Ok - I promised some solutions to this problem. Here is the first one:
2x + y = 7 – 2y
5y – x = 5 – 4x
If (x,y) is a solution to the above system of equations, what is the value of (y+1)/x ?
(A) -11
(B) -1/2
© 2
(D) 20
Solution using the elimination method: We begin by making sure that the two equations are “lined up” properly. We do this by adding 2y to each side of the first equation, and adding 4x to each side of the second equation.
2x + 3y = 7
3x + 5y = 5
We will now multiply each side of the first equation by 5, and each side of the second equation by -3.
5(2x + 3y) = (7)(5)
-3(3x + 5y) = (5)(-3)
Do not forget to distribute correctly on the left. Add the two equations.
10x + 15y = 35
- 9x – 15y = -15
x = 20
Using the first equation in the solution to find y, we have
2⋅20 + 3y = 7
40 + 3y = 7
3y = 7 – 40 = -33
y = (-33)/3 = -11
So (y+1)/x = (-11+1)/20 = -10/20 = -1/2, choice (B).
**Remarks:/b We chose to use 5 and -3 because multiplying by these numbers makes the y column “match up” so that when we add the two equations in the next step the y term vanishes. We could have also used -5 and 3.
(2) If we wanted to find y first instead of x we would multiply the two equations by 3 and -2 (or -3 and 2). In general, if you are only looking for one variable, try to eliminate the one you are not looking for. In this case we need to find both so it doesn’t matter which we find first.
(3) We chose to multiply by a negative number so that we could add the equations instead of subtracting them. We could have also multiplied the first equation by 5, the second by 3, and subtracted the two equations, but a computational error is more likely to occur this way.