Algebra Mastery
And then solve for x:
9x = 36
→
x=4
Then you can solve for y by plugging the value of x into either equation:
6x – 2y = 20
6(4) – 2y = 20 →
→
– 2y = –4 →
y=2
Sometimes you must multiply one of the equations by a negative number to
eliminate a set of variables:
24 – 2y = 20
3x + 2y = 16
6x + 2y = 20
When these equations are written in column form, it is clear that you cannot add
them to eliminate y:
Confidence Quotation
“Impossible is a word
only to be found in the
dictionary of fools.”
—Napoleon Bonaparte,
French general and
politician
3x + 2y = 16
+ 6x + 2y = 20
9x + 4y = 36
Multiply the second equation by –1:
–1(6x + 2y = 20) →
–6x – 2y = –20
Now add them together:
3x + 2y = 16
+ –6x – 2y = –20
–3x
= –4
4
As you can see, x = . When plugged into either equation, you will find that
3
y = 6.
This solution method is very important on the SAT, as we will discuss in the next
section. Therefore, you must also be able to add or subtract equations when the
variables are not equal:
3x – 5y = –41
Write the equations in a column fashion, where all of the variables line up:
8x + y = 34
8x + y = 34
3x – 5y = –41
If we add them together now, no variables are eliminated.
8x + y = 34
+ 3x – 5y = –41
11x – 4y = –7
Chapter Six
211
Algebra Mastery
To eliminate one set of variables, they must be equal in the two equations. To do
this, multiple one or both equations by any value that creates equal variables. You
can multiply the top equation by 3 and the bottom by –8:
3(8x + y = 34) → 24x + 3y = 102
–8(3x – 5y = –41) → –24x + 40y = 328
Or you can simply multiply the first equation by 5:
5(8x + y = 34) →
40x + 5y = 170
3x – 5y = –41
Both of these solutions will result in x = 3 and y = 10.
3. Some systems of equation questions contain three or more equations. You can
also solve these systems by using substitution or addition.
4. A system of equations has no solution when all variables can be eliminated:
A system of equations
has no solution when all
variables are eliminated.
Multiply the second equation by 2:
10x + 4y = 16
–5x – 2y = 20
2(–5x – 2y = 20) →
–10x – 4y = 40
Now attempt to add them together:
10x + 4y = 16
+ –10x – 4y = 40
= 56
212
Because both variables are eliminated, this system of equations does not have a
solution.
The PowerScore SAT Math Bible