The worksheet you’ve uploaded is focused on solving systems of equations by graphing. There are several systems given in different problems, and the goal is to graph the lines and identify their points of intersection as solutions.

Let’s work through the first couple of problems to clarify how to solve them:

Problem 1:

The system of equations is:

1. $y = \frac{2}{3}x - 1$
2. $y = x + 4$

To solve this by graphing:

1. Graph the first equation $y = \frac{2}{3}x - 1$:

   • The slope is $\frac{2}{3}$ (rise over run), and the y-intercept is -1. Start by plotting the point (0, -1), then from there, go up 2 units and over 3 units to find another point. Draw the line.

2. Graph the second equation $y = x + 4$:

   • The slope is 1, and the y-intercept is 4. Start by plotting the point (0, 4), then move up 1 unit and over 1 unit to the right to find another point. Draw the line.

3. Find the intersection: The solution will be the point where the two lines intersect. That point is the solution to the system.

Problem 2:

The system of equations is:

1. $x - y = 7$
2. $x - y = -4$

To graph these:

1. Rearrange both equations to the slope-intercept form $y = mx + b$.

   For the first equation $x - y = 7$:

   • Rearranging: $y = x - 7$. The slope is 1, and the y-intercept is -7.

   For the second equation $x - y = -4$:

   • Rearranging: $y = x + 4$. The slope is 1, and the y-intercept is 4.

2. Graph both lines:

   • The slopes are the same, meaning the lines are parallel. Since they have different y-intercepts, they will never intersect, so there is no solution to this system.

Would you like to continue with more of the problems or get more detailed solutions for these?

Here are five related questions to think about:

1. What is the significance of the slope in a linear equation when graphing?
2. How do you determine if a system has no solution, one solution, or infinitely many solutions?
3. What happens if two equations in a system have the same slope and y-intercept?
4. How would the graphing process change if one of the equations was a vertical or horizontal line?
5. How can you verify graphically if your solution is correct?

Tip: When graphing by hand, using a ruler or straightedge helps ensure your lines are accurate, which makes it easier to identify the point of intersection.