Question

How do you solve the simultaneous equations `7a - 3b = 17` and `2a + b = 16`?

Answer 1

`(a,b) = (5,6)`

Explanation:

[1]`color(white)("XXXX")` `7a-3b=17`
[2]`color(white)("XXXX")` `2a+b =16`

Multiply both sides of [2] by `3` (to get the same coefficient for `b` as in [1])
[3]`color(white)("XXXX")` `6a+3b = 48`

Add [1] and [3]
[4]`color(white)("XXXX")` `13a = 65`

Divide both sides by `13`
[5]`color(white)("XXXX")` `a = 5`

Substitute `5` for `a` in [2]
[6]`color(white)("XXXX")` `2(5) + b =16`

Simplify
[7]`color(white)("XXXX")` `b = 6`

Answer 2

`a=5`

`b=6`

Explanation:

We are given `7a-3b=17` and `2a+b=16`

Let's get the `b`'s equal to each other but opposite in sign by multiplying `2a+b=16` by `3`:

`2a+b=16`

`3*2a+3*b=3*16`

`6a+3b=48`

Now, let's add `6a+3b=48` to `7a-3b=17`

`6a+3b=48`
`7a-3b=17`

`13a=65`

`a=65/13`

`a=5`

Now, substitute `a` into `7a-3b=17`

`7a-3b=17`

`7*5-3b=17`

`35-3b=17`

`-3b=17-35`

`-3b=-18`

`b=6`

Let's check to see if our answers are correct by plugging in the values we found for both `a` and `b` into `2a+b=16`

`2a+b=16`

`2*5+6=16`

`10+6=16`

`16=16`