Question

How do you solve the following system?: `61x -31y =-33 , -3x +7y = 8`

Answer 1

`(x,y)=(17/334,389/334)`

Explanation:

We have:

`{(61x-31y=-33" "" "" "" "" "mathbf(eq. 1)),(-3x+7y=8" "" "" "" "" "" "color(white)(sl)mathbf(eq. 2)):}`

We want to either cancel out the `x` terms or the `y` terms. We can cancel the `x` terms by multiplying `mathbf(eq. 1)` by `3` and `mathbf(eq. 2)` by `61`, and then adding the two.

`{:(color(white)"-.-"183x-93y=-99" "" "" "" "color(white)(sl)mathbf(eq. 1)xx3),(ul(-183x+427y=488" "+)" "" "color(white)(ss)mathbf(eq. 2)xx61),(color(white)("-------------")334y=389" "" "" "" "" "mathbf(eq. 3)):}`

From `mathbf(eq. 3)`, we can solve for `y` to see that

`color(blue)(y=389/334`

We can now plug this value of `y` into either of the original two equations. I will choose `mathbf(eq. 2)` since the numbers will be smaller.

`-3x+7color(blue)y=8" "=>" "-3x+7(color(blue)(389/334))=8`

Solving this equation, we multiply `7` and `389//334`:

`-3x+2723/334=8`

`-3x=8-2723/334`

Find a common denominator.

`-3x=2672/334-2723/334`

`-3x=-51/334`

`x=-51/334(-1/3)`

The negatives will cancel, so `x` will be positive, and note that `51=3xx17`, so the final value of `x` is:

`color(red)(x=17/334`

Written as the ordered pair `(x,y)`, our final answer is

`color(green)(|barul(color(white)(int^int)(x,y)=(17/334,389/334)color(white)(int^int)|))`