Writing the Equation of a Line Given Slope and a Point

Q1. Find an equation of the line with slope $m = \frac{3}{2}$ that passes through the point $(8, -5)$. Write the equation in the form $Ax + By = C$.

Background

Topic: Linear Equations (Point-Slope and Standard Form)

This question tests your ability to write the equation of a line when given a slope and a point, and then convert it to standard form ($Ax + By = C$).

Key Terms and Formulas

• Slope ($m$): The steepness of the line, given as $\frac{3}{2}$.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

• Standard Form: $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ should be positive.

Step-by-Step Guidance

1. Start by writing the point-slope form of the equation using the given point $(8, -5)$ and slope $m = \frac{3}{2}$:

   $y - (-5) = \frac{3}{2}(x - 8)$

2. Simplify $y - (-5)$ to $y + 5$:

   $y + 5 = \frac{3}{2}(x - 8)$

3. Distribute the slope $\frac{3}{2}$ to both terms inside the parentheses:

   $y + 5 = \frac{3}{2}x - \frac{3}{2} \times 8$

4. Calculate $\frac{3}{2} \times 8$ and write the equation with all terms shown:

   $y + 5 = \frac{3}{2}x - 12$

5. To write the equation in standard form $Ax + By = C$, move all terms involving $x$ and $y$ to one side and constants to the other. Multiply both sides by 2 to clear fractions if needed.

Try solving on your own before revealing the answer!