Find an equation for the lines passing through the points (5,-1) (-5,5

To find the equation of the line passing through the points \((5, -1)\) and \((-5, 5)\), we can follow these steps:

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

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

Substituting the given points \((x_1, y_1) = (5, -1)\) and \((x_2, y_2) = (-5, 5)\):

\[
m = \frac{5 - (-1)}{-5 - 5} = \frac{5 + 1}{-10} = \frac{6}{-10} = -\frac{3}{5}
\]

### 2. Use the point-slope form of the equation of a line:
The point-slope form of a line's equation is:

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

Using the slope \(m = -\frac{3}{5}\) and the point \((5, -1)\):

\[
y - (-1) = -\frac{3}{5}(x - 5)
\]

Simplifying:

\[
y + 1 = -\frac{3}{5}(x - 5)
\]

### 3. Distribute and simplify:
\[
y + 1 = -\frac{3}{5}x + 3
\]

Subtract 1 from both sides to get the equation in slope-intercept form:

\[
y = -\frac{3}{5}x + 2
\]

### Final Equation:
The equation of the line passing through the points \((5, -1)\) and \((-5, 5)\) is:

\[
\boxed{y = -\frac{3}{5}x + 2}
\]

Would you like further details on this process or have any other questions?

### Related Questions:
1. How do you find the slope of a line given two points?
2. What is the point-slope form of a linear equation?
3. How do you convert the point-slope form to the slope-intercept form?
4. What does the slope of a line represent in a graph?
5. How can you determine if two lines are parallel?

### Tip:
Always double-check your slope calculation, as it's crucial for determining the correct equation of the line.

Learn how to find the equation of a line passing through the points (5, -1) and (-5, 5) using the slope formula and point-slope form. This problem covers key concepts in analytic geometry and linear equations, suitable for high school students.

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