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`