Question

what is an equation of the line that passes through the points (3, 6) and (-3, 4)?

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)=(3, 6)\) and \((x_2, y_2)=(-3, 4)\). So \( m=\frac{4 - 6}{-3 - 3}=\frac{-2}{-6}=\frac{1}{3} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Let's use the point \((3, 6)\). Substitute \( m = \frac{1}{3} \), \( x_1 = 3 \), and \( y_1 = 6 \) into the formula: \( y - 6=\frac{1}{3}(x - 3) \).

Step3: Simplify the equation

Expand the right - hand side: \( y - 6=\frac{1}{3}x-1 \). Then add 6 to both sides: \( y=\frac{1}{3}x + 5 \). We can also write it in standard form \( x-3y=-15 \) (by multiplying both sides by 3: \( 3y=x + 15\), then \( x-3y=-15 \)) or in other equivalent forms.

Answer:

The equation of the line is \( y=\frac{1}{3}x + 5 \) (or \( x - 3y=-15 \) or other equivalent linear equations)