If `2x + y = 35 and 3x + 4y = 65,` then the value of `(x)/(y)` is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To solve the system of equations given by:

1. \( 2x + y = 35 \) (Equation 1)
2. \( 3x + 4y = 65 \) (Equation 2)

we will follow these steps:

### Step 1: Multiply Equation 1 by 4
We multiply the first equation by 4 to make the coefficients of \( y \) in both equations the same.

\[ 4(2x + y) = 4(35) \]
This simplifies to:
\[ 8x + 4y = 140 \quad \text{(Equation 3)} \]

### Step 2: Write down Equation 2
We will keep the second equation as it is:

\[ 3x + 4y = 65 \quad \text{(Equation 2)} \]

### Step 3: Subtract Equation 2 from Equation 3
Now, we will subtract Equation 2 from Equation 3 to eliminate \( y \):

\[ (8x + 4y) - (3x + 4y) = 140 - 65 \]

This simplifies to:
\[ 8x - 3x + 4y - 4y = 75 \]
\[ 5x = 75 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):

\[ x = \frac{75}{5} = 15 \]

### Step 5: Substitute \( x \) back into Equation 1 to find \( y \)
Now that we have \( x \), we will substitute it back into Equation 1 to find \( y \):

\[ 2(15) + y = 35 \]
\[ 30 + y = 35 \]
\[ y = 35 - 30 = 5 \]

### Step 6: Calculate \( \frac{x}{y} \)
Now we can find the value of \( \frac{x}{y} \):

\[ \frac{x}{y} = \frac{15}{5} = 3 \]

### Conclusion
Thus, the value of \( \frac{x}{y} \) is \( 3 \).

---