The task asks to find linear systems of equations for three different graphs (a, b, and c), and to determine the solution sets for each system.

Step-by-step Approach:

1. Graph (a):

   • There are two lines, which intersect at the point $(-1, 2)$.
   • First line: Slope is negative, passing through points like $(0, 1)$ and $(1, 0)$. This gives the equation of the form $y = -x + 1$.
   • Second line: Slope is positive, passing through points like $(0, -2)$ and $(1, 0)$, resulting in $y = 2x - 2$.
   • The system is: y = -x + 1 \\ y = 2x - 2 \end{cases}$$
   • The solution is $(-1, 2)$, where the two lines intersect.

2. Graph (b):

   • The lines appear to be parallel, meaning they do not intersect (no solution).
   • First line: The slope is negative, passing through $(0, 4)$ and $(1, 3)$, giving the equation $y = -x + 4$.
   • Second line: Also has a negative slope, passing through $(0, 2)$ and $(1, 1)$, which results in $y = -x + 2$.
   • The system is: y = -x + 4 \\ y = -x + 2 \end{cases}$$
   • Since the lines are parallel, there is no solution.

3. Graph (c):

   • The lines coincide, meaning every point on one line is also on the other (infinite solutions).
   • First line: Slope is positive, passing through $(0, 2)$ and $(2, 4)$, giving the equation $y = x + 2$.
   • Second line: It is the same line as the first one, so the equation is also $y = x + 2$.
   • The system is: y = x + 2 \\ y = x + 2 \end{cases}$$
   • The solution is infinite as the lines are identical.

Summary of Solution Sets:

• a) The solution is $(-1, 2)$.
• b) There is no solution.
• c) There are infinitely many solutions.

Would you like further explanation on any of these graphs or concepts?

Related Questions:

1. How do you find the slope of a line from a graph?
2. How can you determine if two lines are parallel by just looking at their equations?
3. What does it mean when a system of equations has no solution?
4. How do we algebraically solve a system of linear equations with two variables?
5. What is the geometric meaning of an infinite number of solutions?

Tip:

Parallel lines have the same slope but different y-intercepts, which is why they never intersect.