Systems of Equations
Sometimes a question will have a lone equation containing
two variables, and using the methods we’ve discussed thus far will
not be enough to solve for the variables. Additional information
is needed, and it must come in the form of another equation.
Say, for example, that a single equation uses the two
variables x and y. Try as you
might, you won’t be able to solve for x or y.
But given another equation with the same two variables x and y,
the values of both variables can be found.
These multiple equations containing the same variables
are called systems of equations. For the Math IIC, there are essentially
two types of systems of equations that you will need to be able
to solve. The first, easier type involves substitution, and the
second type involves manipulating equations simultaneously.
Simply put, the substitution method involves finding the
value of one variable in one equation and then substituting that
value into the other equation to solve for the other variable. Here’s
a straightforward example:
||If x –
4 = y – 3 and 2y = 6, what is x?
In this case, we have two equations. The first equation
contains x and y. The second contains
only y. To solve for x, you must
solve for y in the second equation and then substitute that
value for y in the first equation, eliminating
the second variable from that equation. If 2y =
6, then y = 3, and substituting that into the first
Here is a slightly more complicated example:
3x = y + 5 and 2y –
2= 12k. Solve for x in terms of k.
Again, you cannot solve for x in terms
of k using just the first equation. Instead, you
must solve for y in terms of k in
the second equation and then substitute that value in the first equation
to solve for x.
Then substitute y = 6k +
1 into the equation 3x = y + 5.
Simultaneous equations refer to equations that can be
added or subtracted from each other in order to find a solution.
Consider the following example:
2x + 3y = 5 and –1x –
3y = –7. What is x?
In this particular problem, you can find the value of x by
adding the two equations together:
Here is another example:
2y = 11 and 5x + y = 10. What
is x + y?
By subtracting the second equation from the first,
we find that x + y = 1.
Some test-takers might have seen this problem and been
tempted to immediately start trying to solve for x and y individually.
The better test-taker notices that by subtracting the second equation
from the first, the solution is achieved.
Give this last example a try:
3y = –6 and –4x + 16y =
13. What is the value of y?
The question asks you to solve for y,
which means that you should find a way to eliminate one of the variables
by adding or subtracting the two equations. 4x is
simply twice 2x, so by multiplying the first equation
by 2, you can then add the equations together to find y.
||2 (2x + 3y =
–6) = 4x + 6y = –12
Now add the equations and solve for y.
When you solve for one variable, like we have in this
last example, you can solve for the second variable using either
of the original equations. If the last question had asked you to calculate
the value of xy, for example, you could solve for y,
as above, and then solve for x by substitution
into either equation. Once you know the independent values of x and y, multiply
The simultaneous equations on the Math IIC will all be
this simple. They will have solutions that can be found reasonably
easily by adding or subtracting the equations given. Only as a last
resort should you solve for one variable in terms of the other and
then plug that value into the other equation to solve for the other