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
We have: \[ 2x - 3y = 14 \quad (1) \] \[ 4x + 3y = 10 \quad (2) \]
Step 2: Align the equations for elimination
We can eliminate \(y\) by adding the equations, but we first notice that we can modify Equation 1 to make the coefficients of \(y\) opposite.
We can multiply Equation 1 by 1 (to keep it the same for easy addition): \[ 2x - 3y = 14 \quad (1) \] Equation 2 is kept as is: \[ 4x + 3y = 10 \quad (2) \]
Step 3: Add the equations
Now we'll add the two equations to eliminate \(y\): \[ (2x - 3y) + (4x + 3y) = 14 + 10 \] This simplifies to: \[ 6x = 24 \]
Step 4: Solve for \(x\)
Now divide by 6: \[ x = 4 \]
Step 5: Substitute \(x\) back into one of the original equations
We can substitute \(x = 4\) back into Equation 1: \[ 2(4) - 3y = 14 \] This simplifies to: \[ 8 - 3y = 14 \] Now we isolate \(y\): \[ -3y = 14 - 8 \] \[ -3y = 6 \] \[ y = -2 \]
Solution
The solution to the system of equations is: \[ x = 4, \quad y = -2 \]
Thus, the solution is \( (4, -2) \).