Question

which point on the y - axis is on the line that passes through point r and is parallel to line pq? \\(\bigcirc\\ (0, \frac{2}{3})\\) \\(\bigcirc\\ (0, 4)\\) \\(\bigcirc\\ (\frac{2}{3}, 0)\\) \\(\bigcirc\\ (4, 0)\\)

Explanation:

Step1: Find slope of line PQ

First, identify coordinates: $P(-3, -3)$, $Q(3, -1)$. Slope formula: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$
$\text{slope of } PQ = \frac{-1 - (-3)}{3 - (-3)} = \frac{2}{6} = \frac{1}{3}$

Step2: Use slope for parallel line

Parallel lines have equal slopes, so the line through $R(1,1)$ has slope $\frac{1}{3}$. Use point-slope form $y - y_1 = m(x - x_1)$
$y - 1 = \frac{1}{3}(x - 1)$

Step3: Find y-intercept (x=0)

Substitute $x=0$ into the line equation:
$y - 1 = \frac{1}{3}(0 - 1)$
$y - 1 = -\frac{1}{3}$
$y = 1 - \frac{1}{3} = \frac{2}{3}$

Answer:

$(0, \frac{2}{3})$