What is an equation of the line that passes through the points left parenthesis, 2, comma, minus, 5, right parenthesis(2,−5) and left parenthesis, minus, 8, comma, 5, right parenthesis(−8,5)?

To find the equation of a line passing through two points, \((x_1, y_1)\) and \((x_2, y_2)\), we can use the point-slope form of the line equation. Here's the process:

### Step 1: Find the slope of the line
The slope \( m \) of a line passing through 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 \((2, -5)\) and \((-8, 5)\):

\[
m = \frac{5 - (-5)}{-8 - 2} = \frac{5 + 5}{-8 - 2} = \frac{10}{-10} = -1
\]

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

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

We can use either of the points. Let's use the point \((2, -5)\):

\[
y - (-5) = -1(x - 2)
\]

Simplifying:

\[
y + 5 = -1(x - 2)
\]

Distribute the \(-1\):

\[
y + 5 = -x + 2
\]

Subtract 5 from both sides:

\[
y = -x + 2 - 5
\]

\[
y = -x - 3
\]

### Final equation:
The equation of the line is:

\[
y = -x - 3
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:
1. What would be the equation if the points were different?
2. How do you find the equation of a vertical line?
3. How can you write the equation in standard form?
4. How does the slope of a line affect its graph?
5. How can you verify that the equation is correct by substituting the points?

**Tip:** To quickly identify the slope between two points, remember that a positive slope rises, and a negative slope falls.

What is an equation of the line that passes through the points (2,−5) and (−8,5)?

Learn how to find the equation of a line passing through the points (2,-5) and (-8,5). This problem covers essential algebraic concepts such as the slope formula and point-slope form, making it suitable for students in Grades 8-10.

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