future Value: FV

Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33 What was the annual rate of interest for the savings account?

The compound interest formula:

• Present Value PV = $1,000
• Interest Rate (the value we want): r
• Number of Periods: n = 9
• Future Value: FV = $1,551.33

We are being asked for the Interest Rate: r

So the annual rate of interest is 5%

Check : $1,000 × (1.05) 9 = $1,000 × 1.55133 = $1,551.33

And an example of a Ratio question:

Example: At the start of the year the ratio of boys to girls in a class is 2 : 1 But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3 How many students are there altogether now?

• Number of boys now: b
• Number of girls now: g

The current ratio is 4 : 3

Which can be rearranged to 3b = 4g

At the start of the year there was (b + 4) boys and (g − 2) girls, and the ratio was 2 : 1

b + 4 g − 2 = 2 1

Which can be rearranged to b + 4 = 2(g − 2)

We are being asked for how many students there are altogether now: b + g

There are 12 girls !

And 3b = 4g , so b = 4g/3 = 4 × 12 / 3 = 16 , so there are 16 boys

So there are now 12 girls and 16 boys in the class, making 28 students altogether .

There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3 At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1

And now for some Quadratic Equations :

Example: The product of two consecutive even integers is 168. What are the integers?

Consecutive means one after the other. And they are even , so they could be 2 and 4, or 4 and 6, etc.

We will call the smaller integer n , and so the larger integer must be n+2

And we are told the product (what we get after multiplying) is 168, so we know:

n(n + 2) = 168

We are being asked for the integers

That is a Quadratic Equation , and there are many ways to solve it. Using the Quadratic Equation Solver we get −14 and 12.

Check −14: −14(−14 + 2) = (−14)×(−12) = 168 YES

Check 12: 12(12 + 2) = 12×14 = 168 YES

So there are two solutions: −14 and −12 is one, 12 and 14 is the other.

Note: we could have also tried "guess and check":

• We could try, say, n=10: 10(12) = 120 NO (too small)
• Next we could try n=12: 12(14) = 168 YES

But unless we remember that multiplying two negatives make a positive we might overlook the other solution of (−14)×(−12).

Example: You are an Architect. Your client wants a room twice as long as it is wide. They also want a 3m wide veranda along the long side. Your client has 56 square meters of beautiful marble tiles to cover the whole area. What should the length of the room be?

Let's first make a sketch so we get things right!:

• the length of the room: L
• the width of the room: W
• the total Area including veranda: A
• the width of the room is half its length: W = ½L
• the total area is the (room width + 3) times the length: A = (W+3) × L = 56

We are being asked for the length of the room: L

This is a quadratic equation , there are many ways to solve it, this time let's use factoring :

And so L = 8 or −14

There are two solutions to the quadratic equation, but only one of them is possible since the length of the room cannot be negative!

So the length of the room is 8 m

L = 8, so W = ½L = 4

So the area of the rectangle = (W+3) × L = 7 × 8 = 56

There we are ...

... I hope these examples will help you get the idea of how to handle word questions. Now how about some practice?

Algebra Worksheets

Free worksheets with answer keys.

Enjoy these free printable sheets . Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Distance Formula
• Equation of Circle
• Factor Trinomials Worksheet
• Domain and Range
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Grade 8 Algebra Word Problems

These lessons cover some examples and solutions for algebra word problems that you will commonly encounter in grade 8.

Related Pages More Math Word Problems Algebra Word Problems

How to write algebra word problems into systems of linear equations and solve systems of linear equations using elimination and substitution methods?

Grade 8 Algebra Word Problems How to solve algebra word problems using systems of linear equations?

Example: Devon is going to make 3 shelves for her father. He has a piece of lumber 12 feet long. She wants the top shelf to be half a foot shorter than the middle shelf, and the bottom shelf to be half a foot shorter than twice the length of the top shelf. How long will each shelf be if she uses the entire 12 feet of wood?

Grade 8 Algebra Word Problems - Line Segments

Example: If JK = 7x + 9, JL = 114 and KL = 9x + 9. Find KL.

Grade 8 number word problems - common core How to write word problems into systems of linear equations and solve systems of linear equations using elimination and substitution methods?

Example 1: The sum of two numbers is 361 and the difference between the two numbers is 173. What are the two numbers?

Example 2: There are 356 Grade 8 students at Euclid’s Middle School. Thirty-four more than four times the number of girls is equal to half the number of boys. How many boys are in Grade 8 at Euclid’s Middle School? How many girls?

Example 3: A family member has some five-dollar bills and one-dollar bills in their wallet. Altogether she has 18 bills and a total of $62. How many of each bill does she have?

Example 1: A friend bought 2 boxes of pencils and 8 notebooks for school and it cost him $11. He went back to the store the same day to buy school supplies for his younger brother. He spent $11.25 on 3 boxes of pencils and 5 notebooks. How much would notebooks cost?

