Question

How do you solve the system of equations `3y-4x=-33` and `5x-3y=40.5`?

Answer 1

See a solution process below:

Explanation:

Step 1) Solve each equation for `3y`:

• Equation 1:

`3y - 4x = -33`

`3y - 4x + color(red)(4x) = -33 + color(red)(4x)`

`3y - 0 = -33 + 4x`

`3y = -33 + 4x`

• Equation 2:

`5x - 3y = 40.5`

`5x - color(red)(5x) - 3y = 40.5 - color(red)(5x)`

`0 - 3y = 40.5 - 5x`

`-3y = 40.5 - 5x`

`color(red)(-1) xx -3y = color(red)(-1)(40.5 - 5x)`

`3y = (color(red)(-1) xx 40.5) - (color(red)(-1) xx 5x)`

`3y = -40.5 - (-5x)`

`3y = -40.5 + 5x`

Step 2) Because the left side of both equations are equal we can equate the right side of both equations and solve for `x`:

`-33 + 4x = -40.5 + 5x`

`-33 + color(blue)(40.5) + 4x - color(red)(4x) = -40.5 + color(blue)(40.5) + 5x - color(red)(4x)`

`7.5 + 0 = 0 + (5 - color(red)(4))x`

`7.5 = 1x`

`7.5 = x`

`x = 7.5`

Step 3) Substitute `7.5` for `x` into either of the solutions for the equations in Step 1 and calculate `y`:

`3y = -33 + 4x` becomes:

`3y = -33 + (4 xx 7.5)`

`3y = -33 + 30`

`3y = -3`

`(3y)/color(red)(3) = -3/color(red)(3)`

`(color(red)(cancel(color(black)(3)))y)/cancel(color(red)(3)) = -1`

`y = -1`

The Solution Is:

`x = 7.5` and `y = -1`

Or

`(7.5, -1)`