Finding the Equation of a Straight Line

This problem requires us to find the equation of a straight line given specific information: a point it passes through and its slope.

Understanding the Given Information

• Point: The line passes through the coordinates $(3, 4)$. We denote this point as $(x_1, y_1)$, so $x_1 = 3$ and $y_1 = 4$.
• Slope: The slope of the line is given as 3. We denote the slope as $m$, so $m = 3$.

Key Formula: The Point-Slope Form

The most direct way to find the equation of a line when you have a point and the slope is by using the point-slope form. The formula is:

$$y - y_1 = m(x - x_1)$$

Where:

• $(x_1, y_1)$ are the coordinates of the known point.
• $m$ is the slope of the line.
• $(x, y)$ represents any point on the line.

Step-by-Step Calculation

Let's substitute the given values into the point-slope formula:

1. Substitute the point $(3, 4)$ and slope $m = 3$:
   $$y - 4 = 3(x - 3)$$
2. Distribute the slope on the right side:
   Multiply the 3 by both terms inside the parentheses $(x - 3)$.
   $$y - 4 = 3 \cdot x - 3 \cdot 3$$ $$y - 4 = 3x - 9$$
3. Rearrange the equation into the standard form ($Ax + By + C = 0$):
   To get the standard form, we move all terms to one side of the equation. Subtract $y$ from both sides and add 4 to both sides.
   $$0 = 3x - y - 9 + 4$$ $$0 = 3x - y - 5$$ So, the equation of the straight line is $3x - y - 5 = 0$.

Matching the Correct Option

Now, we compare our derived equation, $3x - y - 5 = 0$, with the given options:

• Option 1: $3x - y - 5 = 0$
• Option 2: $3x - y + 9 = 0$
• Option 3: $x - 3y - 9 = 0$
• Option 4: $3x - y - 9 = 0$

Our calculated equation perfectly matches Option 1.