Identify the solution(s) of the system of equations, if any.
-6x - 4y = 24
8y = -12x - 48
If you rearrange things a bit and make the x term positive, you get
6x+4y = -24
12x+8y = -48
They are the same line, so there are infinitely many solutions. Pick an x, solve for y in either equation, and it works in the other equation as well.
To solve the system of equations, we can use the method of substitution or elimination.
Let's use the method of substitution:
1. Start with the first equation: -6x - 4y = 24.
Solve this equation for x in terms of y:
-6x = 24 + 4y
Divide both sides by -6:
x = (-24 - 4y)/6
Simplify this expression:
x = (-4 - (2/3)y)
2. Take this expression for x and substitute it into the second equation: 8y = -12x - 48.
Replace x with (-4 - (2/3)y):
8y = -12(-4 - (2/3)y) - 48
Distribute -12 to the terms inside the parentheses:
8y = 48 + 8y + (8/3)y - 48
Combine like terms:
0 = (8/3)y + 8y
0 = (8/3 + 8)y
0 = (8/3 + 24/3)y
0 = (32/3)y
Multiply both sides by 3/32 to solve for y:
y = 0
3. Now that we have the value of y, we can substitute it back into the expression for x:
x = (-4 - (2/3)y)
x = (-4 - (2/3)(0))
x = -4
Therefore, the solution to the system of equations is x = -4 and y = 0.