Linear equations are one of the foundational concepts in algebra.
Understanding how to solve them can make all the difference in developing a solid math foundation.
A linear equation typically has an equal sign and involves a mathematical expression that, when simplified, represents a straight line.
These equations help you figure out the value of unknown variables, most commonly represented by letters like “x” or “y.”
In this guide, we’ll break down what a linear equation is, and the rules that define it, and walk you through clear, step-by-step examples to help you solve them confidently.
Whether you’re new to algebra or just need a refresher, this guide will equip you with the tools to master linear equations.
What is a linear equation?
A linear equation is an equation where the variables, usually represented by “x” or “y”, are not raised to any power higher than 1. This means there are no exponents, square roots, or complicated functions involving variables.
Examples of linear expressions include:
- x + 4
- 2x + 4
- 2x + 4y
Non-linear expressions, on the other hand, would look like this:
- x² (because of the exponent)
- √x (because of the square root)
- 2xy + 4 (since variables are being multiplied)
When solving a linear equation, the goal is to isolate the unknown variable on one side of the equation to find its value.
Understanding variables in linear expressions
A variable is a symbol, often “x” or “y,” that represents an unknown value in an equation. In linear expressions:
- Variables cannot have exponents (no x² or x³)
- Variables cannot be multiplied by each other (no xy)
- Variables cannot appear under square roots (no √x)
If any of these rules are broken, the expression is not linear.
4 basic steps for solving linear equations
The process for solving linear equations involves moving terms from one side of the equation to the other, always doing the same operation to both sides to maintain equality.
Here’s a general step-by-step outline:
- Simplify both sides of the equation if necessary.
- Move the terms involving variables to one side.
- Isolate the variable by using addition, subtraction, multiplication, or division.
- Double-check your work by plugging the solution back into the original equation.
Let’s work through a few examples to see these steps in action.
Example 1: Solving a simple linear equation
Find x if:
2x + 4 = 10
In this example, we need to isolate the variable x by following basic algebraic steps. First, we’ll subtract 4 from both sides, then divide by 2 to find x.
| Step | Action | Explanation |
|---|---|---|
| 1 | Subtract 4 from both sides | 2x + 4 – 4 = 10 – 4 → 2x = 6 |
| 2 | Divide both sides by 2 | 2x ÷ 2 = 6 ÷ 2 → x = 3 |
| 3 | Check your work | (2 * 3) + 4 = 10 → 6 + 4 = 10 |
By following these steps, we find that x = 3.
Checking the solution:
Plug x = 3 back into the original equation:
2(3) + 4 = 10
6 + 4 = 10
Correct!
Example 2: Solving with negative numbers
Find x if:
3x – 4 = -10
Dealing with negatives can feel tricky, but the process remains the same. Here, we’ll first add 4 to both sides to isolate the term with x, then divide by 3 to solve for x.
| Step | Action | Explanation |
|---|---|---|
| 1 | Add 4 to both sides | 3x – 4 + 4 = -10 + 4 → 3x = -6 |
| 2 | Divide both sides by 3 | 3x ÷ 3 = -6 ÷ 3 → x = -2 |
| 3 | Check your work | (3 * -2) – 4 = -10 → -6 – 4 = -10 |
Here, the solution is x = -2.
Checking the solution:
Plug x = -2 into the original equation:
3(-2) – 4 = -10
-6 – 4 = -10
Correct!
Example 3: Solving with two variables
Find x if:
4x – 4y = 8
When dealing with two variables, we can still isolate x. In this case, we’ll move the term involving y to the other side and then divide by 4.
| Step | Action | Explanation |
|---|---|---|
| 1 | Add 4y to both sides | 4x – 4y + 4y = 8 + 4y → 4x = 8 + 4y |
| 2 | Divide both sides by 4 | 4x ÷ 4 = (8 + 4y) ÷ 4 → x = 2 + y |
| 3 | Check your work | 4(2 + y) – 4y = 8 → 8 + 4y – 4y = 8 → 8 = 8 |
In this case, we express x as x = 2 + y, since we are working with two variables.
Checking the solution:
Plug x = 2 + y into the original equation:
4(2 + y) – 4y = 8
8 + 4y – 4y = 8
Correct!
Example 4: Solving with constants
Find x if:
x + 3² = 12
In this example, we’re dealing with a constant (3²), which we need to simplify first before solving for x. Once simplified, the equation becomes straightforward, and we can isolate x using basic subtraction.
| Step | Action | Explanation |
|---|---|---|
| 1 | Simplify the constant | x + 9 = 12 |
| 2 | Subtract 9 from both sides | x + 9 – 9 = 12 – 9 → x = 3 |
| 3 | Check your work | 3 + 3² = 12 → 3 + 9 = 12 → 12 = 12 |
The solution to this equation is x = 3.
Checking the solution:
Plug x = 3 into the original equation:
3 + 3² = 12
3 + 9 = 12
Correct!
Why is checking your solution important?
After solving a linear equation, it’s crucial to check your answer by substituting the variable back into the original equation. This ensures that the solution satisfies the equation, eliminating any mistakes made during the process.
Always remember:
- Solving linear equations is about balancing both sides.
- Whatever you do to one side of the equation, you must do to the other.
- Each step should bring you closer to isolating the variable.
Final thoughts
Solving linear equations may seem intimidating at first, but with practice, it becomes much easier.
The key is to follow the steps methodically, keep the equation balanced, and double-check your solutions. Whether you’re working with single variables or multiple ones, the process remains consistent.
Start with simple examples, build your confidence, and soon, you’ll be tackling more complex problems with ease.