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 \(x + 8y = 51\), we can use the method of elimination. Here’s a step-by-step solution:

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

### Step 2: Multiply Equation 2 by 4
To eliminate \(x\), we can multiply Equation 2 by 4:
\[ 4(x + 8y) = 4(51) \]
This gives us:
\[ 4x + 32y = 204 \quad (Equation 3) \]

### Step 3: Subtract Equation 1 from Equation 3
Now we can subtract Equation 1 from Equation 3:
\[ (4x + 32y) - (4x + 5y) = 204 - 42 \]
This simplifies to:
\[ 32y - 5y = 162 \]
So we have:
\[ 27y = 162 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 27:
\[ y = \frac{162}{27} = 6 \]

### Step 5: Substitute \(y\) back into Equation 2
Now that we have \(y\), we can substitute it back into Equation 2 to find \(x\):
\[ x + 8(6) = 51 \]
This simplifies to:
\[ x + 48 = 51 \]

### Step 6: Solve for \(x\)
Now, subtract 48 from both sides:
\[ x = 51 - 48 = 3 \]

### Final Solution
Thus, the solution to the equations is:
\[ x = 3, \quad y = 6 \]