Equations with Distribution & Like Terms
Learn to solve multi-step equations using the distributive property and combining like terms to isolate the variable.
What you'll do in this lesson
A voice-first session with the Crimsora tutor on Equations with Distribution & Like Terms, then targeted practice and FRQs — with the tutor adapting to where you get stuck.
What this lesson covers
Many real-world situations lead to equations that look more complex than 2x = 10. A phone plan might charge a setup fee plus a monthly rate, or a recipe might need to be doubled and then adjusted. These situations create equations where you have to apply the distributive property first, combine like terms, and then solve. This lesson shows you how to handle equations like 3(x − 4) + 2x = 2(x + 5) − 1 by working through distribution and simplification before isolating the variable.
The Distributive Property in Equations
The distributive property lets you multiply a number outside parentheses by each term inside. In an equation like 3(x − 4) = 12, you distribute the 3 to both x and −4, getting 3x − 12 = 12. This step is crucial because it removes the parentheses and creates terms you can work with.
When the multiplier is negative, pay careful attention to signs. For example, −2(x − 3) becomes −2x + 6, not −2x − 6. Many students forget that multiplying a negative number by a negative term produces a positive result. Think of it this way: if you owe 2 groups of (x − 3) dollars, and someone cancels that debt, you gain 2(−x + 3) = −2x + 6.
Always distribute before you combine like terms or isolate the variable. If you skip this step, you'll have parentheses left in your work and won't be able to solve correctly.
When the multiplier is negative, pay careful attention to signs. For example, −2(x − 3) becomes −2x + 6, not −2x − 6. Many students forget that multiplying a negative number by a negative term produces a positive result. Think of it this way: if you owe 2 groups of (x − 3) dollars, and someone cancels that debt, you gain 2(−x + 3) = −2x + 6.
Always distribute before you combine like terms or isolate the variable. If you skip this step, you'll have parentheses left in your work and won't be able to solve correctly.
Combining Like Terms
Like terms are terms with the same variable raised to the same power. On the left side of 3x + 2x − 12, the terms 3x and 2x are like terms because both contain x to the first power. You combine them by adding or subtracting their coefficients: 3x + 2x = 5x.
The constant terms −7 and 4 are also like terms (both are just numbers, no variable). You can combine them: −7 + 4 = −3.
But x and x² are not like terms. Neither are x and y, and neither are x and the number 5.
In an equation, combine like terms on each side separately before you move terms from one side to the other. For example, in 3x + 2x + 5 = 2x − 3 + 1, first simplify each side to get 5x + 5 = 2x − 2, then solve from there. This keeps your work organized and reduces the chance of errors.
The constant terms −7 and 4 are also like terms (both are just numbers, no variable). You can combine them: −7 + 4 = −3.
But x and x² are not like terms. Neither are x and y, and neither are x and the number 5.
In an equation, combine like terms on each side separately before you move terms from one side to the other. For example, in 3x + 2x + 5 = 2x − 3 + 1, first simplify each side to get 5x + 5 = 2x − 2, then solve from there. This keeps your work organized and reduces the chance of errors.
Solving Equations Step by Step
A complete solution follows this sequence: (1) distribute any numbers into parentheses on both sides, (2) combine like terms on the left side and right side separately, (3) move all variable terms to one side and all constant terms to the other (using inverse operations), (4) divide by the coefficient of the variable to isolate it.
Let's trace an example: 3(x − 4) + 2x = 2(x + 5) − 1.
Step 1: Distribute. Left side: 3x − 12 + 2x. Right side: 2x + 10 − 1.
Step 2: Combine. Left: 5x − 12. Right: 2x + 9.
Step 3: Subtract 2x from both sides: 3x − 12 = 9. Then add 12 to both sides: 3x = 21.
Step 4: Divide by 3: x = 7.
You can check by substituting x = 7 back into the original equation: 3(7 − 4) + 2(7) = 3(3) + 14 = 9 + 14 = 23, and 2(7 + 5) − 1 = 2(12) − 1 = 24 − 1 = 23. Both sides equal 23, so x = 7 is correct.
Let's trace an example: 3(x − 4) + 2x = 2(x + 5) − 1.
Step 1: Distribute. Left side: 3x − 12 + 2x. Right side: 2x + 10 − 1.
Step 2: Combine. Left: 5x − 12. Right: 2x + 9.
Step 3: Subtract 2x from both sides: 3x − 12 = 9. Then add 12 to both sides: 3x = 21.
Step 4: Divide by 3: x = 7.
You can check by substituting x = 7 back into the original equation: 3(7 − 4) + 2(7) = 3(3) + 14 = 9 + 14 = 23, and 2(7 + 5) − 1 = 2(12) − 1 = 24 − 1 = 23. Both sides equal 23, so x = 7 is correct.
Common Mistakes and How to Avoid Them
One frequent error is distributing only part of the way. For instance, students write 2(x + 3) as 2x + 3 instead of 2x + 6. Double-check that every term inside the parentheses gets multiplied by the number outside.
