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?

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

\[
\begin{cases}
y = 25x + 10 \quad \text{(1)} \\
y = c^2x + 2c \quad \text{(2)}
\end{cases}
\]

### 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.

Explore how to determine values of 'c' for which a system of equations with variables x and y has infinitely many solutions or no solutions. This problem involves analyzing coefficients and constants in linear equations, suitable for high school 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