How do you solve the following system?: `3x - 2y = 1 , 6x+5y=18`

Step 1) Solve each equation for `6x`:

• Equation 1:

`3x - 2y = 1`

`3x - 2y + color(red)(2y) = 1 + color(red)(2y)`

`3x - 0 = 1 + 2y`

`3x = 1 + 2y`

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

`6x = (color(red)(2) xx 1) + (color(red)(2) xx 2y)`

`6x = 2 + 4y`

• Equation 2:

`6x + 5y = 18`

`6x + 5y - color(red)(5y) = 18 - color(red)(5y)`

`6x + 0 = 18 - 5y`

`6x = 18 - 5y`

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

`2 + 4y = 18 - 5y`

`2 - color(red)(2) + 4y + color(blue)(5y) = 18 - color(red)(2) - 5y + color(blue)(5y)`

`0 + (4 + color(blue)(5))y = 16 - 0`

`9y = 16`

`(9y)/color(red)(9) = 16/color(red)(9)`

`(color(red)(cancel(color(black)(9)))y)/cancel(color(red)(9)) = 16/9`

`y = 16/9`

Step 3) Substitute `16/9` for `y` in either of the equations in Step 1 and solve for `x`:

`6x = 2 + 4y` becomes:

`6x = 2 + 4(16/9)`

`6x = (9/9 xx 2) + 64/9`

`6x = 18/9 + 64/9`

`6x = 82/9`

`6x = 82/9`

`1/6 xx 6x = 1/6 xx 82/9`

`6/6x = 82/54`

`x = 41/27`

The Solution Is:

`x = 41/27` and `y = 16/9`

Or

`(41/27, 16/9)`