Determine the equation of the line that contain the points (-2.5,4) and (5,-2)

Options

A. x+7.5y=2
B. 6x=7.5y=2
C. 12x-15y=-4
D. 12X+15Y=30

Answer

To determine the equation of the line that contains the points (-2.5, 4) and (5, -2), we can follow these steps:

### Step 1: Identify the coordinates of the points
Let the points be:
- \( (x_1, y_1) = (-2.5, 4) \)
- \( (x_2, y_2) = (5, -2) \)

### Step 2: Calculate the slope (m) of the line
The slope \( m \) of the line passing through two points is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values:
\[ m = \frac{-2 - 4}{5 - (-2.5)} = \frac{-6}{5 + 2.5} = \frac{-6}{7.5} \]
To simplify:
\[ m = \frac{-6}{7.5} = \frac{-6 \times 2}{7.5 \times 2} = \frac{-12}{15} = \frac{-4}{5} \]

### Step 3: Use the point-slope form of the equation of a line
The point-slope form is given by:
\[ y - y_1 = m(x - x_1) \]
Using point \( (-2.5, 4) \) and the slope \( m = -\frac{4}{5} \):
\[ y - 4 = -\frac{4}{5}(x - (-2.5)) \]
This simplifies to:
\[ y - 4 = -\frac{4}{5}(x + 2.5) \]

### Step 4: Distribute the slope
Distributing the slope on the right side:
\[ y - 4 = -\frac{4}{5}x - \frac{4}{5} \times 2.5 \]
Calculating \( -\frac{4}{5} \times 2.5 \):
\[ -\frac{4}{5} \times 2.5 = -\frac{4 \times 2.5}{5} = -\frac{10}{5} = -2 \]
So the equation becomes:
\[ y - 4 = -\frac{4}{5}x - 2 \]

### Step 5: Solve for y
Adding 4 to both sides:
\[ y = -\frac{4}{5}x - 2 + 4 \]
This simplifies to:
\[ y = -\frac{4}{5}x + 2 \]

### Step 6: Convert to standard form
To convert to standard form \( Ax + By = C \):
Multiply through by 5 to eliminate the fraction:
\[ 5y = -4x + 10 \]
Rearranging gives:
\[ 4x + 5y = 10 \]

Thus, the equation of the line that contains the points (-2.5, 4) and (5, -2) is:
\[ 4x + 5y = 10 \]

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