Question

How do you solve the system of equations `3x + y = 23` and `4x - y = 19`?

Answer 1

X=6, Y=5

Explanation:

Use simultaneous equations (multiplying both equations, and eliminating either x or y by adding or subtracting to find the value of one, then plugging the value back in to find the other value)

The lowest common multiple of 3 and 4 is 12
`3*4=12` so we have to multiply the first equation by 4
`4*3=12` so we have to multiply the second equation by 3

This is the result:
`12x+4y=92`
`12x-3y=57`

From here as x is the same, you can subtract the bottom equation from the top leaving you with y. After doing this the result is:

`7y=35`
`y=35/7=5`

Then we can plug this value back into the equations, then we can solve to find x.
`3x+y=23`
`3x+5=23`
`3x=18` (-5)
`x=18/3=6`

We have got `x=6` and `y=5`, but we have to check if these values work for both equations so lets check.

`4x-y=19`
`4*6-5=19`
`24-5=19`

So therefore these values fit both equations and we are correct.

Answer 2

The point of intersection is `(6,5)`

Explanation:

You can also use substitution. The resulting values for `x` and `y` are the point of intersection of the two equations.

Equation 1: `3x+y=23`

Equation 2: `4x-y=19`

Solve Equation 1 for `y`.

`3x+y=23`

Subtract `3x` from both sides.

`y=23-3x`

Substitute `23-3x` for `y` in Equation 2 and solve for `x`.

`4x-y=19`

`4x-(23-3x)=19`

Simplify.

`4x-23+3x=19`

Add `23` to both sides.

`4x+3x=19+23`

Simplify.

`7x=42`

Divide both sides by `7`.

`x=42/7`

`x=6`

Substitute `6` for `x` into Equation 1 and solve for `y`.

`3x+y=23`

`3(6)+y=23`

Simplify.

`18+y=23`

Subtract `18` from both sides.

`y=23-18`

`y=5`

The point of intersection is `(6,5)`.

Equation 1:

`3(6)+5=23`

`18+5=23`

`23=23` `sqrt`

Equation 2:

`4(6)-5=19`

`24-5=19`

`19=19` `sqrt`

graph{(y+3x-23)(-y+4x-19)=0 [-15.69, 16.34, -5.83, 10.19]}