• A farm raises cows and chickens. The farmer has a total of 42 animals. One day he counts the legs of all of his animals and realizes he has a total of 114. How many cows does the farmer have? How many chickens?
• The length of a rectangle is 4 times the width. The perimeter of the rectangle is 45 inches. What is the area of the rectangle?
• The sum of the measures of angles x and y is 127". If the measure of angle x is 34° more than half the measure of angle y, what is the measure of each angle?

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Simple Algebra Problems – Easy Exercises with Solutions for Beginners

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Understanding Algebraic Expressions

Breaking down algebra problems, solving algebraic equations, tackling algebra word problems, types of algebraic equations, algebra for different grades.

For instance, solving the equation (3x = 7) for (x) helps us understand how to isolate the variable to find its value.

I always find it fascinating how algebra serves as the foundation for more advanced topics in mathematics and science. Starting with basic problems such as ( $(x-1)^2 = [4\sqrt{(x-4)}]^2$ ) allows us to grasp key concepts and build the skills necessary for tackling more complex challenges.

So whether you’re refreshing your algebra skills or just beginning to explore this mathematical language, let’s dive into some examples and solutions to demystify the subject. Trust me, with a bit of practice, you’ll see algebra not just as a series of problems, but as a powerful tool that helps us solve everyday puzzles.

Simple Algebra Problems and Strategies

When I approach simple algebra problems, one of the first things I do is identify the variable.

The variable is like a placeholder for a number that I’m trying to find—a mystery I’m keen to solve. Typically represented by letters like ( x ) or ( y ), variables allow me to translate real-world situations into algebraic expressions and equations.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like ( x ) or ( y )), and operators (like add, subtract, multiply, and divide). For example, ( 4x + 7 ) is an algebraic expression where ( x ) is the variable and the numbers ( 4 ) and ( 7 ) are terms. It’s important to manipulate these properly to maintain the equation’s balance.

Solving algebra problems often starts with simplifying expressions. Here’s a simple method to follow:

• Combine like terms : Terms that have the same variable can be combined. For instance, ( 3x + 4x = 7x ).
• Isolate the variable : Move the variable to one side of the equation. If the equation is ( 2x + 5 = 13 ), my job is to get ( x ) by itself by subtracting ( 5 ) from both sides, giving me ( 2x = 8 ).

With algebraic equations, the goal is to solve for the variable by performing the same operation on both sides. Here’s a table with an example:

Algebra word problems require translating sentences into equations. If a word problem says “I have six less than twice the number of apples than Bob,” and Bob has ( b ) apples, then I’d write the expression as ( 2b – 6 ).

Understanding these strategies helps me tackle basic algebra problems efficiently. Remember, practice makes perfect, and each problem is an opportunity to improve.

In algebra, we encounter a variety of equation types and each serves a unique role in problem-solving. Here, I’ll brief you about some typical forms.

Linear Equations : These are the simplest form, where the highest power of the variable is one. They take the general form ( ax + b = 0 ), where ( a ) and ( b ) are constants, and ( x ) is the variable. For example, ( 2x + 3 = 0 ) is a linear equation.

Polynomial Equations : Unlike for linear equations, polynomial equations can have variables raised to higher powers. The general form of a polynomial equation is ( $a_nx^n + a_{n-1}x^{n-1} + … + a_2x^2 + a_1x + a_0 = 0$ ). In this equation, ( n ) is the highest power, and ( $a_n$ ), ( $a_{n-1} $), …, ( $a_0$ ) represent the coefficients which can be any real number.

• Binomial Equations : They are a specific type of polynomial where there are exactly two terms. Like ($ x^2 – 4 $), which is also the difference of squares, a common format encountered in factoring.

To understand how equations can be solved by factoring, consider the quadratic equation ( $x^2$ – 5x + 6 = 0 ). I can factor this into ( (x-2)(x-3) = 0 ), which allows me to find the roots of the equation.

Here’s how some equations look when classified by degree:

Remember, identification and proper handling of these equations are essential in algebra as they form the basis for complex problem-solving.

In my experience with algebra, I’ve found that the journey begins as early as the 6th grade, where students get their first taste of this fascinating subject with the introduction of variables representing an unknown quantity.

I’ve created worksheets and activities aimed specifically at making this early transition engaging and educational.

6th Grade :

Moving forward, the complexity of algebraic problems increases:

7th and 8th Grades :

• Mastery of negative numbers: students practice operations like ( -3 – 4 ) or ( -5 $\times$ 2 ).
• Exploring the rules of basic arithmetic operations with negative numbers.
• Worksheets often contain numeric and literal expressions that help solidify their concepts.

Advanced topics like linear algebra are typically reserved for higher education. However, the solid foundation set in these early grades is crucial. I’ve developed materials to encourage students to understand and enjoy algebra’s logic and structure.

