Simultaneous Equations
Key Points
 Simultaneous equations are used when there are two equations with two of
the same unknowns in each.
 Solve by subtraction or substitution.
For Core 1 you need to know how to solve equations simultaneously when one of
the equations is a quadratic.
1. Rearrange the linear equation to make or (or whichever letters are used) the
subject.
2. Substitute into the quadratic equation.
3. Solve by factorisation, completing the square, or the quadratic formula.
4. Feed the values back into the linear equation to find the remaining unknown!
I.
Solving linear equations by subtraction
1. Multiply or divide equation (1) to make the
coefficient the same as equation (2).
2. Eliminate by subtraction equation (1) from equation (2), or vice versa.
3. Solve for .
4. Feed the value for
back into equation (1) and solve for .
For example:
Identify the values for and
if (1)
and (2)
.
Multiply equation (1) by 2 so that the term is the same as equation (2) .
Subtract equation (2) from equation (1).
Solve for .
Feed into equation (1) and solve for .
II.
Solving linear equations by substitution
1. Rearrange equation (1) to make
the subject
2. Eliminate by feeding it into equation (2).
3. Solve for .
4. Feed the value for
back into equation (1) and solve for .
Remember, when you are choosing which equation you are going to rearrange and which
unknown (e.g.
or ) will be the subject, you should try to pick those that will be the easiest to
substitute. For instance, the unknown with the least amount of terms or the simplest terms in
the 2nd equation.
For example:
Identify the values for and
Rearrange equation 1.
if (1)
and (2)
.
Feed into equation 2 and solve for
.
Feed into equation 1 and solve for .
III.
Solving linear/quadradic equations by substitution
You need to know how to solve equations simultaneously when one of the equations is a
quadratic.
1. Rearrange the linear equation to make or (or whichever letters are used) the subject.
2. Substitute this into the quadratic equation.
3. Solve by factorisation, completing the square, or the quadratic formula.
4. Feed the values back into the linear equation to find the other unknown.
5. To check your answers by feeding the values back into the original equations.
Remember, when you are choosing which unknown (e.g. or ) will be the subject you should try
to pick the one that will be the easiest to substitute. For instance, the unknown with the least
amount of terms or the simplest terms in the 2nd equation.
For example:
Identify the values for and
Rearrange equation (1).
Feed into equation (2).
Factorise.
Therefore
or
if (1)
and (2)
.
Substitute
and
into equation (1)
When
When
For example:
Identify the values for and
Rearrange equation (1).
if (1)
and (2)
.
Equation (1) was rearrange to make
the subject for simplicity. There are multiple
in equation (2) which just makes things more complicated.
Feed into equation (2).
Factorise.
Therefore,
or
Substitute
and
into equation (1)
terms
When
When
We could have also used the quadratic formula to solve for . Choose whichever you find
easiest (unless the question tells you to use a certain method)
Therefore,
or,
Substitute
and
into equation (1)
When
When
Notice how both factorisation and formula methods give the same answers. This is another
good way to check your results, if there is time!
MCQ
1. Solve simultaneously:
a)
or
b)
or
c)
or
d)
or
2. Solve simultaneously:
a)
or
b)
or
c)
or
d)
or
3. Solve simultaneously:
a)
b)
c)
and
and
and
d)
4. Find the values of and when
a)
or
b)
or
c)
d)
or
or
5. Find the values of and when
a)
b)
or
or
d)
or
6. Find the values of and when
or
b)
c)
and
or
c)
a)
and
or
or
and
d)
or
7. Find the values of and when
a)
and
or
b)
or
c)
or
d)
or
8. Find the values of and when
a)
or
b)
or
c)
or
d)
or
9. Find where
and
a)
or
b)
or
c)
and
or
intersect.
d)
or
10. Solve simultaneously:
a)
and
or
b)
or
c)
or
d)
or
Answers:
1) d
2) b
3)a
4)a
5)b
6)c
7)a
8)c
9)c
10)d