find an equation in slope-intercept form for the line that passes through the points (-3,-1) and (5,5)

To find the equation of a line in slope-intercept form \( y = mx + b \), we need the slope \( m \) and the y-intercept \( b \).

### Step 1: Find the slope \( m \)
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substitute the points \((-3, -1)\) and \((5, 5)\):
\[
m = \frac{5 - (-1)}{5 - (-3)} = \frac{5 + 1}{5 + 3} = \frac{6}{8} = \frac{3}{4}
\]

### Step 2: Use point-slope form to find \( b \)
Now that we know the slope is \( m = \frac{3}{4} \), use one of the points to find the y-intercept \( b \). Let's use \((5, 5)\).

Substitute the point \((5, 5)\) into the equation \( y = mx + b \):
\[
5 = \frac{3}{4}(5) + b
\]
Simplify:
\[
5 = \frac{15}{4} + b
\]
To solve for \( b \), subtract \( \frac{15}{4} \) from both sides:
\[
5 - \frac{15}{4} = b
\]
Convert \( 5 \) to a fraction with denominator 4:
\[
\frac{20}{4} - \frac{15}{4} = b
\]
Simplify:
\[
b = \frac{5}{4}
\]

### Step 3: Write the equation
Now that we know \( m = \frac{3}{4} \) and \( b = \frac{5}{4} \), the equation of the line is:
\[
y = \frac{3}{4}x + \frac{5}{4}
\]

Would you like further details or clarifications? Here are 5 related questions to explore this topic:

1. How do you convert slope-intercept form to standard form?
2. What is the significance of the slope in real-life applications?
3. How do you find the slope if the line is vertical or horizontal?
4. Can you calculate the equation of a line parallel to the one we just found?
5. How would you find the equation of a perpendicular line?

**Tip**: Always double-check your slope calculation, as small sign errors can lead to incorrect answers.

Learn how to find the equation of a line in slope-intercept form passing through the points (-3,-1) and (5,5). This detailed solution covers the slope formula, linear equations, and step-by-step calculations suitable for Grades 8-10.

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