Question

A system of equations is shown below: x + 3y = 5 (equation 1) 7x − 8y = 6 (equation 2) A student wants to prove that if equation 2 is kept unchanged and equation 1 is replaced with the sum of equation 1 and a multiple of equation 2, the solution to the new system of equations is the same as the solution to the original system of equations. If equation 2 is multiplied by 1, which of the following steps should the student use for the proof? Show that the solution to the system of equations 3x + y = 5 and 8x −7y = 6 is the same as the solution to the given system of equations Show that the solution to the system of equations 8x − 5y = 11 and 7x − 8y = 6 is the same as the solution to the given system of equations Show that the solution to the system of equations 15x + 13y = 17 and 7x − 8y = 6 is the same as the solution to the given system of equations Show that the solution to the system of equations −13x + 15y = 17 and 7x − 8y = 6 is the same as the solution to the given system of equations

Answer 1

8x − 5y = 11 and 7x − 8y = 6

Explanation:

I took the test and got it right.

Answer 2

First Let we solve the Original system of equations:

equation (1):
`x+3y=5`

equation (2):
`7x-8y=6`

Multiplying equation (1) by 7, we get

`7x+21y=35 -->(3)`

`7x-8y=6 --> (2)`

Subtracting,

`29y=29` implies
`y=1`

Then
`x=5-3(1)=2`

Thus the solution of the original equation is
`x=2, y=1.`

Now Let we form the new equation:

Equation 2 is kept unchanged:

Equation (2):
`7x-8y=6`

Equation 1 is replaced with the sum of equation 1 and a multiple of equation 2:

Equation (1):
`8x-5y=11`

Now solve this two equations:
`image`

Multiply (1) by 7 and (2) by 8,

`56x-35y=77`

`56x-64y=48`

Subtracting,
`29y=29` implies
`y=1`

Then x=2.

so the solution for the new system of equation is x=2, y=1.

This Show that the solution to the system of equations 8x − 5y = 11 and 7x − 8y = 6 is the same as the solution to the given system of equations