To solve the system of equations using the elimination method, we start by writing down the equations clearly:
- \(-4x - 6y = -24\) (Equation 1)
- \(4x + 5y = 24\) (Equation 2)
Step 1: Aligning coefficients for elimination
We want to eliminate one of the variables. A good strategy is to eliminate \(x\) by making the coefficients of \(x\) in both equations equal. Notice that the coefficient of \(x\) in Equation 1 is \(-4\) and in Equation 2 is \(4\). We can multiply Equation 2 by 1 (since it's already suitable) and Equation 1 by 1 to keep them as they are, or we can directly consider adding them since they will cancel out with the right setup.
Step 2: Adding the equations
Let's add Equation 1 and Equation 2 directly to eliminate \(x\):
\[ (-4x - 6y) + (4x + 5y) = -24 + 24 \]
This simplifies to:
\[ (-4x + 4x) + (-6y + 5y) = 0 \]
Thus:
\[ 0 - y = 0 \]
This gives us:
\[ -y = 0 \implies y = 0 \]
Step 3: Substituting back to find \(x\)
Now that we have \(y = 0\), we can substitute this value back into one of the original equations to find \(x\). We can use Equation 2 for this:
\[ 4x + 5(0) = 24 \]
This simplifies to:
\[ 4x = 24 \]
Dividing both sides by 4 gives:
\[ x = 6 \]
Step 4: Writing the solution
The solution to the system of equations is:
\[ (x, y) = (6, 0) \]
Conclusion
Thus, the final answer of the system of equations is \((6, 0)\).