https://file.aiquickdraw.com/mathbot/file/1727643202000k1de8zp6.jpg

The image shows a system of two equations where \( p \) is a constant, and the system has no solution. We are asked to find the value of \( p \).

Here is the system of equations from the image:

\[
\frac{7}{8}y - \frac{5}{8}x = \frac{4}{7} + \frac{7}{8}y
\]
\[
\frac{5}{4}x + \frac{7}{4} = py + \frac{15}{4}
\]

### Step 1: Simplify the first equation

The first equation is:
\[
\frac{7}{8}y - \frac{5}{8}x = \frac{4}{7} + \frac{7}{8}y
\]

We can subtract \( \frac{7}{8}y \) from both sides to eliminate \( y \) from the right-hand side:
\[
-\frac{5}{8}x = \frac{4}{7}
\]

Now, solve for \( x \):
\[
x = \frac{4}{7} \times \left(-\frac{8}{5}\right) = -\frac{32}{35}
\]

So, \( x = -\frac{32}{35} \).

### Step 2: Substitute \( x \) into the second equation

The second equation is:
\[
\frac{5}{4}x + \frac{7}{4} = py + \frac{15}{4}
\]

Substitute \( x = -\frac{32}{35} \) into this equation:
\[
\frac{5}{4} \times \left(-\frac{32}{35}\right) + \frac{7}{4} = py + \frac{15}{4}
\]

Simplify the left-hand side:
\[
-\frac{160}{140} + \frac{7}{4} = py + \frac{15}{4}
\]
\[
-\frac{8}{7} + \frac{7}{4} = py + \frac{15}{4}
\]

To combine the terms on the left-hand side, convert them to a common denominator (28):
\[
-\frac{32}{28} + \frac{49}{28} = \frac{17}{28}
\]

Now the equation becomes:
\[
\frac{17}{28} = py + \frac{15}{4}
\]

### Step 3: Condition for no solution

For the system to have no solution, the two equations must represent parallel lines, meaning their slopes must be equal. The slopes are determined by the coefficients of \( x \) and \( y \) in both equations.

From the first equation, the slope is:
\[
\text{slope} = \frac{5}{8}
\]

From the second equation, the slope is related to the coefficient of \( y \), which is \( p \).

Thus, for the system to have no solution, the slopes must not be equal:
\[
\frac{5}{8} \neq p
\]

Thus, \( p \) cannot be \( \frac{5}{8} \), and that is the condition under which the system has no solution.

### Final Answer: \( p = \frac{5}{8} \)

Would you like further clarification on any step or additional details?

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Here are 5 related questions that expand upon the topic:
1. How do you determine if a system of equations has no solution?
2. What does it mean for two lines to be parallel in terms of slopes?
3. Can a system have infinitely many solutions? If yes, under what conditions?
4. How would the solution change if the system had exactly one solution?
5. What role does the determinant of a matrix play in solving systems of linear equations?

**Tip:** When dealing with systems of linear equations, always check the coefficients of variables to determine the nature of the solution.

In the given system of equations, p is a constant. If the system has no solution, what is the value of p?

Learn how to find the value of p in a system of linear equations where the system has no solution. This problem demonstrates key algebraic concepts related to slopes, parallel lines, and systems of equations. Suitable for students in Grades 9-12.

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