Simultaneous equations: special cases

The Maths Habit

1Write down the gradient of 2y = 6x + 1.32Are the lines y = 3x + 2 and 6x − 2y = 5 parallel?Yes: both have gradient 33Solve 4x − 3y = 7 and 3x + 2y = 1.x = 1, y = −1430 coins, all 20p and 50p, are worth £9.30. How many of each?nineteen 20p and eleven 50p

Solve the simultaneous equations.

(1)3x+2y=5
(2)3x+2y=6

No solution. 3x + 2y cannot be 5 and 6 at the same time.

No solution
No pair of values makes both equations true.
Infinitely many solutions
Every pair of values that makes one equation true makes the other true too.

Solving gives 0 = 5. What does this tell you?

  1. AThere is no solution: the lines are parallel.
  2. Bx = 0, y = 5
  3. CThere are infinitely many solutions.
  4. DI have made a mistake.

A: 0 = 5 is never true, so no pair of values fits both equations.

Solve, or show there is no solution or infinitely many.

(1)y=2x + 1
(2)4x−2y=6
Rearrange (1)
(3)−2x+y=1
(3) × 2
(4)−4x+2y=2
(2) + (4)
0=8
Answer no solution
The lines are parallel
xy−2−1123−3−2−112345(1)(2)

Solve, or show there is no solution or infinitely many.

(1)3x−y=2
(2)6x−2y=7
(1) × 2
(3)6x−2y=4
(2) − (3)
0=3
Answer no solution
The lines are parallel

Solve, or show there is no solution or infinitely many.

(1)y=2x + 1
(2)4x−2y=6
Rearrange (1)
(3)−2x+y=1
(3) × 2
(4)−4x+2y=2
(2) + (4)
0=8
Answer no solution
The lines are parallel
xy−2−1123−3−2−112345(1)(2)

Solve, or show there is no solution or infinitely many.

(1)2x+5y=1
(2)4x+10y=7
(1) × 2
(3)4x+10y=2
(2) − (3)
0=5
Answer no solution
The lines are parallel

Solve, or show there is no solution or infinitely many.

(1)x+3y=4
(2)2x+6y=8
(1) × 2
(3)2x+6y=8
(3) − (2)
0=0
Answer infinitely many solutions
The lines are the same line
xy−2−112345−11234(1)(2)

Solve, or show there is no solution or infinitely many.

(1)4x−2y=6
(2)6x−3y=9
(1) × 3
(3)12x−6y=18
(2) × 2
(4)12x−6y=18
(3) − (4)
0=0
Answer infinitely many solutions
The lines are the same line

Solve, or show there is no solution or infinitely many.

(1)x+3y=4
(2)2x+6y=8
(1) × 2
(3)2x+6y=8
(3) − (2)
0=0
Answer infinitely many solutions
The lines are the same line
xy−2−112345−11234(1)(2)

Solve, or show there is no solution or infinitely many.

(1)y=3x − 2
(2)6x−2y=4
Substitute (1) into (2)
6x − 2(3x − 2) = 4
6x − 6x + 4 = 4
4=4
Answer infinitely many solutions
The lines are the same line

One solution, no solution or infinitely many?

a
(1)2x−y=3
(2)4x−2y=6
infinitely many
b
(1)2x−y=3
(2)4x−2y=5
none
c
(1)2x−y=3
(2)x+y=6
one: x = 3, y = 3
d
(1)y=5 − x
(2)x+y=2
none
e
(1)3x+2y=7
(2)6x+4y=14
infinitely many
f
(1)3x+2y=7
(2)2x+3y=8
one: x = 1, y = 2
g
(1)2x+3y=4
(2)4x−3y=−1
one: x = 12, y = 1

Find k so that there is no solution.

(1)3x+ky=7
(2)6x+4y=5
(1) × 2
(3)6x+2ky=14
2k = 4
k = 2
Answer k = 2
Check 14 ≠ 5, so no solution ✓

Find a so that there are infinitely many solutions.

