Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 5: Equations and Inequalities

Lesson 2: Solving Linear Equations I

In this Grade 4 AMC Math lesson from The Art of Problem Solving: Prealgebra, students learn to solve linear equations in one variable by isolating the variable using two core tactics: replacing expressions with equivalent expressions and performing the same operation on both sides of an equation. The lesson covers solving equations with whole numbers, fractions, and mixed numbers, and introduces the concept of solutions as values that make an equation true. Students practice multiple solution methods — including inspection, number lines, and algebraic manipulation — and verify answers by substituting solutions back into the original equation.

Section 1

Equation and Solution

Property

An equation is a statement that two expressions are equal. It may involve one or more variables. A value of the variable that makes an equation true is called a solution of the equation, and the process of finding this value is called solving the equation.

Examples

The statement $x + 5 = 12$ is an equation. The value $x=7$ is a solution because $7 + 5 = 12$ is a true statement.
To check if $y=3$ is a solution to $8y = 24$ , we substitute it in: $8(3) = 24$ . This is true, so $y=3$ is a solution.
Is $z=10$ a solution for $z - 4 = 5$ ? We check: $10 - 4 = 6$ . Since $6$ is not equal to $5$ , $z=10$ is not a solution.

Explanation

Think of an equation as a perfectly balanced scale. A solution is the specific value for the variable that keeps the scale level. Finding that value is what we call solving the equation.

Section 2

Verify a solution of an equation

Property

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

To determine whether a number is a solution to an equation.
Step 1. Substitute the number in for the variable in the equation.
Step 2. Simplify the expressions on both sides of the equation.
Step 3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.

Examples

Is $y=4$ a solution to $5y - 3 = 17$ ? Substitute $y=4$ : $5(4) - 3 = 20 - 3 = 17$ . Since $17=17$ , yes, it is a solution.
Is $x=-3$ a solution to $2x + 8 = x+4$ ? Substitute $x=-3$ : $2(-3) + 8 = 2$ and $-3+4=1$ . Since $2 \neq 1$ , it is not a solution.
Is $a = \frac{1}{2}$ a solution to $8a - 1 = 3$ ? Substitute $a=\frac{1}{2}$ : $8(\frac{1}{2}) - 1 = 4 - 1 = 3$ . Since $3=3$ , yes, it is a solution.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Equation and Solution

Property

Examples

The statement $x + 5 = 12$ is an equation. The value $x=7$ is a solution because $7 + 5 = 12$ is a true statement.
To check if $y=3$ is a solution to $8y = 24$ , we substitute it in: $8(3) = 24$ . This is true, so $y=3$ is a solution.
Is $z=10$ a solution for $z - 4 = 5$ ? We check: $10 - 4 = 6$ . Since $6$ is not equal to $5$ , $z=10$ is not a solution.

Explanation

Think of an equation as a perfectly balanced scale. A solution is the specific value for the variable that keeps the scale level. Finding that value is what we call solving the equation.

Section 2

Verify a solution of an equation

Property

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

Examples

Is $y=4$ a solution to $5y - 3 = 17$ ? Substitute $y=4$ : $5(4) - 3 = 20 - 3 = 17$ . Since $17=17$ , yes, it is a solution.
Is $x=-3$ a solution to $2x + 8 = x+4$ ? Substitute $x=-3$ : $2(-3) + 8 = 2$ and $-3+4=1$ . Since $2 \neq 1$ , it is not a solution.
Is $a = \frac{1}{2}$ a solution to $8a - 1 = 3$ ? Substitute $a=\frac{1}{2}$ : $8(\frac{1}{2}) - 1 = 4 - 1 = 3$ . Since $3=3$ , yes, it is a solution.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Equation and Solution

Property

Examples

The statement $x + 5 = 12$ is an equation. The value $x=7$ is a solution because $7 + 5 = 12$ is a true statement.
To check if $y=3$ is a solution to $8y = 24$ , we substitute it in: $8(3) = 24$ . This is true, so $y=3$ is a solution.
Is $z=10$ a solution for $z - 4 = 5$ ? We check: $10 - 4 = 6$ . Since $6$ is not equal to $5$ , $z=10$ is not a solution.

Explanation

Think of an equation as a perfectly balanced scale. A solution is the specific value for the variable that keeps the scale level. Finding that value is what we call solving the equation.

Section 2

Verify a solution of an equation

Property

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

Examples

Is $y=4$ a solution to $5y - 3 = 17$ ? Substitute $y=4$ : $5(4) - 3 = 20 - 3 = 17$ . Since $17=17$ , yes, it is a solution.
Is $x=-3$ a solution to $2x + 8 = x+4$ ? Substitute $x=-3$ : $2(-3) + 8 = 2$ and $-3+4=1$ . Since $2 \neq 1$ , it is not a solution.
Is $a = \frac{1}{2}$ a solution to $8a - 1 = 3$ ? Substitute $a=\frac{1}{2}$ : $8(\frac{1}{2}) - 1 = 4 - 1 = 3$ . Since $3=3$ , yes, it is a solution.