What is the y-intercept of the line through points (3,-2) and (-1,6) ?

Answer

To find the y-intercept of the line through the points (3, -2) and (-1, 6), we can follow these steps:

### Step 1: Identify the points
The two points given are:
- Point 1: (3, -2) which we will denote as (x1, y1)
- Point 2: (-1, 6) which we will denote as (x2, y2)

### Step 2: Use the slope formula
The slope (m) of the line passing through two points is calculated using the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]
Substituting the values:
\[ m = \frac{6 - (-2)}{-1 - 3} = \frac{6 + 2}{-1 - 3} = \frac{8}{-4} = -2 \]

### Step 3: Write the point-slope form of the line
Using the point-slope form of the equation of a line:
\[ y - y1 = m(x - x1) \]
Substituting the slope and one of the points (let's use (3, -2)):
\[ y - (-2) = -2(x - 3) \]
This simplifies to:
\[ y + 2 = -2(x - 3) \]

### Step 4: Distribute and simplify
Distributing the -2:
\[ y + 2 = -2x + 6 \]
Now, subtract 2 from both sides:
\[ y = -2x + 6 - 2 \]
This simplifies to:
\[ y = -2x + 4 \]

### Step 5: Identify the y-intercept
In the equation of the line \(y = mx + c\), the y-intercept is represented by \(c\). From our equation:
\[ y = -2x + 4 \]
We can see that the y-intercept \(c\) is 4.

### Final Answer
The y-intercept of the line is **4**.

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