Question

How do you solve the system of equations `2x + y = 30` and `4x + 2y = 60`?

Answer 1

An infinite number of solutions exist.

Explanation:

We can start by using substitution.
The first equation solves easily for `y`, so just subtract `2x` from both sides:
`y=-2x+30`

This equals "`y`". Plug in this expression for `y` in the second equation and solve for `x`:
`4x+2(-2x+30)=60`
`4x-4x+60=60`
`0=0`

But wait- the "`x`"s cancel out! What does that mean? Well, there are an infinite number of solutions to this system- so you can't just find one "`x=`" and "`y=`".
So that's the answer: There are an infinite number of solutions.

Also, you can try dividing both sides of the second equation by `2`:
`2x+y=30`, which is actually the exact same line as the first one. When the equations in
a given system of equations represent the same line, it is called a dependent system.