Sometimes, a question will have a lone equation containing twovariables, and using the methods we’ve discussed up until now will notbe 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, then the values of both variables can be found.
These multiple equations containing thesame variables are called systems of equations. For the Math IC, thereare essentially two types of systems of equations that you will need tobe able to solve. The first, easier type involves substitution, and thesecond involves manipulating equations simultaneously.
Substitution
Simply put, substitution is when the valueof one variable is found and then substituted into the other equationto solve for the other variable. It can be as easy as this example:
If
x – 4 =
y – 3 and 2
y = 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 substitute that value for
y in the first equation. If 2
y = 6, then
y = 3, and then
x =
y – 3 + 4 = 3 – 3 + 4 = 4.
Here is a slightly more complicated example.
Suppose 3
x =
y + 5 and 2
y – 2= 12
k. 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 = 6
k + 1 into the equation 3
x =
y + 5.
Simultaneous Equations
Simultaneous equations refer to equations thatcan be added or subtracted from each other in order to find a solution.Consider the following example:
Suppose 2
x + 3
y = 5 and –1
x – 3
y = –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:
6
x + 2
y = 11 and 5
x + y = 10. What is
x + y?
By subtracting the second equation from the first:
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 answer is given.
Give this last example a try:
2
x + 3
y = –6 and –4
x + 16
y = 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. 4
x is simply twice 2
x, 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 wehave in this last example, you can solve for the second variable usingeither of the original equations. If the last question had asked you tocalculate 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, you can multiply them together.
Simultaneous equations on the Math IC willall be this simple. They will have solutions that can be found easilyby adding or subtracting the equations given. Only as a last resortshould you solve for one variable in terms of the other and then plugthat value into the other equation to solve for the second variable.
