Solving Linear Equations: Step-by-Step Guide
A linear equation like 3x + 7 = 22 has one unknown number, x. Solving means getting x all by itself on one side of the equals sign. You do this by undoing the operations one at a time: first undo the adding or subtracting, then undo the multiplying or dividing. Whatever you do to one side, you must do to the other side too, so the equation stays balanced.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Look at the parts of the equation. The equation looks like ax + b = c. The number in front of x is a. The number being added is b. The number on the right side is c.\[ax + b = c\]
- Get the x term by itself. Subtract b from both sides (or add, if b is being subtracted). Now only the x term is left on the left side.\[ax = c - b\]
- Solve for x. Divide both sides by the number in front of x. Now x is alone.\[x = \frac{c-b}{a}\]
- Check your answer. Put your value of x back into the original equation. Both sides should be equal.\[a\left(\tfrac{c-b}{a}\right) + b \stackrel{?}{=} c\]
Worked example
Solve: 3x + 7 = 22
Step 1: In 3x + 7 = 22, the number in front of x is 3, the number added is 7, and the right side is 22.
Step 2: Subtract 7 from both sides: 3x + 7 - 7 = 22 - 7, so 3x = 15.
Step 3: Divide both sides by 3: x = 15 / 3 = 5.
Step 4: The answer is x = 5.
Check: Put x = 5 into 3x + 7. 3(5) + 7 = 15 + 7 = 22. That matches the right side.
Tips and common mistakes
- Always do the same thing to both sides. That keeps the equation balanced.
- Undo adding and subtracting first, then undo multiplying and dividing. It is like PEMDAS backwards.
- If x has a fraction in front of it, multiply both sides by the flipped fraction.
- Negative numbers work the same way. If -3x = 12, divide both sides by -3 to get x = -4.
- If you end up with something true like 5 = 5, then every number is a solution.
- If you end up with something false like 5 = 7, then there is no solution.
- Always check your answer by putting it back in. This catches sign mistakes.