Simultaneous equations: substitution

The Maths Habit

1Simplify 3x + 2(x − 4)5x − 82Solve 5x − 8 = 17x = 53Expand −2(3 − x)−6 + 2x4Solve 2x + 5y = 1 and 3x + 4y = 5x = 3, y = −1

y = 2x and x + y = 12. Find x and y. What did you do?

x = 4, y = 8

y is the same as 2x, so x + 2x = 12

Substitute
Replace a letter with what it is equal to.

Solve the simultaneous equations.

(1)y=2x − 3
(2)3x+2y=22
Substitute (1) into (2)
3x + 2(2x − 3) = 22
3x + 4x − 6 = 22
7x=28
x=4
(1)y=2 × 4 − 3
y=5
x = 4,  y = 5
Check in (2): 12 + 10 = 22 ✓

Solve the simultaneous equations.

(1)y=5 − x
(2)4x−3y=13
Substitute (1) into (2)
4x − 3(5 − x) = 13
4x − 15 + 3x = 13
7x=28
x=4
(1)y=5 − 4
y=1
x = 4,  y = 1
Check in (2): 16 − 3 = 13 ✓

Solve the simultaneous equations.

(1)y=2x − 3
(2)3x+2y=22
Substitute (1) into (2)
3x + 2(2x − 3) = 22
3x + 4x − 6 = 22
7x=28
x=4
(1)y=2 × 4 − 3
y=5
x = 4,  y = 5
Check in (2): 12 + 10 = 22 ✓

Solve the simultaneous equations.

(1)y=3x + 2
(2)5x−y=4
Substitute (1) into (2)
5x − (3x + 2) = 4
5x − 3x − 2 = 4
2x=6
x=3
(1)y=3 × 3 + 2
y=11
x = 3,  y = 11
Check in (2): 15 − 11 = 4 ✓

Substitute y = 2x − 5 into 3x − 2y = 4. Which line is right?

A 3x − 4x − 10 = 4B 3x + 4x − 10 = 4C 3x − 4x + 10 = 4D 3x − 4x − 5 = 4

C

Which method would you use first: substitution or elimination?

Ay = 4x + 1 and 2x + 3y = 17SubstitutionB3x + 2y = 5 and 5x − 2y = −13Elimination: addCx = 3y and 2x + y = 14Substitution

By substitution

(1)y=3x − 2
(2)5x−2y=−1
Substitute (1) into (2)
5x − 2(3x − 2) = −1
5x − 6x + 4 = −1
−x=−5
x=5
(1)y=3 × 5 − 2
y=13
x = 5,  y = 13
Check in (2): 25 − 26 = −1 ✓

By elimination

(1)y=3x − 2
(2)5x−2y=−1
Rearrange (1)
(3)3x−y=2
(3) × 2
(4)6x−2y=4
(4) − (2)
x=5
(1)y=3 × 5 − 2
y=13
x = 5,  y = 13
Check in (2): 25 − 26 = −1 ✓

Which needed less work, and how could you tell before starting?

By substitution

(1)y=3x − 2
(2)5x−2y=−1
Substitute (1) into (2)
5x − 2(3x − 2) = −1
5x − 6x + 4 = −1
−x=−5
x=5
(1)y=3 × 5 − 2
y=13
x = 5,  y = 13
Check in (2): 25 − 26 = −1 ✓

Choose a method, then solve.

(1)3x+2y=11
(2)x−2y=9
(1) + (2)
4x=20
x=5
(2)5−2y=9
−2y=4
y=−2
x = 5,  y = −2
Check in (1): 15 − 4 = 11 ✓

Simultaneous equations: substitution worksheet, page 1

Answer sheet

  1. Solve x = 2y + 1 and 3x − 4y = 9.

    x = 7, y = 3

  2. y = x + 1,  z = 2y and x + y + z = 19. Find x, y and z.

    x = 4, y = 5, z = 10

  3. y = 3x − 1 and 2x + ky = 24 have the solution x = 2. Find k.

    k = 4

  1. Solve the simultaneous equations.
    (1)y=4x − 1(2)2x + y=2

    x = 12, y = 1

  2. y = 4 − x is substituted into 5x − 2y = 6. Which line is right? A  5x − 8 − 2x = 6B  5x − 8 + 2x = 6C  5x − 4 + x = 6

    B

  3. Solve the simultaneous equations.
    (1)y=2x − 5(2)3x + 2y=25

    x = 5, y = 5
Exit tickets to print