Question

Use the given conditions to write an equation for the line in​ point-slope form and in​ slope-intercept form.
Passing through left parenthesis 8 comma 6 right parenthesis with​ x-intercept negative 2
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Part 1
Write an equation for the line in​ point-slope form.
  
enter your response here
​(Simplify your answer. Use integers or fractions for any numbers in the​ equation.)

Answer 1

To find the equation of the line in point-slope form, we need the slope of the line and a point on the line. We are given the point \( (8, 6) \) and the x-intercept, which is \( -2 \).

### Step 1: Find the slope
The x-intercept is the point where the line crosses the x-axis, which is at \( (-2, 0) \).

To find the slope \( m \), we use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):

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

Using the points \( (8, 6) \) and \( (-2, 0) \):

\[
m = \frac{0 - 6}{-2 - 8} = \frac{-6}{-10} = \frac{3}{5}
\]

### Step 2: Write the equation in point-slope form
Point-slope form of a line is given by the equation:

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

Using the point \( (8, 6) \) and the slope \( \frac{3}{5} \), we get:

\[
y - 6 = \frac{3}{5}(x - 8)
\]

This is the equation of the line in point-slope form.

### Part 1 Final Answer
\[
y - 6 = \frac{3}{5}(x - 8)
\]

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### Step 3: Write the equation in slope-intercept form
Now, we'll convert the point-slope form into slope-intercept form \( y = mx + b \).

Starting with:

\[
y - 6 = \frac{3}{5}(x - 8)
\]

Distributing the slope on the right side:

\[
y - 6 = \frac{3}{5}x - \frac{3}{5} \cdot 8
\]
\[
y - 6 = \frac{3}{5}x - \frac{24}{5}
\]

Now, add 6 to both sides to solve for \( y \):

\[
y = \frac{3}{5}x - \frac{24}{5} + 6
\]

Convert 6 to a fraction with a denominator of 5:

\[
6 = \frac{30}{5}
\]

Now combine the fractions:

\[
y = \frac{3}{5}x - \frac{24}{5} + \frac{30}{5}
\]
\[
y = \frac{3}{5}x + \frac{6}{5}
\]

### Part 2 Final Answer
\[
y = \frac{3}{5}x + \frac{6}{5}
\]

In summary:
- Point-slope form: \( y - 6 = \frac{3}{5}(x - 8) \)
- Slope-intercept form: \( y = \frac{3}{5}x + \frac{6}{5} \)