Another mistake is mishandling negative multipliers. −3(x − 2) should give −3x + 6. If you get −3x − 6, you distributed the 3 correctly but forgot that (−3)(−2) = +6.
A third error is combining unlike terms. You cannot simplify 3x + 5 by adding them together. The 3x and 5 are different types of terms.
Also, remember that when you combine like terms, you only add or subtract coefficients; the variable part stays the same. Writing 3x + 2x = 5x² is wrong. The answer is 5x.
Finally, always check your solution in the original equation, not in a simplified version. Mistakes made early can carry through, and checking confirms whether your answer is truly correct.
Another mistake is mishandling negative multipliers. −3(x − 2) should give −3x + 6. If you get −3x − 6, you distributed the 3 correctly but forgot that (−3)(−2) = +6.
A third error is combining unlike terms. You cannot simplify 3x + 5 by adding them together. The 3x and 5 are different types of terms.
Also, remember that when you combine like terms, you only add or subtract coefficients; the variable part stays the same. Writing 3x + 2x = 5x² is wrong. The answer is 5x.
Finally, always check your solution in the original equation, not in a simplified version. Mistakes made early can carry through, and checking confirms whether your answer is truly correct.
Key terms
- Distributive Property.
- A property that states a(b + c) = ab + ac. It allows you to multiply a number by a sum or difference by distributing the multiplication to each term inside the parentheses.
- Like Terms.
- Terms that have the same variable (or variables) raised to the same power. For example, 3x and 5x are like terms, but 3x and 3x² are not.
- Combining Like Terms.
- The process of adding or subtracting like terms by adding or subtracting their coefficients while keeping the variable part unchanged. For example, 3x + 5x = 8x.
- Coefficient.
- The number that multiplies a variable. In the term 7x, the coefficient is 7.
- Variable.
- A letter (usually x, y, or z) that represents an unknown number in an equation or expression.
- Constant.
- A number that does not change or have a variable attached to it. In the expression 5x + 3, the constant is 3.
- Inverse Operation.
- An operation that undoes another operation. Addition and subtraction are inverse operations, as are multiplication and division.
Worked example
Solve: −2(x + 3) + 5x = 4(x − 2) + 6
Step 1: Distribute on both sides.
Left side: −2(x + 3) + 5x = −2x − 6 + 5x Right side: 4(x − 2) + 6 = 4x − 8 + 6
Our equation is now: −2x − 6 + 5x = 4x − 8 + 6
Step 2: Combine like terms on each side.
Left side: −2x + 5x − 6 = 3x − 6 Right side: 4x − 8 + 6 = 4x − 2
Our equation is now: 3x − 6 = 4x − 2
Step 3: Collect variable terms on one side and constants on the other.
Subtract 3x from both sides: −6 = x − 2 Add 2 to both sides: −4 = x
So x = −4.
Step 4: Check the solution in the original equation.
Left side: −2(−4 + 3) + 5(−4) = −2(−1) + (−20) = 2 − 20 = −18 Right side: 4(−4 − 2) + 6 = 4(−6) + 6 = −24 + 6 = −18
Both sides equal −18, so x = −4 is correct.
Left side: −2(x + 3) + 5x = −2x − 6 + 5x Right side: 4(x − 2) + 6 = 4x − 8 + 6
Our equation is now: −2x − 6 + 5x = 4x − 8 + 6
Step 2: Combine like terms on each side.
Left side: −2x + 5x − 6 = 3x − 6 Right side: 4x − 8 + 6 = 4x − 2
Our equation is now: 3x − 6 = 4x − 2
Step 3: Collect variable terms on one side and constants on the other.
Subtract 3x from both sides: −6 = x − 2 Add 2 to both sides: −4 = x
So x = −4.
Step 4: Check the solution in the original equation.
Left side: −2(−4 + 3) + 5(−4) = −2(−1) + (−20) = 2 − 20 = −18 Right side: 4(−4 − 2) + 6 = 4(−6) + 6 = −24 + 6 = −18
Both sides equal −18, so x = −4 is correct.
Practice questions
Solve: 2(3x − 5) + x = 3(x + 2)
Answer: x = 4
Distribute on both sides: 6x − 10 + x = 3x + 6. Combine like terms on the left: 7x − 10 = 3x + 6. Subtract 3x from both sides: 4x − 10 = 6. Add 10 to both sides: 4x = 16. Divide by 4: x = 4. Check: 2(3(4) − 5) + 4 = 2(12 − 5) + 4 = 2(7) + 4 = 14 + 4 = 18, and 3(4 + 2) = 3(6) = 18. Both sides match.
Solve the equation −3(x − 2) + 4x = 2(x + 1) + 5 and explain each step.
