Question

Part 1. How many solutions will the following system of linear equations have?

–2x + 4y = –7
y = -1/2x + 5

A.) 0
B.) 1
C.) 2
D.) infinite

Part 2. For which system of equations is (5, 3) the solution?

A.) 3x – 2y = 9
3x + 2y = 14

B.) x – y = –2
4x – 3y = 11

C.) –2x – y = –13
x + 2y = –11

D.) 2x – y = 7
2x + 7y = 31

Part 3. Solve the system of equations.
8x – 3y = 1
–2x + 3y = 11

A.) (–1, –3)
B.) (–1, 3)
C.) (2, 5)
D.) (5, 2)

Answer 1

–2x + 4y = –7 . . . . . (1)
y = -1/2x + 5 . . . . . . (2)
Putting (2) into (1), we have:
-2x + 4(-1/2x + 5) = -7
-2x - 2x + 5 = -7
-4x = -7 - 5 = -12
x = -12/-4 = 3
y = -1/2(3) + 5 = -3/2 + 5 = 7/2
Hence, the system has one solution.

Answer 2

In part 1, we are given with two equations and according to the degrees of freedom rule, two equations should solve two unknowns. The number of solutions is a pair of two variables. Hence the answer is B
In part 2, this is just a matter of substitution or in a long way solve each expressions for their x and y. The answer should be D.
In part 3, we can use a calculator to solve the system of equations. Using addition to eliminate y, too, the answer is C.