Year 11 Standard · Unit 3 Sequences, formulae and proportion · Lesson 6 of 8

How do you run a formula backwards?

A formula is a chain of steps done to one letter. This lesson asks how to run that chain the other way, so that a different letter becomes the subject.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: solve 3x + 5 = 20.
    Answerx = 5.
  2. From Year 8: what is the inverse of squaring? Evaluate it for 49.
    AnswerThe square root; the positive root of 49 is 7.
  3. From Year 9: solve (x − 2) ÷ 4 = 3.
    Answerx = 14.
Words used in this lesson
subject
the letter on its own on one side of a formula. For example: c in c = 4d + 1.
formula
a rule written with letters that links quantities. For example: area = length × width.
inverse operation
the operation that undoes another. For example: ÷ undoes ×.
rearrange
to change a formula into another form with the same meaning. For example: from y = x + 5 to x = y − 5.
substitute
to replace a letter by a number. For example: put m = 3 into 2m + 1.

Can you run the oven chart backwards?

A cook in Seoul has an oven manual that lists temperatures in degrees Fahrenheit. The conversion is F = 9⁄5C + 32. The manual says to set the oven to 392 F for a loaf, but the oven dial is in Celsius.

Predict first: the formula gives F from C. Which steps would you do to get C from F, and in which order? Write your plan before you look at the figure, which runs a two-step formula forwards and then backwards on the same numbers.

2
3
1
5y at the end of the top row
1x at the end of the bottom row

a = 2, b = 3, x = 1: forwards 1 × 2 = 2, then 2 + 3 = 5. Backwards 5 − 3 = 2, then 2 ÷ 2 = 1.

Show me the steps
  1. a = 2, b = 3, x = 1: forwards 1 × 2 = 2 then 2 + 3 = 5. Backwards 5 − 3 = 2 then 2 ÷ 2 = 1.
  2. a = 5, b = 2, x = 3: forwards 3 × 5 = 15 then 15 + 2 = 17. Backwards 17 − 2 = 15 then 15 ÷ 5 = 3.
  3. a = 3, b = 9, x = 8: forwards 8 × 3 = 24 then 24 + 9 = 33. Backwards 33 − 9 = 24 then 24 ÷ 3 = 8.
  4. Any a, b and x: the backwards row does the opposite operations in the opposite order, so it always returns to x. That is x = (y − b) ÷ a.

Undo the steps in reverse order. To make a letter the subject, list what the formula does to that letter, from the first step to the last. Then undo those steps from the last to the first, doing the same thing to both sides each time.

An operation and its inverse: add and subtract, multiply and divide, square and square root. Undoing a square gives two values, ±√; a length or a time keeps only the positive one.

Two common mistakes. Dividing only part of a side: from y = 3x + 6 the next step is (y − 6) ÷ 3, not y − 6 ÷ 3. And taking the root of one term: from y = x² + 4, x = √(y − 4), not √y − 4.

Worked examples

1. Make x the subject of y = 4x − 7.

  1. The formula multiplies x by 4, then subtracts 7. Undo in reverse order: add 7 first, y + 7 = 4x.
  2. Then divide by 4: x = (y + 7) ÷ 4.

2. Make r the subject of C = 2πr.

  1. The formula multiplies r by 2π. The inverse is dividing by 2π.
  2. So r = C ÷ (2π).

3. Make u the subject of v² = u² + 2as, with u positive.

  1. Subtract 2as from both sides: v² − 2as = u².
  2. Take the positive square root: u = √(v² − 2as).

Your go. The first step is given. Make t the subject of s = ½gt², with t positive. The first step is to multiply both sides by 2, giving 2s = gt².

  1. 2s = gt².
  2. Now undo the multiplication by g, then undo the squaring.
Check the answer

Divide by g: 2s ÷ g = t². Take the positive root: t = √(2s ÷ g).

Check before you go on

1. Which is x as the subject of y = (x + 8) ÷ 5?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The perimeter of a rectangle is P = 2(l + w). A rectangle has P = 38 and w = 7. What is l?

3. A car's speed follows v = u + at. Find a when v = 30, u = 6 and t = 4.

Level 2 · Building

4. The area of a trapezium is A = ½h(a + b). Find b when A = 60, h = 8 and a = 5.

5. The kinetic energy of a body is E = ½mv². Find v (positive) when E = 90 and m = 5.

Level 3 · Stretch

6. In a right-angled triangle a² + b² = c². Find b when c = 17 and a = 8.

7. A temperature in Fahrenheit is F = 9⁄5C + 32. Find C when F = 77.

Something to think about

In v² = u² + 2as, making v the subject gives v = ±√(u² + 2as). When would you keep only the positive root, and when must you keep both?

One way to think about it

If v is a length or a speed, it cannot be negative, so keep the positive root. If v is a velocity, the sign carries a direction, and both roots can be correct answers. The same algebra gives both; the context decides which to keep.

Exit ticket

8. Which is r as the subject of A = πr² (r positive)?