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

1) Rearrange your equations to make them easier to use. The easiest way to do this is to isolate ONE variable that has the coefficient of 1.

`3x - 2y = 4`

`x = -4y + 3`

2) Since you now know what the x variable is in terms of y, you can go ahead and substitute it into the first equation.

`3 (-4y + 3) - 2y = 4`

3) Simplify

`3 (-4y + 3) - 2y = 4`

`-12y + 9 - 2y = 4`

`-14y + 9 = 4`

`-14y = -5`

`y = 5/14`

4) Substitute the solution you just got to get the second variable.

`x + 4y = 3`

`x + 4 (5/14) = 3`

`x + 10/7 = 3`

`x = 11/7`

`3x-2y=4`
`x+4y=3`

Using the second equation, derive a value for `x`.

`x+4y=3`

Subtract `4y` from each side.

`x=3-4y`

In the first equation, substitute `x` with `color(red)((3-4y))`.

`3x-2y=4`

`3color(red)((3-4y))-2y=4`

Open the brackets and simplify. The product of a positive and a negative is a negative.

`9-12y-2y=4`

`9-14y=4`

Subtract `9` from each side.

`-14y=-5`

Divide both sides by `-14`.

`y=5/14`

In the second equation, substitute `y` with `color(blue)(5/14)`.

`x+4y=3`

`x+4(color(blue)(5/14))=3`

Open the brackets and simplify.

`x+2cancel4xx5/(7cancel14)=3`

`x+10/7=3`

Subtract `10/7` from each side.

`x=3-10/7`

`x=21/7-10/7`

`x=11/7=1 4/7`