Simplifying Linear Expressions: Step-by-Step Guide
Simplifying an expression means rewriting it in a shorter form that has the same value. You do this by first multiplying out any parentheses (this is called distributing) and then combining like terms. Like terms are terms with the same letter, such as 6x and 5x, or plain numbers such as -12 and 7.
Math Racer skill guide · Middle School · Grades 6-8
How to solve it, step by step
- Distribute to remove parentheses. Multiply the number outside the parentheses by every term inside. Pay close attention to plus and minus signs.\[k(ax+b) = kax + kb\]
- Group the like terms. Find the terms with the same letter, and find the plain numbers. You can move terms around, but keep each term's sign with it.\[ax + bx + c + d = (a+b)x + (c+d)\]
- Combine the like terms. Add or subtract the numbers in front of the letter. Combine the plain numbers separately.\[(a+b)x + (c+d)\]
- Write the simplified expression. Put the term with the letter first and the plain number last. Check that there are no more like terms to combine.\[mx + n\]
Worked example
Simplify: 3(2x - 4) + 5x + 7
Step 1: Distribute the 3: 3(2x - 4) = 6x - 12. Now the expression is 6x - 12 + 5x + 7.
Step 2: Group like terms: (6x + 5x) + (-12 + 7).
Step 3: Combine: 6x + 5x = 11x, and -12 + 7 = -5.
Step 4: The simplified expression is 11x - 5.
Check: Try x = 1 in the original: 3(2 - 4) + 5 + 7 = -6 + 12 = 6. Try x = 1 in 11x - 5: 11 - 5 = 6. They match.
Tips and common mistakes
- Distribute before you combine like terms. Skipping this is the most common mistake.
- When you distribute a negative number, every sign inside flips: -2(x - 3) = -2x + 6.
- Only terms with the same letter are like terms. 3x and 5 are not like terms.
- The sign in front of a term belongs to that term. Keep it with the term when you move things around.
- To check your work, put a small number in for x in both the original and your answer. Both should give the same result.
- Write the answer with the x term first and the plain number last, like 11x - 5.