Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

Each wrong solve still reveals one true constant

Nora solved the equation $\frac{2}{3}x+a=-x+b$ in $x$ , but she misread $a$ as $0$ and got $x=18$ . Dana solved the same equation, but she misread $b$ as $0$ and got $x=-6$ . What is the solution of the original equation when it is solved correctly?

Expressions

Your answer

How to solve

Strategy Work Backwards — Work backwards from each wrong answer: a solution makes its equation true, so substituting it into the misread equation leaves one unknown, the constant that was read correctly. With both constants found, solve forward.

1STEP 1

Nora's equation gives b

Nora solved (2/3)x=-x+b and got x=18. Substitute.

2/3 × 18=-18+b → 12=-18+b → b=30

2STEP 2

Dana's equation gives a

Dana solved (2/3)x+a=-x and got x=-6. Substitute.

2/3×(-6)+a=6 → -4+a=6 → a=10

3STEP 3

Write the true equation

Put both true constants back.

(2/3)x+10=-x+30

4STEP 4

Solve it

Multiply both sides by 3, then collect terms.

2x+30=-3x+90 → 5x=60 → x=12

Answer

x=12

At x=12: left side 2/3 × 12+10=18; right side -12+30=18. Equal.

Takeaway

A wrong answer isn't useless: plug it into the equation that was actually solved and it hands you a true number.

Nora's equation gives b
Dana's equation gives a
Write the true equation
Solve it

Where next?

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▶ Practice — 12 problems

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