9th grade math - Algebra

Understanding Linear Equations

A Follow-up to Algebraic Expressions and Equations

Now that you’ve learned about algebraic expressions and how to solve basic equations, it’s time to explore a powerful part of algebra: linear equations. These equations are used all the time in real life—from figuring out prices and measurements to predicting outcomes in science and business.

In this article, you’ll learn what a linear equation is, how it looks, how to solve it, and how to check your solution.

What Is a Linear Equation?

A linear equation is an equation where the variable (like x or y) is not raised to any power other than 1. In other words, there are no exponents, no square roots, and no variables multiplied together. The graph of a linear equation is always a straight line, which is why it’s called “linear.”

A linear equation usually looks like this:

ax + b = c

Where:

  • x is the variable (what you’re trying to solve for)
  • a is the coefficient (a number multiplied by the variable)
  • b is a constant added or subtracted
  • c is the number on the other side of the equation

Examples of Linear Equations:

  • 2x + 3 = 11
  • x - 5 = 9
  • 4x = 16
  • x/2 + 1 = 5

The Goal: Solving for the Variable

To solve a linear equation, you want to get the variable (x) by itself on one side of the equation. You do this by performing operations that “undo” what’s being done to x.

You can:

  • Add or subtract the same number from both sides
  • Multiply or divide both sides by the same number

The key rule is:
Whatever you do to one side, you must also do to the other side.

Step-by-Step Examples

Example 1: Solve x + 7 = 12

Step 1: Subtract 7 from both sides
x = 5

Check: 5 + 7 = 12 ✅

Example 2: Solve 2x = 10

Step 1: Divide both sides by 2
x = 5

Check: 2 × 5 = 10 ✅

Example 3: Solve 3x - 4 = 11

Step 1: Add 4 to both sides
3x = 15

Step 2: Divide by 3
x = 5

Check: 3(5) - 4 = 11 ✅

Example 4: Solve x/4 + 2 = 5

Step 1: Subtract 2 from both sides
x/4 = 3

Step 2: Multiply both sides by 4
x = 12

Check: 12 ÷ 4 + 2 = 5 ✅

Solving Linear Equations in Two Steps

Most linear equations have two steps:

  1. Eliminate any constants (using addition or subtraction)
  2. Get rid of the coefficient (using multiplication or division)

Example: 5x + 3 = 18

Step 1: Subtract 3 → 5x = 15

Step 2: Divide by 5 → x = 3

Special Cases

1. No solution

If you simplify an equation and get something false, like 0 = 4, the equation has no solution.

2. Infinite solutions

If both sides are the same, like x + 3 = x + 3, then every value of x works. The equation has infinitely many solutions.

Tips for Solving Linear Equations

  • Always do the same thing to both sides of the equation
  • Keep your equation balanced
  • Combine like terms before solving, if needed
  • Check your answer by plugging it back into the original equation
  • Watch out for negative numbers and fractions

Practice Problems

Try solving these:

  1. x - 8 = 2
  2. 4x + 1 = 13
  3. x/5 - 3 = 1
  4. 7x = 35

Free Printable Algebra Worksheet: Solving Linear Equations

Now that you understand how to solve and graph linear equations, it’s time to explore what happens when we’re not looking for exact equality. In the next article, we’ll dive into algebraic inequalities, where instead of equal signs, you’ll work with greater than, less than, and other comparisons. These are powerful tools for solving real-world problems where multiple solutions are possible.