What is the y-intercept of the line in the standard (x,y) coordinate plane that goes through the points (-3,6) and (3,2) ?

Options

A. 0
B. 2
C. 4
D. 6

Answer

To find the y-intercept of the line that passes through the points (-3, 6) and (3, 2), we will follow these steps:

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

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

Here, we can assign:
- \( (x_1, y_1) = (-3, 6) \)
- \( (x_2, y_2) = (3, 2) \)

Substituting these values into the formula:

\[ m = \frac{2 - 6}{3 - (-3)} = \frac{-4}{3 + 3} = \frac{-4}{6} = -\frac{2}{3} \]

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

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

Using the point \((-3, 6)\) and the slope \( m = -\frac{2}{3} \):

\[ y - 6 = -\frac{2}{3}(x - (-3)) \]

This simplifies to:

\[ y - 6 = -\frac{2}{3}(x + 3) \]

### Step 3: Distribute the slope
Now, distribute \(-\frac{2}{3}\):

\[ y - 6 = -\frac{2}{3}x - 2 \]

### Step 4: Solve for \( y \)
Add 6 to both sides to isolate \( y \):

\[ y = -\frac{2}{3}x - 2 + 6 \]

This simplifies to:

\[ y = -\frac{2}{3}x + 4 \]

### Step 5: Identify the y-intercept
In the equation \( y = mx + c \), \( c \) represents the y-intercept. From our equation, we see that:

\[ c = 4 \]

Thus, the y-intercept of the line is \( 4 \).

### Final Answer
The y-intercept of the line is \( 4 \).

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