Question

How do solve the following linear system?: `4x-2y=2 , -2x+5y=-3`?

Answer 1

See the entire solution process below:

Explanation:

Step 1) Solve the first equation for `y`:

`4x - 2y = 2`

`4x - color(red)(4x) - 2y = 2 - color(red)(4x)`

`0 - 2y = 2 - 4x`

`-2y = 2 - 4x`

`(-2y)/color(red)(-2) = (2 - 4x)/color(red)(-2)`

`(color(red)(cancel(color(black)(-2)))y)/cancel(color(red)(-2)) = 2/color(red)(-2) - (4x)/color(red)(-2)`

`y = -1 + 2x`

Step 2) Substitute `-1 + 2x` for `y` in the second equation and solve for `x`:

`-2x + 5y = -3` becomes:

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

`-2x + (5 * -1) + (5 * 2x) = -3`

`-2x - 5 + 10x = -3`

`10x - 2x - 5 = -3`

`(10 - 2)x - 5 = -3`

`8x - 5 = -3`

`8x - 5 + color(red)(5) = -3 + color(red)(5)`

`8x - 0 = 2`

`8x = 2`

`(8x)/color(red)(8) = 2/color(red)(8)`

`(color(red)(cancel(color(black)(8)))x)/cancel(color(red)(8)) = 1/4`

`x = 1/4`

Step 3) Subsitute `1/4` for `x` in the solution to the first equation at the end of Step 1 and calculate `y`:

`y = -1 + 2x` becomes:

`y = -1 + (2 * 1/4)`

`y = -1 + 1/2`

`y = -1/2`

The solution is: `x = 1/4` and `y = -1/2` or `(1/4, -1/2)`