https://r2.erweima.ai/mathbot/image/1726002238889qd3etjt1.jpg

Let's analyze the given equations from the image:

\[
\frac{x + y}{x \cdot y} = 5 \tag{1}
\]
\[
\frac{x + z}{x \cdot z} = 4 \tag{2}
\]
\[
\frac{y + z}{y \cdot z} = 7 \tag{3}
\]

The goal is to find \( y \). Let's solve these equations step by step.

### Step 1: Rearrange the equations
From the first equation, multiply both sides by \( x \cdot y \) to clear the fraction:
\[
x + y = 5x \cdot y
\]
This simplifies to:
\[
x + y = 5xy
\]
Rearranging it:
\[
5xy - x - y = 0 \tag{4}
\]

Similarly, from the second equation:
\[
x + z = 4x \cdot z
\]
This simplifies to:
\[
x + z = 4xz
\]
Rearranging it:
\[
4xz - x - z = 0 \tag{5}
\]

And from the third equation:
\[
y + z = 7y \cdot z
\]
This simplifies to:
\[
y + z = 7yz
\]
Rearranging it:
\[
7yz - y - z = 0 \tag{6}
\]

### Step 2: Solve the system of equations
Now, let's solve these equations to find \( y \). I will now perform the algebraic steps.The system of equations provides two solutions: 

1. \( x = 0 \), \( y = 0 \), and \( z = 0 \), which is trivial and does not apply here since \( y \) cannot be zero based on the context of the problem.

2. \( x = 1 \), \( y = \frac{1}{4} \), and \( z = \frac{1}{3} \).

Thus, the correct value of \( y \) is \( \frac{1}{4} \), which corresponds to option \( D \).

Would you like further clarification on the steps? Here are some related questions:

1. How can we check for extraneous solutions in this type of problem?
2. What would happen if any of the variables \( x \), \( y \), or \( z \) were zero?
3. Can we apply similar methods to non-linear equations?
4. How would this process change if the equations involved cube roots?
5. What is the significance of unique vs. multiple solutions?

**Tip:** When solving systems of equations, always check for any trivial or extraneous solutions that might arise!

Given the system of equations involving variables x, y, and z: 
\[
\frac{x + y}{x \cdot y} = 5,
\frac{x + z}{x \cdot z} = 4,
\frac{y + z}{y \cdot z} = 7
\]
The goal is to find the value of y.

Learn how to solve a system of rational equations involving three variables, x, y, and z. This problem covers key algebraic concepts such as rational equations and system of equations, guiding you through the process of solving for y. Suitable for high school students.

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