Year 11 Standard · Unit 3 Sequences, formulae and proportion · Lesson 7 of 8

What do you do when the letter you want is in two places?

A formula such as y = ax + bx cannot be undone one step at a time. Collect the terms that hold the subject, take it out as a common factor, and then divide.

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 lesson 6: make x the subject of y = 4x + 9.
    Answerx = (y − 9) ÷ 4.
  2. From Year 8: factorize 6x + 15.
    Answer3(2x + 5).
  3. From Year 9: solve 5x − 7 = 2x + 8.
    Answerx = 5. 3x = 15.
Words used in this lesson
collect
bring terms together on one side of a formula. For example: put the t terms on the left.
factorize
write as a product with a common factor taken out. For example: 10k + 15 = 5(2k + 3).
common factor
a letter or number that divides every term. For example: m in km + 3m.
denominator
the bottom number or expression of a fraction. For example: the w − 1 in 4 ÷ (w − 1).
reciprocal
1 divided by a number. For example: 1⁄8 is the reciprocal of 8.

When are two plans equal?

A gym in Johannesburg offers two plans. Plan A costs 40 a month plus 5 for every class. Plan B costs 10 a month plus 8 for every class. A new member wants to know after how many classes in a month the two plans cost the same.

Predict first: will plan A be cheaper after about 5 classes, 10 classes or 15 classes? Write down your guess and write an equation with n for the number of classes. Notice that n appears on both sides. Then use the figure to see why two terms in one letter can become one.

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3
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10total area y = ax + bx
2y divided by (a + b)

a = 2, b = 3, x = 2: the areas are 4 and 6, the whole is 10, and 10 ÷ 5 = 2, which is x.

Show me the steps
  1. a = 2, b = 3, x = 2: the areas are 4 and 6, the whole is 10, and 10 ÷ 5 = 2.
  2. a = 5, b = 1, x = 3: the areas are 15 and 3, the whole is 18 = 6 × 3, and 18 ÷ 6 = 3.
  3. a = 1, b = 6, x = 4: the areas are 4 and 24, the whole is 28 = 7 × 4, and 28 ÷ 7 = 4.
  4. Any a, b and x: ax + bx = (a + b)x, so x = y ÷ (a + b). Factorizing is the step that turns two x's into one.

Collect, factorize, divide. When the subject appears in more than one place, (1) get every term that contains it on one side and everything else on the other, (2) take the subject out as a common factor, (3) divide by the bracket.

If the subject is in a fraction, multiply both sides by the denominator first so that the fraction is gone.

Two common mistakes. Dividing by one term only: from x(a + b) = y the next step is to divide by the whole bracket, x = y ÷ (a + b), not y ÷ a + b. And moving a term without changing its sign: from 4 − x = 3x the −x goes across as +x, giving 4 = 3x + x.

Worked examples

1. Make x the subject of ax + b = cx + d.

  1. Collect the x terms on the left and the others on the right: ax − cx = d − b.
  2. Factorize: x(a − c) = d − b. Divide: x = (d − b) ÷ (a − c).

2. Make p the subject of T = pq + p.

  1. Both terms on the right contain p, so factorize: T = p(q + 1).
  2. Divide by (q + 1): p = T ÷ (q + 1).

3. Make x the subject of y = (3x + 1) ÷ (x − 2).

  1. Multiply both sides by (x − 2): y(x − 2) = 3x + 1. Expand: xy − 2y = 3x + 1.
  2. Collect the x terms on the left: xy − 3x = 2y + 1. Factorize: x(y − 3) = 2y + 1.
  3. Divide: x = (2y + 1) ÷ (y − 3).

Your go. The first step is given. Make a the subject of ab = c − ad. The first step is to collect the a terms on the left: ab + ad = c.

  1. ab + ad = c.
  2. Take a out as a common factor, then divide by the bracket.
Check the answer

a(b + d) = c, so a = c ÷ (b + d). Check with a = 2, b = 3, d = 1: ab = 6 and c − ad = 8 − 2 = 6 when c = 8, and 8 ÷ 4 = 2. ✓

Check before you go on

1. Which is x as the subject of k(x − 2) = x?

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 formula is ax + bx = c. Find x when a = 8, b = 4 and c = 72.

3. The formula is px − q = rx + s. Find x when p = 7, q = 4, r = 3 and s = 24.

Level 2 · Building

4. The formula is y = (x + 3) ÷ (x − 1). Find x when y = 3.

5. A = P(1 + rt) gives the total after simple interest. Find P when A = 1560, r = 0.1 and t = 2.

Level 3 · Stretch

6. The lens formula is 1/f = 1/u + 1/v. Find v when f = 2 and u = 6.

7. The distance formula is s = ut + ½at². Find a when s = 40, u = 6 and t = 4.

Something to think about

Try to make x the subject of y = x² + 3x. What stops you? Which step from this lesson fails, and what kind of equation have you got?

One way to think about it

Factorizing gives y = x(x + 3), but the bracket still contains x, so dividing by it cannot leave x on its own. The formula is a quadratic in x. It needs a method for quadratics, such as completing the square or the quadratic formula from Year 10: x² + 3x − y = 0 gives x = (−3 ± √(9 + 4y)) ÷ 2.

Exit ticket

8. Which is a as the subject of ab − c = ad + e?