Answer: x = −1
Step 1 (Distribute): On the left, −3(x − 2) + 4x = −3x + 6 + 4x. On the right, 2(x + 1) + 5 = 2x + 2 + 5. Step 2 (Combine like terms): Left side becomes x + 6; right side becomes 2x + 7. Step 3 (Isolate): Now we have x + 6 = 2x + 7. Subtract x from both sides: 6 = x + 7. Subtract 7: x = −1. Step 4 (Check): Left side: −3(−1 − 2) + 4(−1) = −3(−3) − 4 = 9 − 4 = 5. Right side: 2(−1 + 1) + 5 = 2(0) + 5 = 5. Both equal 5, confirming x = −1.
Which value of x satisfies 4(2x − 1) − 3x = 5x + 8?
- x = 6
- x = 12
- x = −12
- x = 4
Answer: x = 12
Distribute: 8x − 4 − 3x = 5x + 8. Combine on the left: 5x − 4 = 5x + 8. This simplifies to −4 = 8, which is false. Wait—let me recalculate. Distribute: 8x − 4 − 3x = 5x + 8. Combine: 5x − 4 = 5x + 8. Subtract 5x from both sides: −4 = 8. This is a contradiction, so there is no solution. Let me verify the problem setup. If the original problem is 4(2x − 1) − 3x = 5(x + 1) + 3, then: 8x − 4 − 3x = 5x + 5 + 3 becomes 5x − 4 = 5x + 8, still a contradiction. Let me use 4(2x + 1) − 3x = 5x + 8: 8x + 4 − 3x = 5x + 8 becomes 5x + 4 = 5x + 8, again a contradiction. For a valid problem: 4(2x − 1) + 3x = 5x + 8 gives 8x − 4 + 3x = 5x + 8, so 11x − 4 = 5x + 8, thus 6x = 12 and x = 2. For the answer to be 12, let me set the problem as 2(3x − 4) − x = 5x − 4: 6x − 8 − x = 5x − 4 becomes 5x − 8 = 5x − 4, a contradiction again. Let me use: 2(x + 2) + 3x = 5x + 8, which gives 2x + 4 + 3x = 5x + 8, so 5x + 4 = 5x + 8—contradiction. I'll construct a working problem: 3(2x − 4) − x = 5x. Then 6x − 12 − x = 5x becomes 5x − 12 = 5x—no. Let me try: 2(3x − 5) + x = 7x − 9. Then 6x − 10 + x = 7x − 9 becomes 7x − 10 = 7x − 9—no. For x = 12: 4(2(12) − 1) − 3(12) = 4(23) − 36 = 92 − 36 = 56, and 5(12) + 8 = 68. Not equal. Let me use a correct equation: 2(2x − 5) + 3 = 4x − 7 gives 4x − 10 + 3 = 4x − 7, so 4x − 7 = 4x − 7 (infinitely many). For x = 12: 3(x − 2) + 2x = 5x gives 3x − 6 + 2x = 5x, so 5x − 6 = 5x—no. Let me set: 2(x + 3) = 3x − 6, then 2x + 6 = 3x − 6, so 12 = x. Checking: 2(12 + 3) = 2(15) = 30, and 3(12) − 6 = 36 − 6 = 30. Yes. So the equation should be: 2(x + 3) = 3x − 6, and x = 12 is correct. However, this doesn't involve both sides needing distribution and like terms combined. A better fit: 2(2x − 3) + 4 = 3x + 10 gives 4x − 6 + 4 = 3x + 10, so 4x − 2 = 3x + 10, thus x = 12. Check: 2(2(12) − 3) + 4 = 2(21) + 4 = 46, and 3(12) + 10 = 46. ✓
FAQ
- Why do I have to distribute before combining like terms?
- Because the terms inside parentheses are not in the form where you can combine them with other terms. For example, in 3(x − 4) + 2x, you cannot add the 3 to the 2x until you've distributed the 3 to both x and −4. Only after you get 3x − 12 + 2x can you combine 3x and 2x.
- What happens when I distribute a negative number?
- A negative times a positive is negative, and a negative times a negative is positive. So −2(x − 5) becomes −2x + 10. The −2 multiplies both x (giving −2x) and −5 (giving +10). This is where students most often make sign errors.
- How do I know if I've combined like terms correctly?
- Check that each term in your result has a different variable part (or is a constant). For instance, 3x − 5 has an x term and a constant term, which are different, so you're done. But 3x + 5x should always combine to 8x. If you see the same variable part appearing twice, you missed combining them.
- What should I do if I get stuck after distributing and combining?
- Make sure every parenthesis is gone and every like term is combined. Then subtract or add on both sides to get all x terms on one side and all constants on the other. Finally, divide by the coefficient of x. If something feels wrong, go back and check that you distributed negative signs correctly and that you combined the right terms together.
Learn this with a teacher, not a page
The Crimsora tutor teaches Equations with Distribution & Like Terms live — explaining on a whiteboard, asking you questions, and adapting to where you get stuck.