Question

How do you solve the system `y = 2x - 3` and `y = -x + 3`?

Answer 1

You may set the two `y`'s equal to each other:

Explanation:

`2x-3=-x+3`. Now add `3` to both sides:

`2x-cancel3+cancel3=-x+3+3`. Now add `x`:

`2x+x=cancelx-cancelx+6->3x=6->x=2`

Plug this into both equations (one for check!):
`y=2*2-3=1`
`y=-2+3=1`
(they agree).

Answer 2

`(2,1)`

Explanation:

Label the equations.

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

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

Since both equations give `color(red)(y)` expressed in terms of x we can equate the right sides.

`rArr2x-3=-x+3`

add x to both sides.

`2x+x-3=cancel(-x)cancel(+x)+3`

`rArr3x-3=3`

add 3 to both sides.

`3xcancel(-3)cancel(+3)=3+3`

`rArr3x=6`

divide both sides by 3

`(cancel(3) x)/cancel(3)=6/3`

`rArrx=2`

Substitute this value into either ( 1 ) or ( 2 ) and evaluate for y

`"Substituting in " (1)`

`x=2toy=(2xx2)-3=4-3=1`

`color(blue)"As a check"`

`"Substituting in " (2)`

`x=2toy=-2+3=1to" true"`

`rArr" the point of intersection " =(2,1)`
graph{(y-2x+3)(y+x-3)=0 [-10, 10, -5, 5]}