Question

How do you solve using the elimination method `3x + 4y = 5`, `6x + 8y = 10`?

Answer 1

This system of equations has an infinite number of solutions.

Explanation:

Your system of equations looks like this

`{(3x + 4y = 5), (6x + 8y = 10) :}`

Notice that if you multiply the first equation by `(-2)` and then add the left-hand sides and the right-hand sides separately, you will get

`{(3x + 4y = 5 | * (-2)), (6x + 8y = 10) :}`

`{(-6x - 8y = -10), (6x " "+ 8y = " "10) :}`
`stackrel("---------------------------------------------")`
`" "0 " "+ 0 " "= 0`

Since you end up with `0=0`, which is an equality that does not depends on any variable, the system of equations will have an infinite number of solutions.

You can think of a system of equations as describing two lines. In your case, both equations decribe the same line, because if you multiply the first equation by `2` you get the second equation

`{(6x + 8y = 10), (6x + 8y = 10) :}`

This implies that you have an infinite number of `(x,y)` points that correspond to this line.