Simultaneous Equations
Simultaneous equations and linear equations, after studying this section, you will be able to:
 solve simultaneous linear equations by substitution
 solve simultaneous linear equations by substitution
 solve simultaneous linear equations by elimination
 solve simultaneous linear equations using straight line graphs
If an equation has two unknowns, such as 2y +
x = 20, it cannot have unique solutions. Two unknowns require two
equations which are solved at the sametime (simultaneously) − but even
then two equations involving two unknowns do not always give unique
solutions.
The video below works through examples of
simultaneous equations. The stepbystep example shows how to group like
terms and then add or subtract to remove one of the unknowns, to leave
one unknown to be solved.
This method is called solution by substitution.
It involves what it says − substitution − using
one of the equations to get an expression of the form ‘y = …’ or ‘x = …’
and substituting this into the other equation. This gives an equation
with just one unknown, which can be solved in the usual way. This value
is then substituted in one or other of the original equations, giving an
equation with one unknown.
Example
Solve the two simultaneous equations:
2y + x = 8 [1]
1 + y = 2x [2]
from [2] y = 2x 1 ← subtract 1 from each side
Substituting this value for y into [1] gives:
2(2x – 1) + x = 8
4x – 2 + x = 8 ← expand the brackets
5x – 2 = 8 ←tidy up
5x = 10 ←Add 2 to each side
x = 2 ←By dividing both sides by 5 the value of x is found.
Substitute the value of x into y = 2x – 1 gives
y = 4  1 = 3
So x = 2 and y = 3
NOTE:
 It is a good idea to label each equation. It helps you explain what you are doing − and may gain you method marks.
 This value of x can be substituted into equation [1] or [2], or into the expression for y: y = 2x − 1.
 Choose the one that is easiest!
 As a check, substitute the values back into each of the two starting equations.
The second method is called solution by elimination.
NOTE:The method is not quite as hard as it first seems, but it helps if you know why it works.
It works because of two properties of equations: Multiplying (or dividing) the expression on each side by the same number does not alter the equation.

Adding two equations produces another valid equation:
e.g. 2x = x + 10 (x = 10) and x − 3 = 7 (x also = 10).
Adding the equations gives 2x + x − 3 = x + 10 + 7 (x also = 10).
The object is to manipulate the two equations
so that, when combined, either the x term or the y term is eliminated
(hence the name) − the resulting equation with just one unknown can then
be solved:
Here
we will manipulate one of the equations so that when it is combined
with the other equation either the x or y terms will drop out. In this
example the x term will drop out giving a solution for y. This is then
substituted into one of the otiginal equations.
Label your equations so you know which one your are working with at each stage.
Equation [1] is 2y + x = 8
Equation [2] is 1 + y = 2x
Rearrange one equation so it is similar to the other.
[2] y – 2x = 1
also 2 x [1] gives 4y + 2x = 16 which we call [3]
[2] y – 2x = 1
[3] 4y +2x = 16
[2] + [3] gives 5y = 15
so y = 3
substituting y = 3 into [1] gives 1 + (3) = 2x
so 2x = 4, giving x = 2 and y = 3
Solving simultaneous linear equations using straight line graphs
The 2 lines represent the equations '4x  6y = 4' and '2x + 2y = 6'.
There is only one point the two equations cross.
Because the graphs of 4x  6y = 12 and 2x + 2y = 6 are straight lines, they are called linear equations.
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