Write an equation for a line passing through the points (5,1) and (-1,3)

To find the equation of the line passing through the points \((5, 1)\) and \((-1, 3)\), we will use the same steps as before.

### 1. Calculate the Slope \(m\):

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

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

Substituting the points \((5, 1)\) and \((-1, 3)\):

\[
m = \frac{3 - 1}{-1 - 5} = \frac{2}{-6} = -\frac{1}{3}
\]

### 2. Write the Equation Using the Point-Slope Form:

Now, we'll use the point-slope form of the equation:

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

Using the slope \(m = -\frac{1}{3}\) and one of the points, say \((5, 1)\):

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

### 3. Convert to Slope-Intercept Form:

Now, expand and simplify to express the equation in slope-intercept form \(y = mx + b\):

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

Add 1 to both sides:

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

Convert 1 to a fraction with a denominator of 3:

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

Simplify:

\[
y = -\frac{1}{3}x + \frac{8}{3}
\]

So, the equation of the line is:

\[
y = -\frac{1}{3}x + \frac{8}{3}
\]

Would you like any more details or have any questions?

Here are 5 related questions:
1. How would you find the y-intercept from a given line equation?
2. How can you find the equation of a line perpendicular to this one?
3. What is the distance between the points \((5, 1)\) and \((-1, 3)\)?
4. Can you write the equation of this line in standard form \(Ax + By = C\)?
5. How do you determine if a point lies on a given line?

**Tip:** When converting to slope-intercept form, simplify fractions carefully to ensure accuracy.

Learn how to find the equation of a line passing through the points (5, 1) and (-1, 3) using the slope formula and point-slope form. This problem covers key geometric concepts related to linear equations, including slope calculation and converting to slope-intercept form. Suitable for students in Grades 7-9.

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