Question

How do you determine whether a linear system has one solution, many solutions, or no solution when given 3x - y = -6 and x + y = 2?

Answer 1

One solution, (-1,3)

Explanation:

Although there are multiple ways one could solve this, I will begin with the algerbraic path to finding this solution. (Skip to bottom for shortcut)

We begin with two equations...
`3x-y =-6` and
`x+y=2`
Lets first put them into `y=mx +b` format
1)`3x-y =-6` Given
2)`-y =-6- 3x` Subtraction property of Equality
3) `y=6+3x` Multiply both side by -1 to turn y positive
4)`y=3x+6` Commutative Property
Now the second equation
1)`x+y=2` Given
2)`y=-x+2` Subtract x from both sides.
Now that both are in y=mx+b format, set the equations equal to eachother.
`y=-x+2=3x+6`
`-x+2=3x+6`
Then solve for x.....
However there is a faster way to do this... using the process of elimination...
`3x-y =-6`
`x+y=2`
_______ Add these equations together
4x=-4
x=-1
Then subsitute the x=-1 into any of the other equations to find the y value....