Remember, algebra is a tool that helps us quantify and solve problems, both numerical and abstract. My goal is to make learning these concepts, from numbers to numeric operations, as accessible as possible, while always maintaining a friendly approach to education.

I’ve walked through various simple algebra problems to help establish a foundational understanding of algebraic concepts. Through practice, you’ll find that these problems become more intuitive, allowing you to tackle more complex equations with confidence.

Remember, the key steps in solving any algebra problem include:

• Identifying variables and what they represent.
• Setting up the equation that reflects the problem statement.
• Applying algebraic rules such as the distributive property ($a(b + c) = ab + ac$), combining like terms, and inverse operations.
• Checking your solutions by substituting them back into the original equations to ensure they work.

As you continue to engage with algebra, consistently revisiting these steps will deepen your understanding and increase your proficiency. Don’t get discouraged by mistakes; they’re an important part of the learning process.

I hope that the straightforward problems I’ve presented have made algebra feel more manageable and a little less daunting. Happy solving!

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Algebra Questions

Algebra questions are provided here, with answers, for students based on Class 6 and 7 syllabi. The questions are prepared as per NCERT (CBSE) guidelines. Solving these questions will help students understand the concept very well in an easy way. Learn expressions and equations of algebra here.

Click here to understand more about algebra .

Algebra Questions for Class 6

Basic algebra deals with finding the unknown value using variables.

1. Soldiers are marching in a parade. There are 10 soldiers in a row. What is the rule which gives the number of soldiers, given the number of rows?

Solution: Let n be the number of rows

Number of soldiers in a row = 10

Total number of soldiers = number of soldiers in a row × number of rows

2. Leela is Radha’s younger sister. Leela is 4 years younger than Radha. Write Leela’s age with respect to Radha’s age. Take Radha’s age to be x years.

Solution: Let Radha’s age be x years

Leela’s age = 4 years younger than Radha

= (x – 4) years

3. If a is the side-length of the equilateral triangle, then the perimeter of the triangle will be?

Solution: Side of equilateral triangle = a

The perimeter of triangle = sum of all its three sides

Since we know that an equilateral triangle has all its sides equal.

Perimeter of equilateral triangle = a+a+a = 3a

4. Give expressions for:

(i) p multiplied by 7

(ii) p divided by 7

5. If Sam age is x years. Then, what is the age of Sam after 7 years?

Solution: Sam’s present age = x

After 7 years,

Sam’s age = x+7 years

Also, read: Algebra For Class 6

Algebra Questions for Class 7

1. Express when x and y are both squared and added.

Solution: x 2 + y 2

2. Identify the terms in 1 + x + x 2

Solution: Given, 1 + x + x 2

Terms: 1, x, x 2

3. Write the coefficients of terms y and y 2 in the given expression 13 – y + 5y 2 .

Solution: The coefficient of y = -1 and of y 2 = 5

4. Find which are the like and unlike terms.

(i) 12x, 7x

(ii) 3xy, 3y

(i) 12x and 7x both have the same algebraic factors, i.e. x.

Hence, both are like terms.

(ii) 3xy have factors 3,x and y

Whereas 3y have factors 3 and y.

Hence, both have different algebraic factors, thus are unlike terms.

5. Classify into monomials, binomials and trinomials.

(i) 3x – 5y

(ii) 100z 2

(iii) x 2 -2x+3

(i) 3x-5y is a binomial because it consists of two terms (i.e.) 3x and 5y.

(ii) 100z 2 is monomial since there is only one term.

(iii) x 2 -2x+3 is trinomial, since x 2 , -2x and 3 are three different terms.

(iv) 32 is a monomial since it represents only a single term.

Also, read: Algebra Problems

Addition and Subtraction of Algebraic Expression

6. Collect like terms and simplify the expression:

12m 2 – 9m + 5m – 4m 2 – 7m + 10

Solution: Rearranging terms, we have:

12m 2 – 4m 2 + 5m – 9m – 7m + 10

= (12 – 4) m 2 + (5 – 9 – 7) m + 10

= 8m 2 + (– 4 – 7) m + 10

= 8m 2 + (–11) m + 10

= 8m 2 –11 m + 10

7. Subtract 24ab – 10b – 18a from 30ab + 12b + 14a.

Solution: 30ab + 12b + 14a – (24ab – 10b – 18a)

= 30ab + 12b + 14a – 24ab + 10b + 18a

= 30ab – 24ab + 12b + 10b + 14a + 18a

= 6ab + 22b + 32a

8. Find the value of x 2 -5x+8, for x = 2.

Solution: Given: x 2 -5x+8

By putting x =2, we get;

= (2) 2 – 5(2) + 8

= 4 – 10 + 8

Practice Questions

Solve the following problems:

• Find the coefficients of x and x 2 and the terms for the polynomial x 2 +4x-12.
• Solve: 2x 2 -4x-2 = 0

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