Questions: Find an equation of the line with the given slope that passes through the given point. Write the equation in the form Ax+By=C. m=3/2,(9,-5) The equation of the line in the form Ax+By=C is □ (Simplify your answer. Use integers or fractions for any numbers in the equation.)

$Find an equation of the line with the given slope that passes through the given point. Write the equation in the form Ax+By=C. m=3/2,(9,-5) The equation of the line in the form Ax+By=C is □ (Simplify your answer. Use integers or fractions for any numbers in the equation.)$

Transcript text: Find an equation of the line with the given slope that passes through the given point. Write the equation in the form $\mathrm{Ax}+\mathrm{By}=\mathrm{C}$. \[ m=\frac{3}{2},(9,-5) \] The equation of the line in the form $A x+B y=C$ is $\square$ (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Solution

Solution Steps

Step 1: Identify the Given Information

We are given the slope $ m = \frac{3}{2} $ and a point $ (9, -5) $ through which the line passes.

Step 2: Use the Point-Slope Form

The point-slope form of the equation of a line is given by: \[ y - y_1 = m(x - x_1) \] Substituting the values: \[ y - (-5) = \frac{3}{2}(x - 9) \] This simplifies to: \[ y + 5 = \frac{3}{2}(x - 9) \]

Step 3: Distribute and Rearrange

Distributing the slope on the right side: \[ y + 5 = \frac{3}{2}x - \frac{27}{2} \] Next, we isolate $ y $: \[ y = \frac{3}{2}x - \frac{27}{2} - 5 \] Converting $ 5 $ to a fraction: \[ 5 = \frac{10}{2} \] Thus, we have: \[ y = \frac{3}{2}x - \frac{27}{2} - \frac{10}{2} \] This simplifies to: \[ y = \frac{3}{2}x - \frac{37}{2} \]

Step 4: Convert to Standard Form

To convert to standard form $ Ax + By = C $, we rearrange the equation: \[ \frac{3}{2}x - y = \frac{37}{2} \] Multiplying through by $ 2 $ to eliminate the fractions: \[ 3x - 2y = 37 \]