11
5x + 2y = 22
4x + y = 17
In the system of equations above, what is the value
of x + y ?
A) 5
B) 4
C) 3
D) 2
To solve by elimination, I would (1) multiply the second equation by -2, (2) add the equations, (3) solve for y, (4) substitute back in to find x, and (5) add x and y.
(1)
5x + 2y = 22
-2[4x + y = 17]
5x + 2y = 22
-8x + -2y = -34
(2)
[5x + 2y = 22]
+[-8x + -2y = -34]
-3x + 0 = -12
(3)
x = 4
(4)
5(4) + 2y = 22
2y=2
y=1
(5) x + y = 5
This is not terribly difficult, but there is a much quicker way to solve the problem. Simply subtract the second equation from the first.
[5x + 2y = 22]
-[4x + y = 17]
x + y = 5
It is rather interesting how often the SAT pulls this little trick, e.g., October 2018.
191
2x + 3y = 1200
3x + 2y = 1300
Based on the system of equations above, what is the value of 5x + 5y?
Once again, it is possible to solve for x and y, but adding the equations yields
[2x + 3y = 1200]
+[3x + 2y = 1300]
5x + 5y = 2500.
This trick doesn't always work. Sometimes the question asks for the value of either x or y by itself, making it necessary to solve for the individual variable. Nevertheless, any time a question asks for the sum or difference of the variables or multiples of the variables (x + y, x - y, 2x + 3y) it's worth looking to see what happens when the equations are simply added or subtracted.
