o solve this system, you could do the following steps: 3x+4y=48 6x+2y=60 (6 points)

1 answer

To solve the system of equations given by \(3x + 4y = 48\) and \(6x + 2y = 60\), we can use the method of substitution or elimination. Here, we will use the elimination method for clarity.

Step 1: Write down the equations

  1. \(3x + 4y = 48\) , (Equation 1)
  2. \(6x + 2y = 60\) , (Equation 2)

Step 2: Simplify Equation 2

Notice that Equation 2 can be simplified by dividing all terms by 2:

\[ 3x + y = 30 \]

Now we have a new system:

  1. \(3x + 4y = 48\) , (Equation 1)
  2. \(3x + y = 30\) , (Equation 3)

Step 3: Eliminate \(x\)

We can eliminate \(x\) by subtracting Equation 3 from Equation 1.

\[ (3x + 4y) - (3x + y) = 48 - 30 \]

This simplifies to:

\[ 4y - y = 18 \implies 3y = 18 \]

Step 4: Solve for \(y\)

Divide both sides by 3:

\[ y = 6 \]

Step 5: Substitute \(y\) back into one of the equations

We can substitute \(y = 6\) back into Equation 3 to solve for \(x\):

\[ 3x + 6 = 30 \]

Subtract 6 from both sides:

\[ 3x = 24 \]

Divide by 3:

\[ x = 8 \]

Step 6: Write the solution

The solution to the system of equations is:

\[ (x, y) = (8, 6) \]

Step 7: Check the solution

To ensure our solution is correct, we can substitute \(x = 8\) and \(y = 6\) back into the original equations.

For Equation 1:

\[ 3(8) + 4(6) = 24 + 24 = 48 \quad \text{(True)} \]

For Equation 2:

\[ 6(8) + 2(6) = 48 + 12 = 60 \quad \text{(True)} \]

Since both original equations are satisfied, our solution is verified. Thus, the final solution is:

\[ \boxed{(8, 6)} \]