Equations with Variables on Both Sides
Some equations have x on both sides. Move the x terms to one side and the numbers to the other. Then solve.
What you'll be able to do
- Solve equations with x on both sides.
- Distribute before you solve.
- Clear fractions and decimals first.
- Write an equation from a story and solve it.
- Check an answer by putting it back in.
The big idea
Keep it balanced. An equation is like a balance. Do the same thing to both sides, and it stays true.
An example. Solve 6x − 4 = 2x + 12. Subtract 2x from both sides: 4x − 4 = 12. Add 4: 4x = 16. Divide by 4: x = 4.
Check it. Put x = 4 back in. The left side is 6(4) − 4 = 20. The right side is 2(4) + 12 = 20. They match.
Parentheses first. In 3(x + 4), the 3 multiplies both parts. So 3(x + 4) = 3x + 12. A minus in front changes every sign: −(2x − 3) = −2x + 3.
Fractions. Multiply every term by a number that clears the fractions. For x/4 + 3 = x/6 + 5, multiply each term by 12. That gives 3x + 36 = 2x + 60.
Decimals. Multiply every term by 10 or 100. Then work with whole numbers.
Stories. Two costs are equal when their equations are equal. Write one side for each, then solve.
Words you need
- Variable
- A letter, like x, that stands for a number you don't know yet.
- Equation
- Two sides with an equals sign between them, like 2x + 1 = 7.
- Solution
- The number that makes the equation true.
- Like terms
- Terms with the same letter, like 4x and 2x. You can add them.
- Distribute
- Multiply the number outside the parentheses by each part inside.
Steps that keep the balance
| Step | Example |
|---|---|
| Distribute | 3(x + 4) = 3x + 12 |
| Clear fractions | x/2 + 1 → multiply each term by 2 |
| Move x terms to one side | Subtract 2x from both sides |
| Move numbers to the other side | Add 4 to both sides |
| Divide | 4x = 16, so x = 4 |
Worked examples
Example 1
Solve 6x − 4 = 2x + 12.
- Subtract 2x from both sides: 4x − 4 = 12.
- Add 4 to both sides: 4x = 16.
- Divide by 4: x = 4.
Answer: x = 4
Example 2
Solve 3(x + 4) = 5x − 2.
- Distribute: 3x + 12 = 5x − 2.
- Subtract 3x: 12 = 2x − 2.
- Add 2: 14 = 2x. So x = 7.
Answer: x = 7
Example 3
Solve x/2 + 1 = x/3 + 3.
- Multiply every term by 6: 3x + 6 = 2x + 18.
- Subtract 2x: x + 6 = 18.
- Subtract 6: x = 12.
Answer: x = 12
Common mistakes
Moving a term without changing its sign. To move +2x across, subtract 2x from both sides.
Losing a sign when you distribute a negative. −2(x − 6) = −2x + 12, because −2 × −6 = +12.
Subtracting a group without changing every sign. 5x − (2x − 3) = 5x − 2x + 3 = 3x + 3.
Clearing fractions in only some terms. Multiply every term on both sides by the same number.
Practice: 20 questions
11 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Solve 4 − 5x = 2x + 32.
- Gym A charges $28 plus $8 a class. Gym B charges $12 a class. After how many classes do they cost the same? Word problem
- Lucía solves 6x + 5 = 2x − 7 and gets x = 3. Check her answer. Find the right one.
- Plant A is 12 cm tall and grows 3 cm a week. Plant B is 30 cm tall and grows 1.5 cm a week. When are they the same height? Word problem
- Tank A holds 90 liters and drains 4 liters a minute. Tank B holds 30 liters and fills 2 liters a minute. After how many minutes do they hold the same amount? Word problem
Tier 2 Intermediate One more step to think through
- Ines is 3 times as old as her cousin. In 8 years, she will be twice as old as her cousin. How old is her cousin now? Word problem
- Solve −2(x − 6) = 5x − 9.
- Kofi rewrites 5x − (2x − 3) = x + 15 as 3x − 3 = x + 15. He gets x = 9. Find his mistake.
- Plan A costs $25 a month plus $0.10 a text. Plan B costs $15 a month plus $0.35 a text. For how many texts do they cost the same? Word problem
- Solve x/4 + 3 = x/6 + 5.
Tier 3 Advanced Several steps - find the hidden structure
- Solve 4(x − 3) − 2(x + 5) = 3(x − 4).
- Solve (x − 2)/3 = (2x + 1)/5.
