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Simultaneous equations
Simultaneous means at the same time
To solve two equations simultaneously is to find a point which is on both graphs.
This is the point of intersection of the two graph lines
y
Example
y = 2x + 4
Solve these simultaneous equations
y=?
y = 2x + 4
y = -0.5x + 10
or
y = -0.5x + 10
Find the point of intersection of the lines
y = 2x + 4 and y = -0.5x + 10
x
x=?
How?
Substitution method: put one equation “into” the other equation
Elimination method: add or subtract the two equations to eliminate one variable
Elimination method
Linear equations can be written in different ways:
eg
y = 2x – 3
y – 2x = -3
y – 2x + 3 = 0
If both equations are written in the same form the elimination method is best
2x – y = 8
x +y=4
Equations
Eliminate y
2x – y = 8
Add
x +y=4
3x
= 12
x
= 4 One solution
x +y=4
4+y=4
y=0
Substitute
Other solution
2x – y = 8
Check!
2x4 – 0 = 8?
8 = 8 Yes
Steps:
1) Line up the terms: x, y, constant, and =
2) Select one variable to eliminate
whatever variable has the same number
in each equation – scaling may be needed!
3) Do we add or subtract the equations?
Add if signs different, Subtract if same sign
4) Solve to find one variable
5) Substitute this variable back into one equation
to find the other variable
Solution: x = 4, y = 0 or as a coordinate (4,0)
6) Substitute both solutions into the other original
equation to check answer is correct
Elimination method A
Eg 1)
3x + y = 13
2x – y = 7
Theta
p39 Ex 5.2
p40 Ex 5.3
Eg 2)
12x – y = 21
3x + 7y = 27
Eg 3)
5x + 2y = -2
4x + 3y = 4
Substitution method
If one equation is written y = or x = the substitution method is best
y = 4x + 5
y = -2x +6
2y + 3x – 18 = 0
Theta
y + 7x – 27 = 0
y = 4x – 12
y = 3x
p40 Ex 5.4
Equations
Steps:
y = 4x + 5
1) Substitute the first equation in place of y in 2nd
y + 7x – 27 = 0
2) Select one variable to eliminate
4x + 5 + 7x – 27 = 0
11x – 22 = 0
11x = 22
x=2
Simplify and solve (brackets may be needed)
y = 4x + 5
y=4x2+5
y=8+5
y = 13
3) Substitute this variable back into one equation
to get the other variable
y + 7x – 27 = 0
13 + 7x2 – 27 = 0?
13 + 14 – 27 = 0
27-27 = 0 Yes!
4) Substitute both solutions into the other original
equation to check answer is correct
One solution
Other solution
Solution: x = 2 and y = 13 or coordinate (2, 13)
Applications
The entrance fees to the zoo are $12 for adults, and $8 for children
One day 16 people go to the zoo, paying a total of $164
Applications
p41 Ex 5.5
How many adults and children went to the zoo?
 Define variables
x = adults y = children
 Form equations
Total people x + y = 16
Total cost for adults = 12x and 8y for children
Total combined cost = 12x + 8y = $164
Equations
x + y = 16
12x + 8y = 164
Solution
9 adults
7 children
 Solve
8x + 8y = 128
12x + 8y = 164
-4x = -36
One solution
x=9
Substitute
x + y = 16
9 + y = 16
so y = 7
Check
12x + 8y = 164
12x9 + 8x7 = 164?
108 + 56 = 164 Yes!
So solution is correct