Math Problem Statement

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

Solution

To determine the value of pp 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=pw6 + 7r = pw 7r5w=117r - 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: 7rpw=67r - pw = -6

  2. The second equation is already in standard form: 7r5w=117r - 5w = 11

So the system now is: 7rpw=67r - pw = -6 7r5w=117r - 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 rr and ww are proportional, but their constants are not.

  1. For the coefficients to be proportional, we need: 77=p5p=5\frac{7}{7} = \frac{-p}{-5} \Rightarrow p = 5

  2. Substitute p=5p = 5 and check the constants:

    • The modified first equation with p=5p = 5 becomes: 7r5w=67r - 5w = -6
    • The second equation is: 7r5w=117r - 5w = 11

Since the left-hand sides are the same but the right-hand sides are different (611-6 \neq 11), the system has no solution when p=5p = 5.

Answer

The value of pp that makes the system inconsistent (no solution) is: p=5p = 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 pp 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.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Inconsistency in Systems
Parallel Lines

Formulas

For no solution: Proportional coefficients with different constants

Theorems

Conditions for Parallel Lines in Linear Systems

Suitable Grade Level

Grades 10-12