Find the equation of the line passing through the point (2,-3) and having its slope `5/4` :

Options

A. 4x-5y=20
B. 3x-2y=5
C. 5x-5y=20
D. none of these

Answer

To find the equation of the line passing through the point (2, -3) with a slope of \( \frac{5}{4} \), we can use the point-slope form of the equation of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

where \( m \) is the slope and \( (x_1, y_1) \) is the point through which the line passes.

### Step 1: Identify the given values
- The slope \( m = \frac{5}{4} \)
- The point \( (x_1, y_1) = (2, -3) \)

### Step 2: Substitute the values into the point-slope formula
Substituting the values into the formula, we have:

\[ y - (-3) = \frac{5}{4}(x - 2) \]

This simplifies to:

\[ y + 3 = \frac{5}{4}(x - 2) \]

### Step 3: Distribute the slope on the right side
Now, we distribute \( \frac{5}{4} \) on the right side:

\[ y + 3 = \frac{5}{4}x - \frac{5}{4} \cdot 2 \]

Calculating \( \frac{5}{4} \cdot 2 \):

\[ \frac{5}{4} \cdot 2 = \frac{10}{4} = \frac{5}{2} \]

So we have:

\[ y + 3 = \frac{5}{4}x - \frac{5}{2} \]

### Step 4: Isolate \( y \)
Next, we isolate \( y \) by subtracting 3 from both sides:

\[ y = \frac{5}{4}x - \frac{5}{2} - 3 \]

To combine the constants, we convert 3 to a fraction with a denominator of 2:

\[ 3 = \frac{6}{2} \]

So:

\[ y = \frac{5}{4}x - \frac{5}{2} - \frac{6}{2} \]

This simplifies to:

\[ y = \frac{5}{4}x - \frac{11}{2} \]

### Step 5: Convert to standard form
To convert this to standard form \( Ax + By = C \), we can multiply through by 4 to eliminate the fraction:

\[ 4y = 5x - 22 \]

Rearranging gives us:

\[ -5x + 4y = -22 \]

Multiplying through by -1 to make the leading coefficient positive:

\[ 5x - 4y = 22 \]

### Final Equation
Thus, the equation of the line is:

\[ 5x - 4y = 22 \]