Solve for x and y:
`4/x+5y=7,3/x+4y=5`

Answer

To solve the equations \( \frac{4}{x} + 5y = 7 \) and \( \frac{3}{x} + 4y = 5 \), we will follow these steps:

### Step 1: Substitute \( \frac{1}{x} \) with a new variable
Let \( u = \frac{1}{x} \). Then, we can rewrite the equations as follows:
1. \( 4u + 5y = 7 \)
2. \( 3u + 4y = 5 \)

### Step 2: Multiply the equations to eliminate \( u \)
To eliminate \( u \), we can multiply the first equation by 3 and the second equation by 4:
1. \( 3(4u + 5y) = 3(7) \) gives us \( 12u + 15y = 21 \)
2. \( 4(3u + 4y) = 4(5) \) gives us \( 12u + 16y = 20 \)

### Step 3: Subtract the second equation from the first
Now, we subtract the second equation from the first:
\[ (12u + 15y) - (12u + 16y) = 21 - 20 \]
This simplifies to:
\[ 15y - 16y = 1 \]
\[ -y = 1 \]

### Step 4: Solve for \( y \)
From \( -y = 1 \), we find:
\[ y = -1 \]

### Step 5: Substitute \( y \) back to find \( u \)
Now, we substitute \( y = -1 \) back into one of the original equations. Let's use the first equation:
\[ 4u + 5(-1) = 7 \]
This simplifies to:
\[ 4u - 5 = 7 \]
Adding 5 to both sides gives:
\[ 4u = 12 \]
Dividing both sides by 4 gives:
\[ u = 3 \]

### Step 6: Find \( x \) from \( u \)
Recall that \( u = \frac{1}{x} \). Therefore:
\[ \frac{1}{x} = 3 \]
Taking the reciprocal gives:
\[ x = \frac{1}{3} \]

### Final Answer
Thus, the solution for the equations is:
\[ x = \frac{1}{3}, \quad y = -1 \]