Math Problem Statement

We have a system of equations with the variables x and y and a constant c.

open curly brackets table attributes columnalign left end attributes row cell y equals 25 x plus 10 end cell row cell y equals c squared x plus 2 c end cell end table close a) For what value of c does the system of equations have infinitely many solutions? b) For what value of c does the system of equations have no solutions?

Solution

To solve the problem, we'll analyze the system of equations given:

y = 25x + 10 \quad \text{(1)} \\ y = c^2x + 2c \quad \text{(2)} \end{cases}$$ ### Part (a): Finding the value of $$c$$ for infinitely many solutions For the system to have infinitely many solutions, the two equations must represent the same line. This occurs when the coefficients of $$x$$ and the constants in both equations are equal. **Step 1: Compare the coefficients of $$x$$ in both equations.** $$25 = c^2$$ This implies: $$c^2 = 25$$ Taking the square root on both sides: $$c = \pm 5$$ **Step 2: Compare the constant terms in both equations.** Substitute $$c = 5$$ and $$c = -5$$ into the constant term to ensure the equations are identical. - If $$c = 5$$, the second equation becomes: $$y = 25x + 10$$ which is identical to the first equation. - If $$c = -5$$, the second equation becomes: $$y = 25x - 10$$ which is **not** identical to the first equation. **Conclusion:** The system of equations has infinitely many solutions when $$c = 5$$. ### Part (b): Finding the value of $$c$$ for no solutions For the system to have no solutions, the lines must be parallel but not coincident, meaning they have the same slope but different y-intercepts. **Step 1: Ensure the slopes are equal but intercepts are different.** We already found that for the slopes to be equal: $$c^2 = 25 \quad \Rightarrow \quad c = \pm 5$$ **Step 2: Check the constant terms.** - If $$c = 5$$, the second equation is identical to the first, so it has infinitely many solutions, not no solutions. - If $$c = -5$$, the second equation becomes $$y = 25x - 10$$. The slopes are equal, but the y-intercepts differ (10 vs. -10). **Conclusion:** The system of equations has no solutions when $$c = -5$$. ### Final Answers: a) The system has infinitely many solutions when $$c = 5$$. b) The system has no solutions when $$c = -5$$. Would you like further details or have any questions? Here are five related questions: 1. How do you find the point of intersection between two lines? 2. What does it mean for two lines to be parallel? 3. How does the value of $$c$$ affect the graph of the line in this system? 4. Can two non-parallel lines ever have more than one point of intersection? 5. What is the general condition for a system of linear equations to be inconsistent? **Tip:** When solving systems of linear equations, always compare the coefficients of $$x$$ and the constants to determine the relationship between the lines represented by the equations.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Infinite Solutions
No Solutions

Formulas

-

Theorems

-

Suitable Grade Level

Grades 10-12

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