Question

quiz instructions
question 1
write the equation of a line through (3,1), parallel to $y = \frac{2}{3}x + 1$
$y = \square x + \square$
question 2
write the equation of a line through (-3,5), parallel to $y = -x$
$y = \square x + \square$

Explanation:

Question 1

Step1: Identify parallel line slope

Parallel lines have equal slopes. The given line is $y=\frac{2}{3}x + 1$, so slope $m=\frac{2}{3}$.

Step2: Use point-slope form

Point-slope formula: $y - y_1 = m(x - x_1)$. Substitute $(x_1,y_1)=(3,1)$ and $m=\frac{2}{3}$:
$y - 1 = \frac{2}{3}(x - 3)$

Step3: Simplify to slope-intercept form

Expand and isolate $y$:
$y - 1 = \frac{2}{3}x - 2$
$y = \frac{2}{3}x - 2 + 1$
$y = \frac{2}{3}x - 1$

Question 2

Step1: Identify parallel line slope

The given line is $y=-x$, which is $y=-1x+0$, so slope $m=-1$.

Step2: Use point-slope form

Substitute $(x_1,y_1)=(-3,5)$ and $m=-1$ into $y - y_1 = m(x - x_1)$:
$y - 5 = -1(x - (-3))$

Step3: Simplify to slope-intercept form

Expand and isolate $y$:
$y - 5 = -x - 3$
$y = -x - 3 + 5$
$y = -x + 2$

Answer:

Question 1: $y=\frac{2}{3}x + (-1)$
Question 2: $y=-1x + 2$