PAIRS OF EQUATIONS
Consider the following information!
Two coffees & three doughnuts cost 290p.
Three coffees & one doughnut cost 260p.
How much is each item?
Coffee = 70p & doughnut = 50p !!
Check
2c + 3d = 140 + 150 = 290.
3c + d =
210 + 50 = 260.
DON’T
COPY !!
Combining Equations
Suppose a banana costs
20p and a kiwi costs 30p.
Using b and k as shorthand.
5b + 3k = 190
3b + 2k = 120
ADDING
SUBTRACTING
MULTIPLYING
5b + 3k = 190
3b + 2k = 120
5b + 3k = 190
3b + 2k = 120
3b + 2k = 120
X5
8b + 5k = 310
2b + k = 70
15b + 10k = 600
Eliminating a Letter
Ex1
Ex2
3a + b = 13
a+b=7
x + y = 13
x-y= 5
add
2x = 18
x=9
use top equation
9 + y = 13
y=4
Also 9 – 4 = 5
Note: y and –y cancel by adding
Subtract 
2a = 6
a=3
use 2nd equation
3+b=7
b=4
Also 3x3 + 4 = 13
Note: b and b cancel by subtracting
Ex3
Ex4
3p + q = 3
3p + 4q = 21
c + 2d = 1
3c – 2d = 19
add
4c = 20
c=5
Subtract 
use top equation
use top equation
5 + 2d = 1
2d = -4
d = -2
Also
3x5 – 2x(-2)
= 15 – (-4) = 19
Note: 2d and –2d cancel by adding
3q = 18
q=6
3p + 6 = 3
3p = -3
p = -1
Also
3x(-1) + 4x6
= -3 + 24 = 21
Note: 3p and 3p cancel by subtracting
Changing One Equation
Consider
3x + 2y = 4
x – 4y = 6
add
4x – 2y = 10
3x + 2y = 4
x – 4y = 6
sub 
2x + 6y = -2
We have not managed to eliminate x or y!!
To get round this we double all of the top equation !
over
Ex5
Ex6
3x + 2y = 4 ( x2 )
x – 4y = 6
now get
5x + 3y = -4
2x + y = -2
now get
6x + 4y = 8
x – 4y = 6
add
( x3 )
7x = 14
x=2
use top equation
6 + 2y = 4
2y = -2
y = -1
also
x – 4y
= 2 – (-4) = 6
5x + 3y = -4
6x + 3y = -6
sub 
x = -2
use top equation
-10 + 3y = -4
3y = 6
y=2
also
2x + y
= -4 + 2 = -2
Ex7
Ex8
a + 5b = -1
3a – b = 13 ( x5 )
now get
4c + 3d = 8
2c + d = 6
now get
a + 5b = -1
15a – 5b = 65
add
( x2 )
16a = 64
a=4
use top equation
4 + 5b = -1
5b = -5
b = -1
also
3a – b
= 12 – (-1) = 13
4c + 3d = 8
4c + 2d = 12
sub 
d = -4
use 2nd equation
2c - 4 = 6
2c = 10
c=5
also
4c + 3d
= 20 +(-12) = 8
Changing Both Equations
Consider
3x - 2y = 4
2x + 5y = 6
To match x terms need to mult by 1.5.
To match y terms need to mult by 2.5.
AVOID
DECIMALS
We do this as follows …….
over
Ex 9
Ex 10
3x - 2y = 4
2x + 5y = 9
(x5)
(x2)
now get
5a + 3b = 1
4a + 5b = 6
now get
15x - 10y = 20
4x + 10y = 18
add
19x = 38
x=2
Use 2nd equation
4 + 5y = 9
5y = 5
y=1
also
3x – 2y
=6–2=4
20a + 12b = 4
20a + 25b = 30
sub 
13b = 26
b=2
Use top equation
5a + 6 = 1
5a = -5
a = -1
also
4a + 3b
= -4 + 10 = 6
(x4)
(x5)
Ex 11
Ex 12
5m - 4n = 4 (x3)
2m + 3n = 8.5 (x4)
now get
5p + 3q = 0
4p + 5q = -2.6
now get
15m - 12n = 12
8m + 12n = 34
add
23m = 46
m=2
Use 2nd equation
4 + 3n = 8.5
3n = 4.5
n = 1.5
also
5m – 4n
= 10 – 6 = 4
20p + 12q = 0
20p + 25q = -13
sub 
13q = -13
q = -1
Use top equation
5p - 3 = 0
5p = 3
p = 0.6
also
4p + 5q
= 2.4 - 5 = -2.6
(x4)
(x5)
Using Pairs of Equations
Ex13
Suppose 3cones plus 2 lollies cost £1.90 while 5 cones and a
lolly cost £2.35. How much is each?
*************
Let a cone cost c, a lolly cost l and prices be in pence!
Then
3c + 2l = 190
5c + l = 235 ( x2)
now get
3c + 2l = 190
10c + 2l = 470
sub 
7c = 280
c = 40
Use 2nd equation
200 + l = 235
l = 35
Conclusion
Cones are 40p & lollies are 35p