(1)ax+2y=1
(2)6x+4y=2
(1) × 2
(3)2ax+4y=2
2a = 6
a = 3
Answer a = 3
Check 2 = 2, the same line ✓

Find k so that there is no solution.

(1)3x+ky=7
(2)6x+4y=5
(1) × 2
(3)6x+2ky=14
2k = 4
k = 2
Answer k = 2
Check 14 ≠ 5, so no solution ✓

Find k so that there is no solution. Can any k give infinitely many?

(1)4x+ky=3
(2)2x+3y=5
(2) × 2
(3)4x+6y=10
k = 6
Answer k = 6
Infinitely many? No: that needs 3 = 10

£5000 is split between two accounts, both paying simple interest for one year: one 3%, the other 5%. Together they earn £196. How much is in each?

Let a = £ at 3%, b = £ at 5%
(1)a+b=5000
3% of a + 5% of b = 196, × 100
(2)3a+5b=19 600
(1) × 3
(3)3a+3b=15 000
(2) − (3)
2b=4600
b=2300
(1)a+2300=5000
a=2700
Answer £2700 at 3%, £2300 at 5%
Check 81 + 115 = 196 ✓

A parallelogram has angles (3x + y)° and (x + 4y)° opposite each other, and (x + 2y)° next to (3x + y)°. Find its angles.

Opposite angles: 3x + y = x + 4y
(1)2x−3y=0
Next to each other: 4x + 3y = 180
(2)4x+3y=180
(1) + (2)
6x=180
x=30
(1)60−3y=0
−3y=−60
y=20
Answer x = 30, y = 20
The angles 110°, 110°, 70°, 70°
(3x + y)°(x + 4y)°(x + 2y)°

£5000 is split between two accounts, both paying simple interest for one year: one 3%, the other 5%. Together they earn £196. How much is in each?

Let a = £ at 3%, b = £ at 5%
(1)a+b=5000
3% of a + 5% of b = 196, × 100
(2)3a+5b=19 600
(1) × 3
(3)3a+3b=15 000
(2) − (3)
2b=4600
b=2300
(1)a+2300=5000
a=2700
Answer £2700 at 3%, £2300 at 5%
Check 81 + 115 = 196 ✓

Ana has £10 more than Ben. 20% of Ana’s money and 50% of Ben’s add up to £23. How much does each have?

Let a = Ana’s £, b = Ben’s £
(1)a−b=10
20% of a + 50% of b = 23, × 10
(2)2a+5b=230
(1) × 2
(3)2a−2b=20
(2) − (3)
7b=210
b=30
(1)a−30=10
a=40
Answer Ana £40, Ben £30
Check 8 + 15 = 23 ✓

Simultaneous equations: special cases worksheet, page 1

Answer sheet

  1. Solve 31x + 17y = 113 and 17x + 31y = 127 without multiplying. Add the whole equations, then subtract them.

    x = 2, y = 3

  2. Find c so that 2x + 3y = c and 4x + 6y = 10 have infinitely many solutions. What happens for any other c?

    c = 5; no solution

  3. Write a second equation to go with 3x − y = 4 so the pair has (a) no solution, (b) infinitely many solutions, (c) the solution x = 2, y = 2.

    e.g. (a) 3x − y = 1 (b) 6x − 2y = 8 (c) x + y = 4

  1. How many solutions do 6x − 3y = 9 and y = 2x − 3 have?

    infinitely many (the same line)

  2. How many solutions do y = 4x − 1 and 8x − 2y = 5 have?

    none (2 = 5: parallel lines)

  3. Solve 4x + 3y = 9 and 2x − 3y = 0.

    x = 32, y = 1

  4. Printer A for 5 minutes and printer B for 3 minutes print 181 pages. A for 3 minutes and B for 4 minutes print 146 pages. How many pages a minute does each print?

    A 26, B 17 pages a minute
Exit tickets to print