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!