Let's go through the steps to solve the system of equations:
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Equations to solve: \[ 3x + 4y = 48 \quad (1) \] \[ 6x + 2y = 60 \quad (2) \]
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Choose the best method to solve:
Based on the equations, the elimination method could be effective, especially since the coefficients of \(x\) in equations (1) and (2) can be manipulated to facilitate cancellation. -
Get the x variables to cancel out:
To eliminate \(x\), we can manipulate (1) and (2) to have the same coefficient for \(x\). We can multiply equation (1) by 2:\[ 2(3x + 4y = 48) \rightarrow 6x + 8y = 96 \quad (3) \]
Now we have: \[ 6x + 8y = 96 \quad (3) \] \[ 6x + 2y = 60 \quad (2) \]
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Line them up and subtract the equations together to eliminate \(x\):
Now, we will subtract equation (2) from equation (3): \[ (6x + 8y) - (6x + 2y) = 96 - 60 \] This gives: \[ 6y = 36 \] -
Solve for \(y\):
Divide both sides by 6: \[ y = 6 \] -
Solve for \(x\):
Now, substitute \(y = 6\) back into one of the original equations, say (1): \[ 3x + 4(6) = 48 \] \[ 3x + 24 = 48 \] Subtract 24 from both sides: \[ 3x = 24 \] Divide by 3: \[ x = 8 \]
Final Answers:
- \(x = 8\)
- \(y = 6\)
This method using elimination correctly yields the solution of the system \( (8, 6) \).