A line passes through the points (-4,11) and (5,8). Write an equation of the line.

To write the equation of a line that passes through two points, we use the **slope-intercept form** of the equation of a line:

\[
y = mx + b
\]

Where:
- \( m \) is the slope of the line.
- \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \)).

### Step 1: Calculate the slope \( m \)

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

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

For the points \((-4, 11)\) and \((5, 8)\):
\[
m = \frac{8 - 11}{5 - (-4)} = \frac{-3}{9} = -\frac{1}{3}
\]

### Step 2: Use the point-slope form to find the equation of the line

We can use the point-slope form of the equation of a line:

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

Substitute the slope \( m = -\frac{1}{3} \) and one of the points, say \((-4, 11)\):

\[
y - 11 = -\frac{1}{3}(x + 4)
\]

### Step 3: Simplify to the slope-intercept form

Now simplify the equation:

\[
y - 11 = -\frac{1}{3}(x + 4)
\]
\[
y - 11 = -\frac{1}{3}x - \frac{4}{3}
\]
\[
y = -\frac{1}{3}x - \frac{4}{3} + 11
\]

Convert 11 to a fraction with a denominator of 3:

\[
y = -\frac{1}{3}x - \frac{4}{3} + \frac{33}{3}
\]
\[
y = -\frac{1}{3}x + \frac{29}{3}
\]

### Final Answer:
The equation of the line is:

\[
y = -\frac{1}{3}x + \frac{29}{3}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you find the equation of a line given the slope and a point?
2. What is the equation of a line parallel to \( y = -\frac{1}{3}x + \frac{29}{3} \)?
3. How do you calculate the y-intercept of a line when given two points?
4. How can you verify if a point lies on a given line?
5. How does the slope of a line relate to its steepness?

**Tip:** When working with fractions in linear equations, it helps to convert all constants to the same denominator before simplifying.

Learn how to find the equation of a line passing through the points (-4, 11) and (5, 8). This solution demonstrates the use of the slope formula and slope-intercept form, suitable for students in Grades 9-11.

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