Question

How do you determine the solution in terms of a system of linear equations for `5x + 2y = 5`, `3x + y = 2`?

Answer 1

`x=-1`
`y=5`

Explanation:

You can solve for the solution using elimination or substitution. Personally, I think it will be easier to use substitution for this problem, so that's what I will use.

First, let's look at the second equation. You will notice that it is easy to isolate `y` here.
`3x+y=2`
`3x+y-3x=2-3x`
`color(red)(y=2-3x)`

Now that we've isolated `y` in the second equation, we will now substitute this into `y` in the first equation.
`5x+2color(red)y=5`
`5x+2color(red)((2-3x))=5`
`5x+4-6x=5`
`4-x+x=5+x`
`4=5+x`
`4-5=5+x-5`
`-1=x`
`color(blue)(x=-1)`

Now that we have solved for the value of `x`, let's go back to the equation earlier where we isolated `y`.
`color(red)(y=2-3)color(blue)x`
`y=2-3color(blue)((-1))`
`y=2+3`
`color(red)(y=5)`

So the solutions are: `color(blue)(x=-1)` and `color(red)(y=5)`.

Checking
In case you are unsure of your answer, you can check this by substituting the solutions in both of the equations.
`5color(blue)x+2color(red)y=5`
`5color(blue)((-1))+2color(red)((5))=5`
`-5+10=5`
`color(magenta)(5=5)`

`3color(blue)x+color(red)y=2`
`3color(blue)((-1))+color(red)((5))=2`
`-3+5=2`
`color(magenta)(2=2)`