linear equations

Question

A system of equations is given below.
2x + 7y = 1
-3x – 4y = 5

Create an equivalent system of equations by replacing the first equation by multiplying the first equation by an integer other than 1, and adding it to the second equation.
Use any method to solve the equivalent system of equations (the new first equation with the original second equation).
Prove that the solution for the equivalent system is the same as the solution for the original system of equations.

Answer 1

Easiest way is to multiply by 3:

3(2x+7y) = 3(1)

6x + 21y = 3

 

Now you can use substitution, or elimination:

 6x + 21y = 3

-3x -   4y = 5

 

Multiply bottom equation by 2

  6x + 21y =  3

-6x  -   8y  =10

0      +13y = 13

 

y = 1

Substitute y = 1 into the new equation 1:

6x + 21(1) = 3

6x = -18

x = -3

 

Substitute y = 1 into original equation 2:

-3x - 4(1) = 5

-3x = 9

x = -3

Solution works for both, now check original equation 1:

2x + 7(1) = 1

2x = -6

x = -3

Last step, substitute both values into both versions of equation 1 to check for truth:

2(-3) + 7(1) = 1

-6 + 7 = 1

1 = 1 ; check

6(-3) + 21(1) = 3

-18 + 21 = 3 

3 = 3 ; check