Find the equation of the line that passes through (1,9) and (7,-4)

To find the equation of the line that passes through the points \((1, 9)\) and \((7, -4)\), we can use the point-slope form of a line:

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

where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope of the line.

### Step 1: Find the Slope
The slope \(m\) is calculated using the formula:

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

Substituting the given points \((x_1, y_1) = (1, 9)\) and \((x_2, y_2) = (7, -4)\):

\[
m = \frac{-4 - 9}{7 - 1} = \frac{-13}{6}
\]

### Step 2: Use Point-Slope Form
Using the point-slope form with the point \((1, 9)\) and the slope \(m = \frac{-13}{6}\):

\[
y - 9 = \frac{-13}{6}(x - 1)
\]

### Step 3: Simplify to Slope-Intercept Form
Now, let's simplify this equation to the slope-intercept form \(y = mx + b\):

\[
y - 9 = \frac{-13}{6}x + \frac{13}{6}
\]

Adding 9 to both sides:

\[
y = \frac{-13}{6}x + \frac{13}{6} + 9
\]

Convert 9 to a fraction with a denominator of 6:

\[
y = \frac{-13}{6}x + \frac{13}{6} + \frac{54}{6}
\]

\[
y = \frac{-13}{6}x + \frac{67}{6}
\]

### Final Equation
So, the equation of the line in slope-intercept form is:

\[
y = \frac{-13}{6}x + \frac{67}{6}
\]

Would you like more details on any of these steps, or do you have any other questions?

Here are 5 related questions:
1. How would you find the equation of a line if given a different pair of points?
2. What is the significance of the slope in the context of a line?
3. How do you convert between point-slope form and slope-intercept form?
4. What does the y-intercept represent in the equation of a line?
5. How can you verify that a point lies on a given line?

**Tip:** Always double-check your slope calculation, as it's a common source of errors when finding the equation of a line.

Learn how to find the equation of a line passing through the points (1, 9) and (7, -4) using the point-slope form of a line. This problem involves calculating slope, applying the point-slope form, and converting it to slope-intercept form. Suitable for Grades 7-9 students learning about linear equations and slope.

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