The graph of a line in the xy-plane passes through the points `(5,4) and (3, (1)/(2)).` Which of the following equations describes the line ?

Options

A. `y = 7/4x + (19)/(4)`
B. `y =- 7/4x - (19)/(4)`
C. `y = 7/4x - (19)/(4)`
D. `y = 4/6 x + 8/7`

Explanation

To find the equation of the line that passes through the points (5, 4) and (3, 1/2), we will follow these steps:

### Step 1: Identify the points
We have two points:
- Point 1 (x1, y1) = (5, 4)
- Point 2 (x2, y2) = (3, 1/2)

### Step 2: Calculate the slope (m)
The formula for the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by:
\[ m = \frac{y2 - y1}{x2 - x1} \]

Substituting the values:
\[ m = \frac{\frac{1}{2} - 4}{3 - 5} \]

Calculating the numerator:
\[ \frac{1}{2} - 4 = \frac{1}{2} - \frac{8}{2} = \frac{1 - 8}{2} = \frac{-7}{2} \]

Calculating the denominator:
\[ 3 - 5 = -2 \]

Now substituting back into the slope formula:
\[ m = \frac{\frac{-7}{2}}{-2} = \frac{-7}{2} \times \frac{-1}{2} = \frac{7}{4} \]

### Step 3: Use the slope-intercept form
The slope-intercept form of a line is given by:
\[ y = mx + c \]
where m is the slope and c is the y-intercept.

### Step 4: Find the y-intercept (c)
We can use one of the points to find c. Let's use point (5, 4):
\[ 4 = \frac{7}{4} \cdot 5 + c \]

Calculating:
\[ 4 = \frac{35}{4} + c \]

To isolate c, we subtract \(\frac{35}{4}\) from both sides:
\[ c = 4 - \frac{35}{4} \]
Convert 4 to a fraction:
\[ c = \frac{16}{4} - \frac{35}{4} = \frac{16 - 35}{4} = \frac{-19}{4} \]

### Step 5: Write the equation of the line
Now that we have both m and c, we can write the equation of the line:
\[ y = \frac{7}{4}x - \frac{19}{4} \]

### Final Answer
The equation that describes the line is:
\[ y = \frac{7}{4}x - \frac{19}{4} \]

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