The y-intercept of the line through the two points whose coordinates are (5,-2) and (1,3) is

Options

A. `-(5)/(4)`
B. `(5)/(4)`
C. `(17)/(4)`
D. 7

Explanation

To find the y-intercept of the line passing through the points (5, -2) and (1, 3), we can follow these steps:

### Step 1: Identify the points
The two points given are:
- Point 1: \( (x_1, y_1) = (5, -2) \)
- Point 2: \( (x_2, y_2) = (1, 3) \)

### Step 2: Calculate the slope (m)
The formula for the slope \( m \) of a line through two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values:
\[ m = \frac{3 - (-2)}{1 - 5} = \frac{3 + 2}{1 - 5} = \frac{5}{-4} = -\frac{5}{4} \]

### Step 3: Use the point-slope form of the line
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Using point \( (5, -2) \) and the slope we calculated:
\[ y - (-2) = -\frac{5}{4}(x - 5) \]
This simplifies to:
\[ y + 2 = -\frac{5}{4}(x - 5) \]

### Step 4: Distribute and rearrange
Now, distribute the slope on the right side:
\[ y + 2 = -\frac{5}{4}x + \frac{25}{4} \]
Next, subtract 2 from both sides:
\[ y = -\frac{5}{4}x + \frac{25}{4} - 2 \]
Convert 2 into quarters:
\[ 2 = \frac{8}{4} \]
So we have:
\[ y = -\frac{5}{4}x + \frac{25}{4} - \frac{8}{4} \]
Combine the constants:
\[ y = -\frac{5}{4}x + \frac{17}{4} \]

### Step 5: Identify the y-intercept
The equation of the line is now in the slope-intercept form \( y = mx + c \), where \( c \) is the y-intercept. Here, \( c = \frac{17}{4} \).

### Final Answer
The y-intercept of the line is \( \frac{17}{4} \).

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