`1/5x+1/4y=2`
px+2y=16
In the system of linear equations above , p is a constant . If the system has an infinite number of solutions, what is the value of p ?

Options

A. `8/5`
B. `5/8`
C. `3/5`
D. `1/8`

Answer

To find the value of \( p \) such that the system of linear equations has an infinite number of solutions, we need to analyze the given equations:

1. **Equations Given:**
\[ \frac{1}{5}x + \frac{1}{4}y = 2 \quad \text{(Equation 1)} \]
\[ px + 2y = 16 \quad \text{(Equation 2)} \]

2. **Convert to General Form:**
We can rewrite both equations in the general form \( ax + by + c = 0 \).

- For Equation 1:
\[ \frac{1}{5}x + \frac{1}{4}y - 2 = 0 \]
- For Equation 2:
\[ px + 2y - 16 = 0 \]

3. **Identify Coefficients:**
From the equations, we identify the coefficients:
- For Equation 1: \( a_1 = \frac{1}{5}, b_1 = \frac{1}{4}, c_1 = -2 \)
- For Equation 2: \( a_2 = p, b_2 = 2, c_2 = -16 \)

4. **Condition for Infinite Solutions:**
For the system to have an infinite number of solutions, the following condition must hold:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

5. **Set Up Ratios:**
We can set up the ratios:
\[ \frac{\frac{1}{5}}{p} = \frac{\frac{1}{4}}{2} = \frac{-2}{-16} \]

6. **Simplify the Ratios:**
- The rightmost ratio simplifies to:
\[ \frac{-2}{-16} = \frac{1}{8} \]
- The middle ratio simplifies to:
\[ \frac{\frac{1}{4}}{2} = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

7. **Set Up the First Ratio:**
Now we focus on the first ratio:
\[ \frac{\frac{1}{5}}{p} = \frac{1}{8} \]

8. **Cross Multiply:**
Cross multiplying gives:
\[ 1 \cdot 8 = 5p \]
Thus, we have:
\[ 8 = 5p \]

9. **Solve for \( p \):**
Dividing both sides by 5:
\[ p = \frac{8}{5} \]

10. **Final Answer:**
The value of \( p \) is:
\[ \boxed{\frac{8}{5}} \]