To solve the system of equations using the elimination method, we have the following equations:
- \( 2x - 3y = 14 \) (Equation 1)
- \( 4x + 3y = 10 \) (Equation 2)
Step 1: Align the equations for elimination.
We can eliminate \(y\) by adding both equations together. To do this, we first make the coefficients of \(y\) have the same absolute value. We can achieve this by multiplying Equation 1 by 1 and Equation 2 by 1.
Step 2: Modify the equations if necessary.
From the first equation, we can leave it as is:
\[ 2x - 3y = 14 \quad \text{(Multiply by 1)} \] From the second equation:
\[ 4x + 3y = 10 \quad \text{(Multiply by 1)} \]
Step 3: Add Equations to eliminate \(y\).
Now we can add both equations together to eliminate \(y\):
\[ (2x - 3y) + (4x + 3y) = 14 + 10 \]
This simplifies to:
\[ 6x = 24 \]
Step 4: Solve for \(x\).
Now, divide both sides by 6:
\[ x = 4 \]
Step 5: Substitute \(x\) back into one of the original equations to find \(y\).
We can use Equation 1:
\[ 2(4) - 3y = 14 \]
This simplifies to:
\[ 8 - 3y = 14 \]
Now, isolate \(y\):
\[ -3y = 14 - 8 \] \[ -3y = 6 \] \[ y = -2 \]
Step 6: State the solution.
Thus, the solution to the system of equations is:
\[ (x, y) = (4, -2) \]
Final answer:
\[ \boxed{(4, -2)} \]
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