How do you solve `2 x + 3 y = -7` and `2 x - 7 y = 7` using substitution?

Arrange the first equation:

`2x = -3y - 7`

Now put this into the second equation:

`-3y - 7 - 7y = 7`

`-10y - 7 = 7`

`-10 y = 14`

`y = -14/10` or

`y = -7/5`

Put this into the first original equation

`2x + 3times(-7/5) = -7`

`2x -21/5 = -7`

`2x = 21/5 - 7`

`2x = (21-35)/5`

`2x = -14/5`

`x = -14/10`

`x = -7/5`

Your solution is `x=-7/5` and `y=-7/5`

Step 1) We can solve both equations for `2x` which is a common term:

Equation 1)

`2x + 3y = -7`

`2x + 3y - color(red)(3y) = -7 - color(red)(3y)`

`2x + 0 = -7 - 3y`

`2x = -7 - 3y`

Equation 2)

`2x - 7y = 7`

`2x - 7y + color(red)(7y) = 7 + color(red)(7y)`

`2x - 0 = 7 + 7y`

`2x = 7 + 7y`

Step 2) Equate the right sides of each equation and solve for `y`:

`-7 - 3y = 7 + 7y`

`-color(blue)(7) - 7 - 3y + color(red)(3y) = -color(blue)(7) + 7 + 7y + color(red)(3y)`

`-14 - 0 = 0 + (7 + color(red)(3))y`

`-14 = 10y`

`-14/color(red)(10) = (10y)/color(red)(10)`

`-7/5 = (color(red)(cancel(color(black)(10)))y)/cancel(color(red)(10))`

`-7/5 = y`

`y = -7/5`

Step 3) Substitute `-7/5` for `y` in the solution to either equation in Step 1 and calculate `x`. I will choose the second equation:

`2x = 7 + 7y` becomes:

`2x = 7 + (7 * -7/5)`

`2x = 7 + (-49/5)`

`2x = (7 xx 5/5) - 49/5`

`2x = 35/5 - 49/5`

`2x = -14/5`

`color(red)(1/2) xx 2x = color(red)(1/2) xx -14/5`

`1x = -14/10`

`x = -7/5`

The solution is: `x = -7/5` and `y = -7/5` or `(-7/5, -7/5)`