Two points in a straight line have the coordinates (-3,8) and (6,-4). Find the equation of the line and the intercepts. Equation: y = ? x + ?   a) y - intercept = ?   b) slope = ?   c) x - intercept = ?

To find the equation of the line that passes through the points \((-3, 8)\) and \((6, -4)\), we need to follow these steps:

### Step 1: Find the slope (m)
The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Substituting the given points \((-3, 8)\) and \((6, -4)\):

\[
m = \frac{-4 - 8}{6 - (-3)} = \frac{-12}{9} = -\frac{4}{3}
\]

### Step 2: Find the equation of the line
Using the point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\), and substituting one of the points \((-3, 8)\) and the slope \(m = -\frac{4}{3}\):

\[
y - 8 = -\frac{4}{3}(x + 3)
\]

Now, distribute and simplify:

\[
y - 8 = -\frac{4}{3}x - 4
\]

Add 8 to both sides:

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

So, the equation of the line is:

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

### Step 3: Find the intercepts

**a) \(y\)-intercept:**  
The \(y\)-intercept is the value of \(y\) when \(x = 0\).

Substitute \(x = 0\) into the equation:

\[
y = -\frac{4}{3}(0) + 4 = 4
\]

So, the \(y\)-intercept is \(4\).

**b) Slope:**  
We already calculated the slope \(m = -\frac{4}{3}\).

**c) \(x\)-intercept:**  
The \(x\)-intercept is the value of \(x\) when \(y = 0\).

Set \(y = 0\) in the equation and solve for \(x\):

\[
0 = -\frac{4}{3}x + 4
\]

\[
\frac{4}{3}x = 4
\]

\[
x = \frac{4 \times 3}{4} = 3
\]

So, the \(x\)-intercept is \(3\).

### Summary of the results:
- **Equation:** \( y = -\frac{4}{3}x + 4 \)
- **a) \(y\)-intercept:** \(4\)
- **b) Slope:** \( -\frac{4}{3} \)
- **c) \(x\)-intercept:** \(3\)

Would you like further details or have any other questions? 

Here are 5 related questions you might consider:
1. How do you derive the point-slope form of a linear equation?
2. Can you explain the difference between slope-intercept form and standard form?
3. What is the significance of the slope in a real-world context?
4. How do you find the equation of a line parallel to a given line?
5. How would the equation change if the line were vertical?

**Tip:** Remember that the sign of the slope indicates the direction of the line; a negative slope means the line is decreasing as it moves from left to right.

Discover how to find the equation of a line that passes through the points (-3,8) and (6,-4). This problem covers key concepts in coordinate geometry, including calculating the slope, finding the y-intercept, and determining the x-intercept. Ideal for students in Grades 8-10.

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