Solve the equations `4x+5y = 42 and x+ 8y = 51` for x and y.

Options

A. `x =3, y =6`
B. `x=6, y=9`
C. `x=4, y =9`
D. `x =3, y=8`

Answer

To solve the equations \(4x + 5y = 42\) and \(x + 8y = 51\) for \(x\) and \(y\), we can follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \(4x + 5y = 42\) (Equation 1)
2. \(x + 8y = 51\) (Equation 2)

### Step 2: Equalize the coefficients of \(x\)
To eliminate \(x\), we can multiply Equation 2 by 4 to match the coefficient of \(x\) in Equation 1.

\[ 4(x + 8y) = 4(51) \]
This gives us:
\[ 4x + 32y = 204 \quad \text{(Equation 3)} \]

### Step 3: Subtract Equation 1 from Equation 3
Now we will subtract Equation 1 from Equation 3 to eliminate \(x\):

\[ (4x + 32y) - (4x + 5y) = 204 - 42 \]
This simplifies to:
\[ 32y - 5y = 162 \]
\[ 27y = 162 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 27 to find \(y\):

\[ y = \frac{162}{27} = 6 \]

### Step 5: Substitute \(y\) back into one of the original equations
Now that we have \(y\), we can substitute it back into Equation 2 to find \(x\):

\[ x + 8(6) = 51 \]
\[ x + 48 = 51 \]
Subtract 48 from both sides:
\[ x = 51 - 48 = 3 \]

### Step 6: Write the final solution
Thus, the solution to the equations is:
\[ x = 3, \quad y = 6 \]