To find the equation of the line passing through the points $(1, 4)$ and $(-3, -2)$, we can use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

Where:

• $(x_1, y_1)$ is a point on the line.
• $m$ is the slope of the line.

Step 1: Find the slope

The formula for the slope $m$ 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 of the points $(1, 4)$ and $(-3, -2)$:

$m = \frac{-2 - 4}{-3 - 1} = \frac{-6}{-4} = \frac{3}{2}$

Step 2: Use the point-slope form

Now that we have the slope $m = \frac{3}{2}$, we can use one of the points (let's use $(1, 4)$) in the point-slope form equation:

$y - 4 = \frac{3}{2}(x - 1)$

Step 3: Simplify to slope-intercept form

Now, simplify the equation to get the slope-intercept form $y = mx + b$:

$y - 4 = \frac{3}{2}(x - 1)$ $y - 4 = \frac{3}{2}x - \frac{3}{2}$ $y = \frac{3}{2}x - \frac{3}{2} + 4$ $y = \frac{3}{2}x + \frac{5}{2}$

Thus, the equation of the line in slope-intercept form is:

$y = \frac{3}{2}x + \frac{5}{2}$

Would you like further details or have any questions?

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Here are five related questions:

1. How do you convert a linear equation from point-slope form to standard form?
2. How would the equation change if one point was at the origin?
3. Can you find the equation of a line parallel to this one passing through a different point?
4. How would you find the equation of a line perpendicular to this one?
5. What is the interpretation of the slope $\frac{3}{2}$ in real-world terms?

Tip: Always verify your slope calculation to avoid errors in the final equation.