If `4x + 5y = 83 and 3x : 2y = 21 : 22` then y - x equals

Options

A. 3
B. 4
C. 7
D. 11

Answer

To solve the problem step by step, we will use the given equations:

1. **Given equations**:
- \( 4x + 5y = 83 \) (Equation 1)
- \( \frac{3x}{2y} = \frac{21}{22} \) (Equation 2)

2. **Rewrite Equation 2**:
- From Equation 2, we can cross-multiply to eliminate the fraction:
\[ 3x \cdot 22 = 2y \cdot 21 \]
This simplifies to:
\[ 66x = 42y \]
Dividing both sides by 6 gives:
\[ 11x = 7y \]
Rearranging this gives us:
\[ y = \frac{11}{7}x \quad \text{(Equation 3)} \]

3. **Substituting Equation 3 into Equation 1**:
- Now, we substitute \( y \) from Equation 3 into Equation 1:
\[ 4x + 5\left(\frac{11}{7}x\right) = 83 \]
Simplifying this gives:
\[ 4x + \frac{55}{7}x = 83 \]
To combine the terms, convert \( 4x \) into a fraction with a denominator of 7:
\[ \frac{28}{7}x + \frac{55}{7}x = 83 \]
This simplifies to:
\[ \frac{83}{7}x = 83 \]

4. **Solving for \( x \)**:
- Multiply both sides by 7 to eliminate the fraction:
\[ 83x = 581 \]
Dividing both sides by 83 gives:
\[ x = 7 \]

5. **Finding \( y \)**:
- Now substitute \( x = 7 \) back into Equation 3 to find \( y \):
\[ y = \frac{11}{7}(7) = 11 \]

6. **Calculating \( y - x \)**:
- Now we can find \( y - x \):
\[ y - x = 11 - 7 = 4 \]

Thus, the final answer is:
\[ y - x = 4 \]