- A member deal costs $14 for the first 2 games, then $3.25 for each extra game: 14 + 3.25(x − 2). Non-members pay $4.75 a game. For how many games do both cost the same? Word problem
- A rectangle is 2x + 3 cm long and x cm wide. A square has sides of x + 4 cm. They have the same perimeter. Find x. Word problem
- Wei multiplies 0.2(x + 15) = 0.5x − 3 by 10. He writes 2(x + 15) = 5x − 3. Find his mistake.
Tier 4 Challenge Stretch problems, some above grade level
- Ami's test scores are 84, 90 and 75. She wants her 4th score to equal her new mean. What must she score? Word problem
- The equation 3(x + a) = 2x + 10 has the solution x = 4. Find a.
- A train leaves at 60 km/h. Two hours later, a car follows it at 90 km/h. How many hours after the car leaves does it catch the train? Word problem
- Shops A and B rent bikes for 0.5x + 2 and 0.25x + 5 dollars for x hours. Shops C and D charge 2(x + 1) and x + 12 dollars. Which pair costs the same at more hours? Word problem
- C = (5/9)(F − 32) changes degrees Fahrenheit F to degrees Celsius C. Solve it for F. Then find F when C = 35. Word problem Above grade level
Answers
Tier 1 Foundations
- x = −4. Solve 4 − 5x = 2x + 32. Check x = −4: left side 24, right side 24.
- x = 7. Solve 28 + 8x = 12x. Check x = 7: left side 84, right side 84.
- x = −3. x = 3 gives 23 on the left and −1 on the right, so it is wrong. Subtract 2x and 5 from both sides: 4x = −12. Solve 6x + 5 = 2x − 7. Check x = −3: left side −13, right side −13.
- x = 12. Solve 12 + 3x = 30 + 1.5x.
- x = 10. Solve 90 − 4x = 30 + 2x. Check x = 10: left side 50, right side 50.
Tier 2 Intermediate
- x = 8. Call the cousin's age x. Ines is 3x. In 8 years: 3x + 8 = 2(x + 8). So 3x + 8 = 2x + 16, and x = 8. Check x = 8: left side 32, right side 32.
- x = 3. Solve −2 · (x − 6) = 5x − 9. Check x = 3: left side 6, right side 6.
- x = 6. The minus changes every sign in the group. Minus −3 is plus 3, so 3x + 3 = x + 15. Solve 5x − (2x − 3) = x + 15. Check x = 6: left side 21, right side 21.
- x = 40. Solve 25 + 0.10x = 15 + 0.35x.
- x = 24. Solve x/4 + 3 = x/6 + 5. Check x = 24: left side 9, right side 9.
Tier 3 Advanced
- x = −10. Solve 4 · (x − 3) − 2 · (x + 5) = 3 · (x − 4). Check x = −10: left side −42, right side −42.
- x = −13. Solve (x − 2)/3 = (2x + 1)/5. Check x = −13: left side −5, right side −5.
- x = 5. Solve 14 + 3.25 · (x − 2) = 4.75x.
- x = 5. The rectangle's perimeter is 2(2x + 3) + 2x = 6x + 6. The square's is 4(x + 4) = 4x + 16. So 6x + 6 = 4x + 16, and x = 5. Check x = 5: left side 36, right side 36.
- x = 20. Wei didn't multiply the 3 by 10. It should be 2(x + 15) = 5x − 30. Then 2x + 30 = 5x − 30. Solve 0.2 · (x + 15) = 0.5x − 3.
Tier 4 Challenge
- x = 83. Call the 4th score x. The new mean is (84 + 90 + 75 + x) ÷ 4. Set it equal to x: 249 + x = 4x. So 3x = 249, and x = 83. Check x = 83: left side 83, right side 83.
- a = 2. Put x = 4 into the equation: 3(4 + a) = 18. So 4 + a = 6, and a = 2. Check a = 2: left side 18, right side 18.
- x = 4. Call the car's time x. The train has gone 2 hours more, so its time is x + 2. They meet when the distances match: 60(x + 2) = 90x. So 120 = 30x, and x = 4. Check x = 4: left side 360, right side 360.
- Shops A and B (12 hours, against 10). Shops A and B match when 0.25x = 3, at 12 hours. Shops C and D match when x = 10. 12 is more than 10.
- F = (9/5)C + 32. When C = 35, F = 95. Undo the last step first. Multiply both sides by 9/5: (9/5)C = F − 32. Then add 32: F = (9/5)C + 32. When C = 35, F = 63 + 32 = 95.
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