Let’s solve the problems on the page systematically:

1. Which shows the solution for the following system of equations?

Equations:

1. $y = -4x + 6$
2. $x + 4y = 8$

Solution Steps:

1. Rewrite the second equation in slope-intercept form:
   $x + 4y = 8 \quad \Rightarrow \quad 4y = -x + 8 \quad \Rightarrow \quad y = -\frac{1}{4}x + 2$ So, the equations are: $y = -4x + 6 \quad \text{and} \quad y = -\frac{1}{4}x + 2$

2. These lines intersect, and the solution is the point where the two lines meet.
   Upon examining the graphs, Option B correctly shows the intersection point.

2. Solve the system of equations by graphing.

Equations:

1. $y = \frac{5}{2}x - 5$
2. $x - \frac{3}{2}y = 2$

Solution Steps:

1. Rewrite the second equation in slope-intercept form:
   $x - \frac{3}{2}y = 2 \quad \Rightarrow \quad -\frac{3}{2}y = -x + 2 \quad \Rightarrow \quad y = \frac{2}{3}x - \frac{4}{3}$

   So the system is:
   $y = \frac{5}{2}x - 5 \quad \text{and} \quad y = \frac{2}{3}x - \frac{4}{3}$

2. These two lines intersect at the point $(2, 5)$.
   The correct answer is B. (2, 5).

3. Which point is NOT a solution to the system of equations?

The given equations are:

1. $x + 3y = 12$
2. $\frac{-2}{3}x + 8$

Solution Steps:

1. Substitute each point into both equations to determine if it satisfies the system.

   • Point A $(-3, 10)$:
     Substituting into $x + 3y = 12$:
     $-3 + 3(10) = -3 + 30 = 27 \quad \text{(NOT true)}$ So, $(-3, 10)$ is NOT a solution.

   The correct answer is A. (-3, 10).

Let me know if you'd like a more detailed breakdown of any of these!

Here are five related questions to expand your understanding:

1. How do you solve systems of equations algebraically versus graphically?
2. What is the importance of slope-intercept form in graphing equations?
3. How can you verify the solution to a system of equations using substitution?
4. When do systems of equations have no solution or infinitely many solutions?
5. What are some real-life applications of solving systems of equations?

Tip: Always convert equations to slope-intercept form ($y = mx + b$) for easier graphing.