Bot GPT 4-o mini To solve the system of equations using the elimination method, we have the following equations:
- \( 4x + 8y = 20 \) (Equation 1)
- \(-4x + 2y = -30\) (Equation 2)
The goal is to eliminate one variable by adding or subtracting the equations. Let's first line them up for clarity:
\[ \begin{align*}
- & \quad 4x + 8y = 20 \
- & \quad -4x + 2y = -30 \end{align*} \]
Step 1: Add the equations to eliminate \(x\)
Adding Equation 1 and Equation 2 together:
\[ (4x + 8y) + (-4x + 2y) = 20 + (-30) \]
This simplifies to:
\[ 0 + 10y = -10 \]
Step 2: Solve for \(y\)
\[ 10y = -10 \] \[ y = -1 \]
Step 3: Substitute \(y\) back into one of the original equations to find \(x\)
We'll use Equation 1:
\[ 4x + 8(-1) = 20 \] \[ 4x - 8 = 20 \] \[ 4x = 28 \] \[ x = 7 \]
Final Solution:
The solution to the system of equations is:
\[ (x, y) = (7, -1) \]
Verification:
We can check by substituting \(x = 7\) and \(y = -1\) into both original equations:
For Equation 1: \[ 4(7) + 8(-1) = 28 - 8 = 20 \quad \text{(True)} \]
For Equation 2: \[ -4(7) + 2(-1) = -28 - 2 = -30 \quad \text{(True)} \]
Both equations check out, confirming that the solution is correct: \( (x, y) = (7, -1) \).
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