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 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 step-by-step 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.
NOTE:With this method you need to take particular care
with your algebra. Should your solutions be ‘strange’ fractions such as 9/13
the chances are you’ve made a slip − check your algebra.
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.
Note: Only draw a graph if the question asks you to, it is
usually quicker to work out the point two simultaneous equtions cross
algebraically.
