Question

1. write an equation for a line that passes through point \\((3,5)\\) with a slope of \\(-\frac{2}{3}\\). graph the equation.

what is the equation in general form?

Explanation:

⚡ Using what you learned: Writing Equations of Linear Functions

Step 1: Write the equation in point-slope form

Using the point-slope formula \( y - y_1 = m(x - x_1) \) with point \( (3, 5) \) and slope \( m = -\frac{2}{3} \):

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

Step 2: Convert to general form

Multiply the entire equation by \( 3 \) to clear the fraction:

$$ 3(y - 5) = -2(x - 3) $$

$$ 3y - 15 = -2x + 6 $$

Move all terms to one side to write in general form \( Ax + By + C = 0 \):

$$ 2x + 3y - 21 = 0 $$

Step 3: Identify key points for graphing

Start at the given point \( (3, 5) \).

Using the slope \( m = -\frac{2}{3} \) (down \( 2 \) units, right \( 3 \) units):

• Next point: \( (3 + 3, 5 - 2) = (6, 3) \)

Going in the opposite direction (up \( 2 \) units, left \( 3 \) units):

• Previous point: \( (3 - 3, 5 + 2) = (0, 7) \)

Plot these points—\( (0, 7) \), \( (3, 5) \), and \( (6, 3) \)—on the grid and draw a straight line through them.

Answer:

The equation of the line in General form is:

$$ 2x + 3y - 21 = 0 $$

To graph the equation, plot the points \( (0, 7) \), \( (3, 5) \), and \( (6, 3) \) on the coordinate plane and connect them with a straight line.