# Solving equations

Here you will learn about solving equations, including linear and quadratic algebraic equations, and how to solve them.

Students will first learn about solving equations in grade 8 as a part of expressions and equations, and again in high school as a part of reasoning with equations and inequalities.

Every week, we teach lessons on solving equations to students in schools and districts across the US as part of our online one-on-one math tutoring programs. On this page we’ve broken down everything we’ve learnt about teaching this topic effectively.

## What is solving an equation?

Solving equations is a step-by-step process to find the value of the variable. A variable is the unknown part of an equation, either on the left or right side of the equals sign. Sometimes, you need to solve multi-step equations which contain algebraic expressions.

To do this, you must use the order of operations, which is a systematic approach to equation solving. When you use the order of operations, you first solve any part of an equation located within parentheses. An equation is a mathematical expression that contains an equals sign.

For example,

\begin{aligned}y+6&=11\\\\ 3(x-3)&=12\\\\ \cfrac{2x+2}{4}&=\cfrac{x-3}{3}\\\\ 2x^{2}+3&x-2=0\end{aligned}

There are two sides to an equation, with the left side being equal to the right side. Equations will often involve algebra and contain unknowns, or variables, which you often represent with letters such as x or y.

You can solve simple equations and more complicated equations to work out the value of these unknowns. They could involve fractions, decimals or integers.

## Common Core State Standards

How does this relate to 8 th grade and high school math?

• Grade 8 – Expressions and Equations (8.EE.C.7)
Solve linear equations in one variable.

• High school – Reasoning with Equations and Inequalities (HSA.REI.B.3)
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

## How to solve equations

In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value.

1. Combine like terms.
2. Simplify the equation by using the opposite operation to both sides.
3. Isolate the variable on one side of the equation.

## Solving equations examples

### Example 1: solve equations involving like terms

Solve for x.

5q-4q=9

1. Combine like terms.

Combine the q terms on the left side of the equation. To do this, subtract 4q from both sides.

(5 q-4 q)=9-4 q

The goal is to simplify the equation by combining like terms. Subtracting 4q from both sides helps achieve this.

After you combine like terms, you are left with q=9-4q.

2Simplify the equation by using the opposite operation on both sides.

Add 4q to both sides to isolate q to one side of the equation.

q+4q=9-4q+4q

The objective is to have all the q terms on one side. Adding 4q to both sides accomplishes this.

After you move the variable to one side of the equation, you are left with 5q=9.

3Isolate the variable on one side of the equation.

Divide both sides of the equation by 5 to solve for q.

5q \div 5 = 9 \div 5

Dividing by 5 allows you to isolate q to one side of the equation in order to find the solution. After dividing both sides of the equation by 5, you are left with q=1 \cfrac{4}{5} \, .

### Example 2: solve equations with variables on both sides

Solve for x.

7v=8–8v

Combine like terms.

Simplify the equation by using the opposite operation on both sides and isolate the variable to one side.

### Example 3: solve equations with the distributive property

Solve for x.

3(c-5)-4=2

Combine like terms by using the distributive property.

Simplify the equation by using the opposite operation on both sides.

Isolate the variable to one side of the equation.

### Example 4: solve linear equations

Solve for x.

2x+5=15

Combine like terms by simplifying.

Continue to simplify the equation by using the opposite operation on both sides.

Isolate the variable to one side of the equation and check your work.

### Example 5: solve equations by factoring

Solve the following equation by factoring.

2x^2+3x-20=0

Combine like terms by factoring the equation by grouping.

Simplify the equation by using the opposite operation on both sides.

Isolate the variable for each equation and solve.

### Example 6: solve quadratic equations

x^{2}+2x-5=0

Combine like terms by factoring the quadratic equation when terms are isolated to one side.

Simplify the equation by using the opposite operation on both sides.

Isolate the variable for each equation and solve.

### Teaching tips for solving equations

• Use physical manipulatives like balance scales as a visual aid. Show how you need to keep both sides of the equation balanced, like a scale. Add or subtract the same thing from both sides to keep it balanced when solving. Use this method to practice various types of equations.

• Emphasize the importance of undoing steps to isolate the variable. If you are solving for x and 3 is added to x, subtracting 3 undoes that step and isolates the variable x.

