System of Equations

Question

Solve system using elimination and substitution

x^2 + x −3 =y

2x^2 + 2x = 6 + 2y

Answer 1

Elimination Method:

Step 1: Take the 2nd equation and subtract 6 from both sides. You now have:

2x^2 + 2x -6 = 2y

Step 2: Divide everything by 2. You now have:

x^2 + x −3 =y (Observe that this is the same equation as your top)

Step 3: If you have to show this using elimination, line them up on top of each other and subtract.

x^2 + x −3 =y

-(x^2 + x −3 =y)

And you get 0 = 0 meaning that you have infinitely many solutions.

Substitution Method:

Step 1: Observe that in the first equation, you have y= ... so, replace y in your 2nd equation with the information from your first.

This becomes: 2x^2 + 2x = 6 + 2[x^2 + x −3]

Step 2: Distribute the 2.

2x^2 + 2x = 6 + 2x^2 + 2x - 6

Step 3: Notice that you have 6 and - 6 on the same side so this simplifies

2x^2 + 2x = 2x^2 + 2x

Step 4... observe that left and right are identical!

Thus, there are infinitely many solutions! (You could also subtract everything to one side and notice that you will once again get 0 = 0)