Question

How do you solve the following linear system: # 2x + 4y = 7 , 3x + y = 4 #?

Answer 1

`(9/10, 13/10)`
Refer below for explanation.

Explanation:

First, arrange them like this so you can compare terms:
`2x+4y=7`
`3x+y=4`

When we solve linear systems like this, we always look for ways to cancel one variable. We can see that if we multiply `3x+y=4` by `-4`, we will get a `-4y` term. And then, if we add the new equation (with `-4y` in it) to the other equation, the `4y` and `-4y` will cancel. Watch:
`-4(3x+y=4)->-12x-4y=-16`

`-12xcancel(-4y)=-16`
`+2xcancel(+4y)=7`
`-----`
`-10x=-9`
`x=9/10`

We can now use this `x` value to solve for `y`, like so:
`2x+4y=7`
`2(9/10)+4y=7`
`9/5+4y=7`
`4y=7-9/5`
`4y=26/5`
`y=13/10`

Therefore our solution is `(9/10, 13/10)`. We can confirm this result in several ways. One, we can substitute these values into the original equations:
`2x+4y=7->2(9/10)+4(13/10)=7->7=7`
`3x+y=4->3(9/10)+(13/10)=4->4=4`

We can also look at the graph of these two equations, and confirm that they intersect at the point `(9/10, 13/10)`.

Note that `9/10=0.9` and `13/10=1.3`.