Solving equations
In “x + 7 = 12”, what operation is attached to 7? What inverse operation removes it?
- minus subtract the 7
Finish the sentence: “Whatever I do to one side of an equation, I must …”
Do the same thing to the other side.
5x + 2 = 17 has become 5x = 15. What is the next step?
Divide both sides by 5, giving x = 3.
x - 6 = 10, so x = 4. Identify and correct the error.
They subtracted 6 instead of adding 6. Correct answer: x = 16.
How can you check whether x = 4 solves 2x + 3 = 11?
Substitute 4 for x: 2(4) + 3 = 11, so it works.
What should you do first to solve 3x + 5 = 20? Explain why.
Take 5 away from both sides
If 5x = 35, what must both sides be divided by? Say the reason.
Divide both sides by 5 to undo multiplication by 5.
3(x + 4) = 21 has become x + 4 = 7. What happens next?
Subtract 4 from both sides, giving x = 3.
4x = 28, so x = 24. What should happen?
Divide by 4, not subtract 4. Correct answer: x = 7.
Solve 7x = 42, but every spoken sentence must name the operation used.
Divide both sides by 7 to undo multiplication; x = 6.
For 4(x - 2) = 20, give two possible first moves. Which is more efficient?
You could divid both sides by 4 OR expand the brackets first on the left side.
A student changes 2x + 9 = 17 into 2x = 17. What did they forget?
They forgot to subtract 9 from both sides; it should become 2x = 8.
8x - 5 = 3x + 20 has become 5x - 5 = 20. Give the next two steps.
Add 5 to both sides: 5x = 25. Divide by 5: x = 5.
3(x + 2) = 15 becomes 3x + 2 = 15. Correct the expansion.
3 must multiply both terms: 3x + 6 = 15, so x = 3.
Solve 5x - 8 = 2 and explain why your first step is sensible.
Add 8 to both sides to undo subtraction: 5x = 10; divide by 5: x = 2.
In 7x + 3 = 4x + 18, which x-term would you move first? Why?
Subtract teh 4x from both sides so you do not get a negative.
Which line is balanced?
A: 3x + 4 - 4 = 19
B: 3x + 4 - 4 = 19 - 4
B. The same operation must be applied to both sides.
2(3x - 1) + 4 = 20. Expand correctly, then state the next step.
6x - 2 + 4 = 20, then simplify to 6x + 2 = 20 (or subtract 2 from both sides).
-2x = 14, so x = 7. What sign error occurred?
14 divided by -2 is -7. Correct answer: x = -7.
Solve 4x + 1 = 2x + 13. Then prove the answer by checking both sides.
2x = 12, so x = 6. Check: LHS 25; RHS 25.
Describe the complete process for solving a linear equation without using any numbers.
Simplify both sides move variable terms to one side and constants to the other using inverse operations, isolate the variable, then substitute the solution back in to check.
Explain why “moving a number across and changing its sign” can be misleading.
Nothing literally moves; we apply the same inverse operation to both sides. Showing this protects the balance and reduces sign errors.
(2x + 3)/5 = 7. State the first three steps in order.
Multiply both sides by 5: 2x + 3 = 35; subtract 3: 2x = 32; divide by 2: x = 16.
Student solves 6x + 2 = 2x + 18: “4x + 2 = 18, 4x = 20, x = 5.” Find the incorrect line and finish correctly.
4x + 2 = 18 is correct; subtracting 2 gives 4x = 16, so x = 4.
Create a linear equation whose solution is x = -3.
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