Question

How do you solve `12x-5y=8` and `3x=5/4y+2` using substitution?

Answer 1

See a solution process below:

Explanation:

Step 1) Solve the second equation for `x`:

`3x = 5/4y + 2`

`color(red)(1/3) xx 3x = color(red)(1/3)(5/4y + 2)`

`color(red)(1/color(black)(cancel(color(red)(3)))) xx color(red)(cancel(color(black)(3)))x = (color(red)(1/3) xx 5/4y) + (color(red)(1/3) xx 2)`

`x = 5/12y + 2/3`

Step 2) Substitute `5/12y + 2/3` for `x` in the first equation and solve for `y`:

`12x - 5y = 8` becomes:

`12(5/12y + 2/3) - 5y = 8`

`(12 xx 5/12y) + (12 xx 2/3) - 5y = 8`

`(color(red)(cancel(color(black)(12))) xx 5/color(red)(cancel(color(black)(12)))y) + (color(red)(cancel(color(black)(12)))4 xx 2/color(red)(cancel(color(black)(3)))) - 5y = 8`

`5y + 8 - 5y = 8`

`5y - 5y + 8 = 8`

`0 + 8 = 8`

`8 = 8`

Because `8` does in fact equal `8` these two equations are parallel and the same. They have an infinite number of points the same.