Question

1. What is the value of the y variable in the solution to the following system of equations?
3x − 6y = 3
7x − 5y = −11

A. −2
B. −3
C. x can be any number as there are infinitely many solutions to the system.
D. There is no x value as there is no solution to the system

2. What is the value of the x variable in the solution to the following system of equations?
3x + y = 6
6x + 2y = 8

A. 0
B. 4
C. x can be any number as there are infinitely many solutions to the system.
D. There is no x value as there is no solution to the system.

3. What is the value of the x variable in the solution to the following system of equations?
4x + 2y = 6
x – y = 3

A. −1
B. 5
C. −2
D. 2

4. The three Math Idol judges have been eliminating contestants all day! The number of one-step equations and two-step equations who have been eliminated today is equal to 1120. If three times the number of one-step equations minus twice the number of two-step equations is equal to 1300, how many two-step equations auditioned today?

A. 1300
B. 1120
C. 708
D. 412

Answer 1

Part 1)

we have

$3x-6y=3$ ------> equation A

$7x-5y=-11$ ------> equation B

Multiply by $-7$ the equation A

$-7(3x-6y)=-7*3$

$-21x+42y=-21$ ------> equation C

Multiply by $3$ the equation B

$3*(7x-5y)=-11*3$

$21x-15y=-33$ -------> equation D

Adds equation C and equation D

$-21x+42y=-21 \\21x-15y=-33\\---------- \\ 42y-15y=-21-33\\27y=-54 \\ y=-2$

therefore

the answer Part 1) is the option A

$y=-2$

Part 2)

we have

$3x+y=6$

$y=-3x+6$ ------> equation A

$6x+2y=8$

Simplify Divide by $2$ both sides

$3x+y=4$

$y=-3x+4$ ------> equation B

the lines A and B are parallel lines, because the slope m is equal

so

The system has no solution

therefore

the answer Part 2) is the option D

There is no x value as there is no solution to the system.

Part 3)

we have

$4x+2y=6$ ------> equation A

$x-y=3$

$y=x-3$  ------> equation B

substitute equation B in equation A

$4x+2[x-3]=6$

$6x-6=6$

$6x=12$

$x=2$

therefore

the answer part 3) is the option D

$x=2$

Part 4)

Let

x---------> The number of one-step equations

y---------> The number of two-step equations

we know that

$x+y=1,120$

$x=1,120-y$ -------> equation A

$3x-2y=1,300$ ------> equation B

substitute equation A in equation B

$3[1,120-y]-2y=1,300$

$3,360-5y=1,300$

$5y=3,360-1,300$

$5y=2,060$

$y=412$

therefore

the answer part 4) is  the option D

$y=412$