College Linear Algebra: ### System of Linear Equations: Solutions Based on Parameter \( c \)Consider the system of linear equations:\[ x - 2y = \frac{1}{2} \]\[ -4x + cy = 3c \]Determine the values of \( c \) for which the system has: a) No solution b) Infinitely many solutions c) Exactly one solution#### Explanation:This problem involves analyzing a system of two linear equations to determine the values of parameter \( c \) that will yield different types of solutions. Here’s what each part requires:1. **No Solution**: - For the system to have no solution, the lines represented by the equations must be parallel but not coincident. This happens when the coefficients of \( x \) and \( y \) are proportional, but the constants on the right side are not proportional.2. **Infinitely Many Solutions**: - For the system to have infinitely many solutions, the equations must represent the same line. This occurs when both the coefficients of \( x \) and \( y \), and the constants are proportional.3. **Exactly One Solution**: - For the system to have exactly one solution, the lines must intersect at a single point. This happens when the lines are neither parallel nor coincident.By solving and analyzing the equations accordingly, one can determine the critical values of \( c \) for each case. Let's break down this analysis step-by-step:1. **Transform both equations to standard form for comparison**.2. **Compare coefficients to identify relationships between them**.3. **Substitute back to confirm values of \( c \)**.These steps ensure a thorough understanding of how different values of \( c \) affect the solution set of the system.

Name: College Linear Algebra: ### System of Linear Equations: Solutions Base
Uploaded: 2024-03-05T11:53:49.000Z
Channel: Bartleby
Description: College Linear Algebra: ### System of Linear Equations: Solutions Based on Parameter \( c \)Consider the system of linear equations:\[ x - 2y = \frac{1}{2} \]\[ -4x + cy = 3c \]Determine the values of \( c \) for which the system has: a) No solution