Simultaneous Equations
The Elimination method
If a pair of simultaneous equations contain an x term which are exactly the same, we can solve
them by elimination.
When the signs of the equal terms are
DIFFERENT, we ADD together the two
equations to eliminate x.
The same method can be used if the y – terms
are equal. (Or for that matter any letter term)
1.
ADDING THE TWO EQUATIONS
(i)
Solve
3x + y = 15
(ii)
2x – y = 5
The y-terms in both equations are the same, but the
signs are different. So we add the two equations to
eliminate y.
3x + 2x + y – y = 15 + 5
Then put this value in
equation (i)
5x = 20
x=4
3x + y = 15
(3 x 4) + y = 15
12 + y
y
= 15
= 3
(i)
2.
Solve
3x + 4y = 11
(i)
-3x + 2y = 1
(ii)
The x-terms in both equations are the same, but the
3x – 3xSo+we
4y add
+ 2y
=two
11 +
1
signs are different.
the
equations
to
eliminate x.
6y = 12
Then put this value in
equation (i)
y=2
3x + 4y = 11
3x + (4 x 2) = 11
3x + 8
= 11
3x
= 3
x
= 1
(i)
SUBTRACTING THE TWO EQUATIONS
When the signs of the equal terms are
the SAME, we SUBTRACT together the
two equations to eliminate x.
Once again, the same method can be used
if the y – terms are equal.
(Or for that matter any letter term)
SUBTRACTING THE TWO EQUATIONS
(i)
1. Solve
2x + y = 7
(ii)
x + y= 4
The y-terms in both equations are the same and the
signs are also the SAME. So we subtract the two
equations to eliminate y.
2x – x + y – y = 7 – 4
Then put this value in
x=3
equation (i)
2x + y = 7
(2 x 3) + y = 7
6 +y
y
= 7
= 1
(i)
2.
Solve
3x + 4y = 11
(i)
3x + 2y = 7
(ii)
The x-terms in both equations are the same and the
3xthe
– 3x
+ 4ySo– we
2y subtract
= 11 – 7the two
signs are also
SAME.
equations to eliminate x.
2y = 4
y=2
3x + 4y = 11
3x + (4 x 2) = 11
3x + 8
= 11
3x
= 3
x
= 1
(i)
Simultaneous Equations
If the equal terms in both equations are the same,
but the signs are different.
We add the two equations to eliminate one unknown.
If the equal terms in both equations are the same
and the signs are also the SAME.
We subtract the two equations to eliminate one
unknown.
Simultaneous Equations
If neither the x-term or y-term are the same
the elimination method will not work.
However, it is possible to form a pair of
equations by multiplying one or both equations
by a number.
NB. The multiplication must be applied to all
parts of the equation.
1.
Solve
2x + 3y = 13
x + 2y = 8
(i)
(ii)
Neither x or y terms are the same. But if I multiply equation (ii)
by 2 an equal term can be created with equation (i).
2x + 4y = 16
2x + 3y = 13
The x-terms in both equations are the same and the signs are
also the SAME. So we subtract the two equations to eliminate x.
2x – 2x + 4y – 3y = 16 – 13
y=3
2x + 3y = 13
2x + (3 x 3) = 13
2x + 9 = 13
2x = 4
x = 2
(i)