Equation of the line that passes through the points (-3,8) and (5,-2)

To find the equation of the line passing through the points \((-3, 8)\) and \((5, -2)\), we follow these steps:

### Step 1: Find the slope \(m\)
The slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

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

Substituting the given points \((-3, 8)\) and \((5, -2)\):

\[
m = \frac{-2 - 8}{5 - (-3)} = \frac{-2 - 8}{5 + 3} = \frac{-10}{8} = -\frac{5}{4}
\]

### Step 2: Use the point-slope form of the line equation
The point-slope form is:

\[
y - y_1 = m(x - x_1)
\]

Using the slope \(m = -\frac{5}{4}\) and one of the points, say \((-3, 8)\):

\[
y - 8 = -\frac{5}{4}(x - (-3)) = -\frac{5}{4}(x + 3)
\]

### Step 3: Simplify to the slope-intercept form
Now, distribute the slope:

\[
y - 8 = -\frac{5}{4}(x + 3)
\]
\[
y - 8 = -\frac{5}{4}x - \frac{15}{4}
\]

Next, add 8 to both sides to isolate \(y\):

\[
y = -\frac{5}{4}x - \frac{15}{4} + 8
\]
Convert 8 to a fraction with a denominator of 4:

\[
y = -\frac{5}{4}x - \frac{15}{4} + \frac{32}{4}
\]
\[
y = -\frac{5}{4}x + \frac{17}{4}
\]

Thus, the equation of the line is:

\[
y = -\frac{5}{4}x + \frac{17}{4}
\]

### Final Equation
The equation of the line passing through the points \((-3, 8)\) and \((5, -2)\) is:

\[
y = -\frac{5}{4}x + \frac{17}{4}
\]

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Would you like more details or have any questions?

Here are 5 related questions:
1. How do you calculate the slope if the points are vertical?
2. How would the equation change if the slope were positive?
3. Can you derive the line equation in standard form?
4. How do you check if a point lies on the given line?
5. What if the line were horizontal, what would the equation look like?

**Tip:** Memorizing the point-slope form of a line can simplify the process of finding the equation of a line.

Discover how to find the equation of a line that passes through the points (-3,8) and (5,-2). This tutorial explores key concepts like slope, point-slope form, and slope-intercept form, providing a step-by-step solution for students in Grades 8-10.

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