• Relate equations to real-world, relevant examples for students. For example, word problems about tickets for sports games, cell phone plans, pizza parties, etc. can make the concepts click better.

• Allow time for peer teaching and collaborative problem solving. Having students explain concepts to each other, work through examples on whiteboards, etc. reinforces the process and allows peers to ask clarifying questions. This type of scaffolding would be beneficial for all students, especially English-Language Learners. Provide supervision and feedback during the peer interactions.

### Easy mistakes to make

• Forgetting to distribute or combine like terms
One common mistake is neglecting to distribute a number across parentheses or combine like terms before isolating the variable. This error can lead to an incorrect simplified form of the equation.

• Misapplying the distributive property
Incorrectly distributing a number across terms inside parentheses can result in errors. Students may forget to multiply each term within the parentheses by the distributing number, leading to an inaccurate equation.

• Failing to perform the same operation on both sides
It’s crucial to perform the same operation on both sides of the equation to maintain balance. Forgetting this can result in an imbalanced equation and incorrect solutions.

• Making calculation errors
Simple arithmetic mistakes, such as addition, subtraction, multiplication, or division errors, can occur during the solution process. Checking calculations is essential to avoid errors that may propagate through the steps.

• Ignoring fractions or misapplying operations
When fractions are involved, students may forget to multiply or divide by the common denominator to eliminate them. Misapplying operations on fractions can lead to incorrect solutions or complications in the final answer.

### Practice solving equations questions

1. Solve 4x-2=14.

x=3

x=4

x=14

x=3.5
4x-2=14

4x=16

Divide both sides by 4.

x=4

2. Solve 3x-8=x+6.

x=7

x=4

x=8

x=6
3x-8=x+6

3x=x+14

Subtract x from both sides.

2x=14

Divide both sides by 2.

x=7

3. Solve 3(x+3)=2(x-2).

x=2

x=–3

x=–13

x=13
3(x+3)=2(x-2)

Expanding the parentheses.

3x+9=2x-4

Subtract 9 from both sides.

3x=2x-13

Subtract 2x from both sides.

x=-13

4. Solve \cfrac{2 x+2}{3}=\cfrac{x-3}{2}.

x=2

x=–3

x=–13

x=13
\cfrac{2 x+2}{3}=\cfrac{x-3}{2}

Multiply by 6 (the lowest common denominator) and simplify.

2(2x+2)=3(x-3)

Expand the parentheses.

4x+4=3x-9

Subtract 4 from both sides.

4x=3x-13

Subtract 3x from both sides.

x=-13

5. Solve \cfrac{3 x^{2}}{2}=24.

x=±4

x=±2

x=±3

x=±6
\cfrac{3 x^{2}}{2}=24

Multiply both sides by 2.

3 x^{2}=48

Divide both sides by 3.

x^{2}=16

Square root both sides.

x=\pm 4

6. Solve by factoring:

x^{2}-13 x+30=0.

x=-3, \; x=10

x=-3, \; x=-10

x=3, \; x=10

x=3, \; x=-10
x^{2}-13 x+30=0

Use factoring to find simpler equations.

(x-3)(x-10)=0

Set each set of parentheses equal to zero and solve.

x=3 or x=10

## Solving equations FAQs

What is the first step in solving a simple linear equation?

The first step in solving a simple linear equation is to simplify both sides by combining like terms. This involves adding or subtracting terms to isolate the variable on one side of the equation.

Why is it important to perform the same operation on both sides of the equation when solving it?

Performing the same operation on both sides of the equation maintains the equality. This ensures that any change made to one side is also made to the other, keeping the equation balanced and preserving the solutions.

How do I handle variables on both sides of the equation when solving for a variable?

To handle variables on both sides of the equation, start by combining like terms on each side. Then, move all terms involving the variable to one side by adding or subtracting, and simplify to isolate the variable. Finally, perform any necessary operations to solve for the variable.

What should I do if there are fractions in the equation when solving for a variable?

To deal with fractions in an equation, aim to eliminate them by multiplying both sides of the equation by the least common denominator. This helps simplify the equation and make it easier to isolate the variable. Afterward, proceed with the regular steps of solving the equation.

## Still stuck?

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