Simultaneous Equations

KS4 Mathematics
A5 Simultaneous
equations
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Contents
A5 Simultaneous equations
· AA5.1 Solving simultaneous equations graphically
· AA5.2 The elimination method
· AA5.3 The substitution method
· AA5.4 Simultaneous linear and quadratic equations
· AA5.5 Problems leading to simultaneous equations
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Simultaneous equations
Equations in two unknowns have an infinite number of solution
pairs. For example,
x+y=3
is true when x=1 and y=2
x=3 and y=0
x = ­2 and y=5 and so on ...
y
We can represent the set
3
of solutions on a graph: x+y=3
0 x
3
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Simultaneous equations
Another equation in two unknowns will also have an infinite
number of solution pairs. For example,
y­x=1
is true when x=1 and y=2
x=3 and y=4
x = ­2 and y = ­1 and so on ...
y
This set of solutions can
3
also be represented in a
y­x=1
graph:
0 3 x
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Simultaneous equations
There is one pair of values that solves both these equations:
x+y=3
y­x=1
We can find the pair of values by drawing the lines x + y = 3
and y ­ x = 1 on the same graph.
y y­x=1 The point where the two lines
intersect gives us the solution to both
3
equations.
This is the point (1, 2).
0 3 x
x+y=3 At this point x = 1 and y = 2.
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Simultaneous equations
x+y=3
y­x=1
are called a pair of simultaneous equations.
The values of x and y that solve both equations are x = 1 and
y = 2, as we found by drawing graphs.
We can check this solution by substituting these values into
the original equations.
1+2=3
2­1=1
Both the equations are satisfied and so the solution is correct.
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