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We are given the equation:

\[
-7(px + q) = \frac{42}{23}x + \frac{35}{9}
\]

where \( p \) and \( q \) are constants, and the equation has infinitely many solutions. We need to find the value of \( q \) given that \( p < 0 \).

### Step 1: Condition for Infinitely Many Solutions
For a linear equation to have infinitely many solutions, the two sides of the equation must be identical for all values of \( x \). This means the coefficients of \( x \) on both sides must be equal, and the constant terms on both sides must also be equal.

### Step 2: Expanding and Simplifying the Equation
Expand the left-hand side of the equation:

\[
-7(px + q) = -7p x - 7q
\]

Now, rewrite the equation:

\[
-7p x - 7q = \frac{42}{23}x + \frac{35}{9}
\]

### Step 3: Equating the Coefficients of \( x \)
To have infinitely many solutions, the coefficients of \( x \) on both sides must be equal. So, we set the coefficients of \( x \) equal to each other:

\[
-7p = \frac{42}{23}
\]

Solving for \( p \):

\[
p = \frac{-42}{7 \times 23} = \frac{-6}{23}
\]

### Step 4: Equating the Constant Terms
Next, equate the constant terms on both sides of the equation:

\[
-7q = \frac{35}{9}
\]

Solving for \( q \):

\[
q = \frac{-35}{7 \times 9} = \frac{-5}{9}
\]

Thus, the value of \( q \) is:

\[
q = \frac{-5}{9}
\]

### Conclusion:
The value of \( q \) is \( \frac{-5}{9} \).

Would you like further clarification on any steps, or do you have additional questions?

### Relative Questions:
1. How do we solve linear equations with infinitely many solutions?
2. What conditions must be met for a system of linear equations to have infinitely many solutions?
3. How can we verify that two expressions are equivalent for all values of \( x \)?
4. Why is it important to isolate variables when solving equations?
5. What is the significance of the condition \( p < 0 \) in this problem?

### Tip:
Always double-check the coefficients of both sides of an equation when solving for conditions like infinitely many solutions.

The given equation, where p and q are constants, has infinitely many solutions. If p < 0, what is the value of q?

Learn how to solve a linear equation with infinitely many solutions, where p < 0. This algebraic problem involves expanding and simplifying the equation, equating coefficients, and solving for constants p and q. Suitable for students in Grades 9-12.

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