Question

How do you solve `2x-3y=-2` and `4x+y=24`?

Answer 1

`(5,4)`

Explanation:

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

`4x+color(red)(y)=24to(2)`

`(2)" can be expressed as "color(red)(y)=24-4x`

`color(red)(y)=24-4x" may now be substituted into " (1)`

`rArr2x-3(24-4x)=-2`

`rArr2x-72+12x=-2larr" distributing bracket"`

`rArr14x-72=-2larr" simplifying left side"`

`"add 72 to both sides"`

`14xcancel(-72)cancel(+72)=-2+72`

`rArr14x=70`

`"divide both sides by 14"`

`(cancel(14) x)/cancel(14)=70/14`

`rArrx=5`

`"substitute this value into " y=24-4x" and evaluate"`

`rArry=24-(4xx5)=24-20=4`

`color(blue)"As a check"`

`"substitute these values into " (1)" and if left side"`
`"equals right side then these values are the solution"`

`(2xx5)-(3xx4)=10-12=-2=" right side"`

`rArr"point of intersection "=(5,4)`
graph{(y-2/3x-2/3)(y+4x-24)=0 [-10, 10, -5, 5]}

Answer 2

`x =5 and y=4`

Explanation:

The other contributors have shown the methods of substitution, graphical, and matrices.

Let's use elimination:
`color(white)(...........)2x-3y" " =-2`.............................A
`color(white)(...........)4x+y" " =+24`............................B

We would like to make additive inverses and we note that the `y`-terms already have opposite signs. Make `B` three times bigger:

`B xx3:color(white)(....)12xcolor(blue)(+3y) =+72`............................C
`color(white)(..................)2xcolor(blue)(-3y) =-2`..............................A

`A+C: color(white)(....)14xcolor(white)(......)=70" "larrdiv 14`
`color(white)(.........................)x= 5`

Substitute `5` for `x` in B to find the value of `y`

`4(5) +y =24`
`20+y =24`
`" "y = 4`