https://file.aiquickdraw.com/mathbot/file/17301446545049yr55d7w.jpg

To determine the value of \( p \) such that the system has no solution, we should analyze the conditions for inconsistency in a system of linear equations.

The given system is:
\[
6 + 7r = pw
\]
\[
7r - 5w = 11
\]

Let's rewrite both equations in standard form (Ax + By = C) to identify the conditions for no solution, which occurs when the lines represented by the equations are parallel.

### Step 1: Rewrite the Equations
1. From the first equation:
   \[
   7r - pw = -6
   \]

2. The second equation is already in standard form:
   \[
   7r - 5w = 11
   \]

So the system now is:
\[
7r - pw = -6
\]
\[
7r - 5w = 11
\]

### Step 2: Conditions for No Solution
For the system to have no solution, the two equations must represent parallel lines. This happens when their coefficients of \( r \) and \( w \) are proportional, but their constants are not.

1. For the coefficients to be proportional, we need:
   \[
   \frac{7}{7} = \frac{-p}{-5} \Rightarrow p = 5
   \]

2. Substitute \( p = 5 \) and check the constants:
   - The modified first equation with \( p = 5 \) becomes:
     \[
     7r - 5w = -6
     \]
   - The second equation is:
     \[
     7r - 5w = 11
     \]

Since the left-hand sides are the same but the right-hand sides are different (\(-6 \neq 11\)), the system has no solution when \( p = 5 \).

### Answer
The value of \( p \) that makes the system inconsistent (no solution) is:
\[
p = 5
\]

Would you like a deeper breakdown of any steps, or do you have any questions?

---

Here are 5 related questions:
1. What would be the solution if the system had a unique solution?
2. How can we interpret this system graphically?
3. How do we identify parallel lines in a general linear system?
4. What other values of \( p \) could cause inconsistency if the system were different?
5. How can the elimination method be applied to verify parallelism?

**Tip:** Inconsistent systems occur when the lines represented by the equations are parallel but have different intercepts.

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 that makes a system of equations inconsistent. This problem explores linear systems, parallel lines, and conditions for no solution, suitable for high school math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation