Year 9 · Unit 7 Equations and linear systems · Lesson 5 of 7

Replace one unknown and solve the rest

When one equation says what y is, put that into the other equation. Two unknowns become one, and you already know how to solve one.

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. Solve 3x + 5 = 20.
    Answerx = 5, because 3x = 15.
  2. Expand 4(2x − 3).
    Answer8x − 12.
  3. Write y − 2x = 3 as y = …
    Answery = 2x + 3: add 2x to both sides.
Words used in this lesson
substitution
replacing a letter by an expression. For example: replace y by 3x − 2.
subject
the letter on its own on one side. For example: in y = 3x + 1 the subject is y.
rearrange
rewrite so a different letter is the subject. For example: y − 2x = 3 becomes y = 2x + 3.
brackets
symbols that show a whole expression is multiplied. For example: 3(4x − 1).
back-substitute
put the answer in to find the other unknown. For example: x = 5, so y = 5 + 3.
check
put the pair in both original equations. For example: 10 + 8 = 18.

One fact says what the other is

At a school fair in Manila a juice costs 3 pesos more than a samosa. Two samosas and one juice cost 18 pesos. Call a samosa x pesos and a juice y pesos. That gives two facts: y = x + 3 and 2x + y = 18.

Predict first: the first fact says y and x + 3 are the same price. In the second fact, what could you do with the y to get an equation in x only? Then explore with the figure.

y = 3x − 2 and 2x + y = 13
1
1y from the first equation
3left side of the second
10 short of 13against the target

y = 3x − 2 and 2x + y = 13 at x = 1: y = 1, so the left side of the second is 3, which is 10 short of 13.

Show me the steps
  1. y = 3x − 2 and 2x + y = 13 at x = 1: y = 1 and 2x + y = 3, which is 10 short of 13.
  2. The same system at x = 3: y = 7 and 2x + y = 6 + 7 = 13. Balanced, so (3, 7) solves both.
  3. y = x + 4 and 3x + 2y = 18 at x = 2: y = 6 and 3 × 2 + 2 × 6 = 18. Balanced: (2, 6).
  4. y = −2x + 5 and 3x + 2y = 7 at x = 3: y = −1 and 9 − 2 = 7. Balanced: (3, −1).

Substitution:

1. Make one equation say what one unknown is, for example y = 3x − 2.
2. Replace that unknown in the other equation, in brackets.
3. Solve the one-unknown equation.
4. Put the answer into the first equation to find the other unknown.
5. Check in both original equations.

A common mistake: losing the brackets. Replacing y with 4x − 1 in 3y gives 3(4x − 1) = 12x − 3, not 12x − 1. Test with x = 2: y = 7 and 3y = 21; 12 × 2 − 3 = 21, but 12 × 2 − 1 = 23.

Worked examples

The fair: y = x + 3 and 2x + y = 18. Find the price of each.

  1. Replace y in the second equation: 2x + (x + 3) = 18.
  2. 3x + 3 = 18, so 3x = 15 and x = 5.
  3. Then y = 5 + 3 = 8.
  4. Check: 2 × 5 + 8 = 18 and 8 = 5 + 3. A samosa is 5 pesos and a juice is 8 pesos.

Rearrange first: 3x + y = 14 and 5x − 2y = 5.

  1. Make y the subject of the first: y = 14 − 3x.
  2. Replace in the second: 5x − 2(14 − 3x) = 5.
  3. 5x − 28 + 6x = 5, so 11x = 33 and x = 3.
  4. y = 14 − 9 = 5. Check: 15 − 10 = 5. (3, 5).

Your go. The first step is given. Solve y = 4x − 1 and 2x + 3y = 25.

  1. Replace y in the second equation: 2x + 3(4x − 1) = 25.
  2. Expand, solve for x, find y and check.
Check the answer

2x + 12x − 3 = 25, so 14x = 28 and x = 2. Then y = 4 × 2 − 1 = 7. Check: 4 + 21 = 25. (2, 7).

Check before you go on

1. In the system y = 2x + 5 and 4x + 3y = 33, replacing y in the second equation gives which line?

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. y = 2x + 1 and x + y = 10. Find x.

3. y = 2x − 1 and x + y = 8. Find y.

Level 2 · Building

4. y − 2x = 3 and 3x + 2y = 20. Find x.

5. At a charity bake sale a drink costs 5 taka more than a snack. 2 drinks and 3 snacks cost 60 taka. What does one snack cost, in taka?

Level 3 · Stretch

6. y = 3 − 2x and 5x − 4y = 27. Find x.

7. x = 2y + 1 and 3x − 5y = 7. Find y.

Something to think about

Solve y = 4x − 3 and x + y = 12 by replacing y in the second equation. Then solve it a second way: write y from the second equation (y = 12 − x) and put that into the first. Do you reach the same point?

One way to think about it

The first way: x + 4x − 3 = 12 gives 5x = 15, so x = 3 and y = 9. The second way: 12 − x = 4x − 3 gives 15 = 5x, so x = 3, and y = 12 − 3 = 9. The same point, (3, 9): there is more than one way in, and the choice of which unknown to replace is yours.

Exit ticket

8. Which pair solves y = 3x − 5 and 2x + y = 15?