slope intercept form of (5,2) and   (-2,7)

To find the equation of a line in **slope-intercept form**, \( y = mx + b \), where:
- \( m \) is the slope
- \( b \) is the y-intercept,

we first calculate the slope, \( m \), using the two points \( (5, 2) \) and \( (-2, 7) \).

### Step 1: Calculate the slope
The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the coordinates \( (x_1, y_1) = (5, 2) \) and \( (x_2, y_2) = (-2, 7) \):

\[
m = \frac{7 - 2}{-2 - 5} = \frac{5}{-7} = -\frac{5}{7}
\]

### Step 2: Use the point-slope form to find the equation
Now, substitute the slope \( m = -\frac{5}{7} \) into the slope-intercept form \( y = mx + b \). To find \( b \), use one of the points. Let's use \( (5, 2) \):

\[
2 = -\frac{5}{7}(5) + b
\]

\[
2 = -\frac{25}{7} + b
\]

### Step 3: Solve for \( b \)
To isolate \( b \), add \( \frac{25}{7} \) to both sides:

\[
2 + \frac{25}{7} = b
\]

Convert 2 to a fraction with the same denominator:

\[
\frac{14}{7} + \frac{25}{7} = b
\]

\[
b = \frac{39}{7}
\]

### Step 4: Write the final equation
Now that we have the slope \( m = -\frac{5}{7} \) and the y-intercept \( b = \frac{39}{7} \), the equation of the line is:

\[
y = -\frac{5}{7}x + \frac{39}{7}
\]

Let me know if you want further details or have any questions!

Here are 5 related questions:
1. How can you verify if a point lies on a given line?
2. How do you convert the slope-intercept form to the standard form of a line?
3. What does the slope tell us about the direction of a line?
4. How would the equation change if the points were switched?
5. Can you find the equation of a parallel line passing through a different point?

**Tip**: The y-intercept, \( b \), represents the point where the line crosses the y-axis, which can help graphing quickly!

Discover how to find the slope-intercept form of a line passing through the points (5,2) and (-2,7). This problem demonstrates key algebraic concepts, including the slope formula and the slope-intercept form of a line. Perfect for students in Grades 8-10.

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