Question

a geometry class is asked to find the equation of a line that is parallel to $y - 3 = -(x + 1)$ and passes through $(4, 2)$. trish states that the parallel line is $y - 2 = -1(x - 4)$, demetri states that the parallel line is $y = -x + 6$. are the students correct? explain. \bigcirc trish is the only student who is correct; the slope should be $-1$, and the line passes through $(4, 2)$. \bigcirc demetri is the only student who is correct; the slope should be $-1$, and the $y$-intercept is $6$. \bigcirc both students are correct; the slope should be $-1$, passing through $(4, 2)$ with a $y$-intercept of $6$. \bigcirc neither student is correct; the slope of the parallel line should be $1$.

Explanation:

Step1: Identify slope of given line

The given line is $y - 3 = -(x + 1)$, which is in point-slope form $y-y_1=m(x-x_1)$. The slope $m=-1$. Parallel lines have equal slopes, so the target line has slope $-1$.

Step2: Verify Trish's equation

Trish's equation is $y - 2 = -1(x - 4)$. This uses point-slope form with slope $-1$ and the point $(4,2)$, which matches the required conditions.

Step3: Convert to slope-intercept form

Simplify Trish's equation:
$y - 2 = -x + 4$
$y = -x + 4 + 2$
$y = -x + 6$
This is exactly Demetri's equation.

Step4: Confirm both equations are valid

Both forms represent the same line: Trish uses point-slope form, Demetri uses slope-intercept form, both with the correct slope and passing through $(4,2)$.

Answer:

Both students are correct; the slope should be -1, passing through (4, 2) with a y-intercept of 6.