Question

How do you solve `3x + y = 7` and `3x - y = 5`?

Answer 1

`x = 2 " and " y = 1`

Explanation:

There are 2 obvious methods which can be used in this example, based on the x and y values in the equations.

ELIMINATION METHOD
The two y terms are additive inverses. This means they have the same value, but opposite signs.
When they are ADDED together, they will make 0.

`" " 3x + y = 7 " A"`
`" " 3x - y = 5 " B"`

`"A + B: " 6x " "= 12" (the y's have been eliminated)"`
`" " x" " = 2`

Substitute into A to find the value of y:
`" " 3(2) + y = 7`
`" " 6 + y = 7`
`" " y = 1`

EQUATING METHOD

The two x-terms are exactly the same. Make the x term the subject.

`3x + y = 7 " and " 3x - y = 5`
`3x " " = 7 - y " " 3x " "=5 + y`

But `3x" is the same as "3x`!!

Therefore `" " 7 - y = 5 + y`
solve for y: `" " 7-5 = 2y`
`" " 2 = 2y " " rArr y = 1`

If `y = 1`, substitute into either equation to find `x`.

`3x = 7 - y rArr 3x = 7-1 = 6`
`3x = 6`
`x = 2`