Simultaneous Equations

1
Simultaneous Equations

• OCR Module 8

2
What are they?

• Simply 2 equations
• With 2 unknowns
• Usually x and y
• To SOLVE the equations means we find values of x
  and y that
• Satisfy BOTH equations work in
• At same time simultaneously

3
Elimination Method
We have the same number of ys in each
2x y 1
A

If we ADD the equations, the ys disappear
3x y 9
B
10
5x
Divide both sides by 5
x
2
Substitute x 2 in equation A
2 x 2 y 1
4 y 1
Answer x 2, y 3
y 3
4
Elimination Method
We have the same number of ys in each
5x y 17
A
-
3x y 11
B
If we SUBTRACT the equations, the ys disappear
6
2x
Divide both sides by 2
x
3
Substitute x 3 in equation A
5 x 3 y 17
15 y 17
Answer x 3, y 2
y 2
5
Elimination Method
We have the same number of xs in each
2x 3y 9
A
-
2x y 7
B
If we SUBTRACT the equations, the xs disappear
2
2y
Divide both sides by 2
1
y
Substitute y 1 in equation A
2x 3 9
2x 6
Answer x 3, y 1
x 3
6
Elimination Method
We have the same number of ys in each
4x - 3y 14
A

2x 3y 16
B
If we ADD the equations, the ys disappear
6x
30
Divide both sides by 6
x
5
Substitute x 5 in equation A
20 3y 14
3y 6
Answer x 5, y 2
y 2
7
Basic steps

• Look at equations
• Same number of xs or ys?
• If the sign is different, ADD the equations
  otherwise subtract tem
• Then have ONE equation
• Solve this
• Substitute answer to get the other
• CHECK by substitution of BOTH answers

8
What if NOT same number of xs or ys?
A
3x y 10
If we multiply A by 2 we get 2y in each
B
5x 2y 17
A
-
6x 2y 20
5x 2y 17
B
x
3
In B
5 x 3 2y 17
Answer x 3, y 1
15 2y 17
y 1
9
What if NOT same number of xs or ys?
A
4x - 2y 8
If we multiply A by 3 we get 6y in each
B
3x 6y 21
A
12x - 6y 24

3x 6y 21
B
15x
45
x 3
In B
3 x 3 6y 21
Answer x 3, y 2
6y 12
y 2
10
if multiplying 1 equation doesnt help?
A
3x 7y 26
Multiply A by 5 B by 3, we get 15x in each
B
5x 2y 24
15x 35y 130
A
-
15x 6y 72
B
Could multiply A by 2 B by 7 to get 14y in each
58
29y
y 2
In B
5x 2 x 2 24
Answer x 4, y 2
5x 20
x 4
11
if multiplying 1 equation doesnt help?
A
3x - 2y 7
Multiply A by 3 B by 2, we get 6y -6y
B
5x 3y 37
9x 6y 21
A

10x 6y 74
B
Could multiply A by 5 B by 3 to get 15x in each
95
19x
x 5
In B
5 x 5 3y 37
Answer x 5, y 4
3y 12
y 4