Of a line passes through the points (5, 3) and (8, -1), at what point will this line intersect the y - axis ?

Options

A. (0,8.33)
B. (0,8.67)
C. (0,9.00)
D. (0,9.67)

Explanation

To find the point where the line passing through the points (5, 3) and (8, -1) intersects the y-axis, we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point 1: \( (x_1, y_1) = (5, 3) \)
- Point 2: \( (x_2, y_2) = (8, -1) \)

### Step 2: Use the Two-Point Formula
The equation of a line passing through two points can be found using the two-point formula:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

### Step 3: Substitute the Values
Substituting the values from our points into the formula:
\[ y - 3 = \frac{-1 - 3}{8 - 5} (x - 5) \]
This simplifies to:
\[ y - 3 = \frac{-4}{3} (x - 5) \]

### Step 4: Distribute and Rearrange
Now, we will distribute and rearrange the equation:
\[ y - 3 = \frac{-4}{3}x + \frac{20}{3} \]
Adding 3 (which is \(\frac{9}{3}\)) to both sides gives:
\[ y = \frac{-4}{3}x + \frac{20}{3} + \frac{9}{3} \]
Combining the constants:
\[ y = \frac{-4}{3}x + \frac{29}{3} \]

### Step 5: Find the Y-Intercept
The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) into the equation:
\[ y = \frac{-4}{3}(0) + \frac{29}{3} \]
This simplifies to:
\[ y = \frac{29}{3} \]

### Step 6: Write the Intersection Point
Thus, the point where the line intersects the y-axis is:
\[ (0, \frac{29}{3}) \]
Calculating \(\frac{29}{3}\) gives approximately \(9.67\).

### Final Answer
The line intersects the y-axis at the point:
\[ (0, 9.67) \]