Math Detours · Grade 7 · Sharpen · 6

Undo it backwards

A two-step equation is a number that was multiplied, then added to. To find it, undo the last step first, like taking off shoes then socks.

Name ____________________Date ____________

Tonight's idea

Look at \(3x + 7 = 22\). It tells a story. Someone started with a number \(x\), multiplied it by 3, then added 7, and ended at 22. To solve the equation, run the story backwards to where it began.

Build it, left to rightUndo it, right to leftx3x3x + 7× 3+ 751522÷ 3− 7
Top: how 22 was built. Bottom: taking it apart again. Each dotted line joins two names for the same number.

The last thing done was "add 7", so undo that first: \(22 - 7 = 15\). That 15 is \(3x\). Now undo "times 3": \(15 \div 3 = 5\). So \(x = 5\).

Why backwards? You put on socks, then shoes. Shoes come off first, because they are on the outside. In \(3x + 7\) the "+ 7" is on the outside.

Each undo is done to both sides of the equation, like the balance you met last year. That keeps it true.

Read what happened to \(x\), in order. Undo the last step first. Subtracting undoes adding, and dividing undoes multiplying. Then check by putting your answer back in.

Watch one: \(\dfrac{x}{4} - 3 = 2\)

The story: \(x\) was divided by 4, then 3 was taken away. Undo the taking away first. Add 3 to both sides: \(\dfrac{x}{4} = 5\). Then multiply both sides by 4: \(x = 20\). Check: \(20 \div 4 - 3 = 2\).

Brackets change the story. In \(4(x - 3) = 20\) the subtracting came first and the multiplying last. So divide first: \(x - 3 = 5\). Then add 3: \(x = 8\).

Try both undo moves on a balance, and see what a wrong move does: mathdetours.com/demos/balance.html.

Your turn

  1. For each equation, say what happened to \(x\) and in what order. Then solve it and check. (a) \(2x + 5 = 19\); (b) \(5x - 4 = 41\); (c) \(\dfrac{x}{3} + 2 = 6\).
  2. Solve: (a) \(3(x + 4) = 27\); (b) \(\dfrac{x - 6}{2} = 5\); (c) \(2x + 9 = 3\).
  3. A taxi charges $4 to start and then $3 for each kilometre. A ride costs $31. Write an equation for the number of kilometres, \(k\), and solve it.
  4. Mia solves \(3x + 6 = 24\) by dividing first. She writes \(x + 6 = 8\), then \(x = 2\). (a) Check her answer in the equation. (b) What went wrong in her first step? (c) Find the right answer two ways: by undoing "+ 6" first, and by dividing first but doing it properly.

Stretch: four steps

I think of a number. I double it, add 3, multiply the result by 5, then subtract 4. I end at 61. Write this as one equation, using brackets. Then undo the four steps in the right order to find my number.

Answers

  1. (a) Times 2, then add 5. \(2x = 14\), \(x = 7\). (b) Times 5, then subtract 4. \(5x = 45\), \(x = 9\). (c) Divide by 3, then add 2. \(\dfrac{x}{3} = 4\), \(x = 12\).
  2. (a) Divide by 3: \(x + 4 = 9\), so \(x = 5\). (b) Multiply by 2: \(x - 6 = 10\), so \(x = 16\). (c) Subtract 9: \(2x = -6\), so \(x = -3\). Check: \(-6 + 9 = 3\).
  3. \(3k + 4 = 31\). Subtract 4: \(3k = 27\). Divide by 3: \(k = 9\) km.
  4. (a) \(3 \times 2 + 6 = 12\), not 24. (b) She divided the \(3x\) and the 24 by 3 but left the 6 alone. Dividing a side by 3 divides every part of it. (c) Undo "+ 6": \(3x = 18\), \(x = 6\). Dividing properly: \(x + 2 = 8\), \(x = 6\). Check: \(18 + 6 = 24\).

Stretch. \(5(2x + 3) - 4 = 61\). Add 4: \(5(2x + 3) = 65\). Divide by 5: \(2x + 3 = 13\). Subtract 3: \(2x = 10\). Halve: \(x = 5\). Check: 5, 10, 13, 65, 61.

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