https://file.aiquickdraw.com/mathbot/file/1727730847677rf1urtom.jpg

Let's work through each part of the problem using the graph shown.

### Part (a): Write a point-slope form of the equation for the line shown.
The point-slope form of a line is given by the formula:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \((x_1, y_1)\) is a point on the line.
- \(m\) is the slope of the line.

From the graph, we can see two points on the line:
- Point 1: \((-5, 4)\)
- Point 2: \((5, 1)\)

First, calculate the slope \(m\) using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 4}{5 - (-5)} = \frac{-3}{10}
\]
So, the slope \(m = -\frac{3}{10}\).

Using the point \((-5, 4)\), the point-slope form is:
\[
y - 4 = -\frac{3}{10}(x + 5)
\]

### Part (b): Estimate the value of the y-intercept of the line.
To estimate the y-intercept, look at where the line crosses the y-axis. The line appears to cross around \(y \approx 2.5\).

### Part (c): Use proportional reasoning to support your conjecture about the value of the y-intercept.
We can use the slope to calculate the y-intercept more precisely. From the point-slope form of the equation in part (a):
\[
y - 4 = -\frac{3}{10}(x + 5)
\]
To convert this to slope-intercept form (\(y = mx + b\)), we simplify:
\[
y - 4 = -\frac{3}{10}x - \frac{3}{2}
\]
\[
y = -\frac{3}{10}x + 4 - \frac{3}{2}
\]
\[
y = -\frac{3}{10}x + \frac{8}{2} - \frac{3}{2} = -\frac{3}{10}x + \frac{5}{2}
\]
Thus, the y-intercept \(b = \frac{5}{2} = 2.5\), which agrees with the estimate.

### Part (d): Rewrite the point-slope form of the equation in slope-intercept form.
We already converted it in part (c):
\[
y = -\frac{3}{10}x + 2.5
\]

Would you like more details on any part?

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### Related Questions:
1. How do you calculate the slope of a line from two points?
2. How can we derive the y-intercept from the equation of a line?
3. How does the point-slope form relate to the slope-intercept form?
4. What are the benefits of using the point-slope form in certain cases?
5. How would the equation change if the slope were positive?

**Tip:** Remember that converting between forms of a line equation (like point-slope and slope-intercept) can give different insights, such as the slope and y-intercept of the line!

Use the graph of the line shown. a. Write a point-slope form of the equation for the line shown. b. Estimate the value of the y-intercept of the line. c. Use proportional reasoning to support your conjecture about the value of the y-intercept. d. Rewrite the point-slope form of the equation in slope-intercept form to check your conjecture.

Learn how to find the equation of a line using point-slope form and slope-intercept form. This problem involves calculating the slope and y-intercept of a line, graphing, and converting between forms of linear equations. Suitable for students in Grades 8-10.

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