To solve the system of equations given by:
- \( 4x - 6y = -48 \) (Equation 1)
- \( -2x + 4y = 30 \) (Equation 2)
We can solve for \(x\) and \(y\) using either substitution or elimination. Here, I will use the elimination method.
First, let's simplify both equations.
Simplifying Equation 1:
-
\( 4x - 6y = -48 \)
We can divide every term by 2:
\[ 2x - 3y = -24 \] (Equation 1 simplified)
Now for Equation 2:
-
\( -2x + 4y = 30 \)
We can also divide every term by 2:
\[ -x + 2y = 15 \] (Equation 2 simplified)
Next, we can multiply Equation 2 by 2 to align the coefficients for \(x\):
\[-2x + 4y = 30 \implies 2x - 4y = -30\] (Equation 2 after multiplying by 2)
Now we have:
- \[ 2x - 3y = -24 \]
- \[ 2x - 4y = -30 \]
Now we can subtract Equation 1 from Equation 2 to eliminate \(x\):
\[ (2x - 4y) - (2x - 3y) = -30 - (-24) \] \[ -4y + 3y = -30 + 24 \] \[ -y = -6 \]
Solving for \(y\):
\[ y = 6 \]
Now that we have \(y\), we can substitute \(y\) back into one of the original equations to find \(x\). Let's use Equation 1 (the original one for simplicity):
\[ 4x - 6(6) = -48 \] \[ 4x - 36 = -48 \] \[ 4x = -48 + 36 \] \[ 4x = -12 \] \[ x = -3 \]
The solution to the system of equations is \( (x, y) = (-3, 6) \).
From the provided response options, the correct one is:
\[ \text{(−3, 6)} \]