https://file.aiquickdraw.com/mathbot/file/17241924594232pipy5hf.jpg

The image shows a set of linear equations in the Cartesian plane, and it asks for the equations of the lines and to plot them. Let's solve the equations first, and then I can describe how the lines would look when plotted.

### Given Equations:
1. \( y = \frac{2}{3}x + \frac{2}{5} \)
2. \( y = \frac{1}{2}x + 4 \)
3. \( y = \frac{1}{3}x + 7 \)
4. \( y = \frac{1}{2}x - 2 \)
5. \( y + 1 = 0 \)  (This is equivalent to \( y = -1 \), which is a horizontal line.)
6. \( y + 3 = 7 \)  (Simplifying gives \( y = 4 \), another horizontal line.)

### Equations in slope-intercept form:
All the equations are already in slope-intercept form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

### Step 1: Identify Slopes and Intercepts

1. **Equation 1:** \( y = \frac{2}{3}x + \frac{2}{5} \)
   - Slope \( m = \frac{2}{3} \)
   - Y-intercept \( c = \frac{2}{5} \)

2. **Equation 2:** \( y = \frac{1}{2}x + 4 \)
   - Slope \( m = \frac{1}{2} \)
   - Y-intercept \( c = 4 \)

3. **Equation 3:** \( y = \frac{1}{3}x + 7 \)
   - Slope \( m = \frac{1}{3} \)
   - Y-intercept \( c = 7 \)

4. **Equation 4:** \( y = \frac{1}{2}x - 2 \)
   - Slope \( m = \frac{1}{2} \)
   - Y-intercept \( c = -2 \)

5. **Equation 5:** \( y = -1 \) 
   - Slope \( m = 0 \) (horizontal line)
   - Y-intercept \( c = -1 \)

6. **Equation 6:** \( y = 4 \)
   - Slope \( m = 0 \) (horizontal line)
   - Y-intercept \( c = 4 \)

### Step 2: Plot the Lines
Each line is plotted on a Cartesian plane based on its slope and y-intercept:

1. **Equation 1**: Starts at \( (0, \frac{2}{5}) \) and has a slope \( m = \frac{2}{3} \), so it rises 2 units for every 3 units it moves to the right.
2. **Equation 2**: Starts at \( (0, 4) \) with a slope \( m = \frac{1}{2} \), rising 1 unit for every 2 units to the right.
3. **Equation 3**: Starts at \( (0, 7) \) with a slope \( m = \frac{1}{3} \), rising 1 unit for every 3 units to the right.
4. **Equation 4**: Starts at \( (0, -2) \) with a slope \( m = \frac{1}{2} \).
5. **Equation 5**: Is a horizontal line at \( y = -1 \).
6. **Equation 6**: Is a horizontal line at \( y = 4 \).

### Visualization:
Since I can't draw directly here, you can imagine or plot the lines:

1. The first line passes through the point \( (0, \frac{2}{5}) \) and rises relatively steeply.
2. The second line passes through \( (0, 4) \) and rises less steeply.
3. The third line passes through \( (0, 7) \) and has a gentler slope.
4. The fourth line passes through \( (0, -2) \) and rises at the same angle as the second line.
5. The fifth and sixth lines are horizontal at \( y = -1 \) and \( y = 4 \), respectively.

If you want to see this graphically, you can use graphing software or even plot them by hand on graph paper.

Would you like me to provide more details or do you have any questions?

Here are some related questions to explore:

1. How do you determine the point of intersection between two lines?
2. What does it mean when two lines have the same slope?
3. How would the graph change if the slope is negative?
4. How do you find the distance between two parallel lines?
5. Can you determine the slope of a line given two points?

**Tip:** When plotting a line, start by marking the y-intercept and then use the slope to determine the next point on the line.

Learn how to solve linear equations in the Cartesian plane with detailed step-by-step solutions. This problem involves identifying slopes, y-intercepts, and plotting lines on the graph. Suitable for students in Grades 7-9.

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