Beyonce is solving a system of equations: 2x – 3y = -2 4x + y = 24

She decides to use the elimination method to find the solution. She multiplies the second equation by 3 and adds it to the first equation to find 14x = 70, showing her that x = 5. Beyonce finds that y = 4.

Step 3: 4x + y = 24

4(5) + y = 24

20 + y = 24
-20 -20

y = 4

Step 1: 3(4x + y = 24) → 12x + 3y = 72
Step 2: 2x – 3y = -2
Solution: (5, 4)

+ 12x + 3y = 72
14x = 70

→ x = 5

Thinking about this process, Beyonce says out loud, “There are lots of ways I could go about solving this problem. I could do the method above or I could multiply the first equation by -2 and add the second.

Step 3: 4x + y = 24

4x + 4 = 24
-4 -4

4x = 20

x = 5

Step 1: -2(2x – 3y = -2) → -4x + 6y = 4
Solution: (5, 4)

Step 2: -4x + 6y = 4
+ 4x + y = 24
7y= 28

→ y = 4

“I seem to find that there is only one solution to the two equations, but I wonder if I will get the same solution if I use a different method?”

Explain how you know that a system od equations has an infinite numbe of solutions.