https://file.aiquickdraw.com/mathbot/file/1727009734152lsacdhge.jpg

Let's analyze the given system of equations:

1. \( 48x - 72y = 30y + 24 \)
2. \( ry = \frac{1}{6} - 16x \)

We need to find the value of \( r \) for which the system has no solution.

### Step-by-Step Solution:

1. **Simplify the first equation:**
   \[
   48x - 72y = 30y + 24
   \]
   Combine like terms:
   \[
   48x - 102y = 24
   \]
   Dividing the entire equation by 6:
   \[
   8x - 17y = 4 \quad \text{(Equation 1)}
   \]

2. **Rearrange the second equation:**
   \[
   ry = \frac{1}{6} - 16x
   \]
   Multiply through by 6 to clear the fraction:
   \[
   6ry = 1 - 96x
   \]
   Rearrange to solve for \( y \):
   \[
   y = \frac{1 - 96x}{6r} \quad \text{(Equation 2)}
   \]

3. **Substitute \( y \) from Equation 2 into Equation 1:**
   \[
   8x - 17 \left( \frac{1 - 96x}{6r} \right) = 4
   \]
   Simplify:
   \[
   8x - \frac{17(1 - 96x)}{6r} = 4
   \]
   Multiply through by \( 6r \) to eliminate the denominator:
   \[
   48xr - 17(1 - 96x) = 24r
   \]
   Expand:
   \[
   48xr - 17 + 1632x = 24r
   \]
   Combine like terms:
   \[
   (48r + 1632)x = 24r + 17
   \]
   \[
   x = \frac{24r + 17}{48r + 1632}
   \]

4. **Determine the condition for no solution:**
   For the system to have no solution, the denominator must be zero, since it would lead to a division by zero, making \( x \) undefined. Set the denominator equal to zero:

   \[
   48r + 1632 = 0
   \]
   \[
   48r = -1632
   \]
   \[
   r = -34
   \]

Therefore, the value of \( r \) for which the system has no solution is \( \boxed{-34} \).

Would you like more details on any step or have any questions?

Here are some related questions to explore:
1. What happens if \( r = -34 \) in the system of equations?
2. How would you solve this system if \( r \) had a different value?
3. Can you find the solution for the system if \( r = 1 \)?
4. What geometric interpretation corresponds to a system with no solution?
5. What is the significance of parallel lines in the context of this problem?

**Tip:** When solving systems of equations, always consider simplifying one equation before substitution or elimination to reduce calculation complexity.

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

Learn how to determine the value of r in a system of equations where no solution exists. This algebra problem focuses on solving linear systems, simplifying equations, and understanding the condition for no solution. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation