Question

what is the equation of a line with a slope of 4 that passes through point (-2,5)?
a. $y = \frac{-1}{4}x - 2$ b. $y = -4x + 5$ c. $y = 4x + 5$ d. $y = 4x + 13$

1. in a coordinate plane, the points (2,4) and (3,-1) are on a line. which of the following must be true?

a. the line crosses the x - axis.
b. the line passes through (0,0).
c. the line stays above the x - axis at all times.
d. the line is parallel to the y - axis.

1. which of the following equations represents the line that passes through the points (2, -6) and (-4, 3)?

a. $y = -\frac{3}{2}x - 7$ b. $y = -\frac{2}{3}x - 3$ c. $y = -\frac{3}{2}x - 3$ d. $y = -\frac{2}{3}x + \frac{1}{3}$

1. what is the equation of the line passing through the points (4, 2) and (6, 3)?

a. $y = -2x$ b. $y = 2x$ c. $y = \frac{1}{2}x$

Explanation:

Question 51

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. Here, $m = 4$ and $(x_1,y_1)=(-2,5)$.

Step2: Substitute values into point - slope form

Substitute $m = 4$, $x_1=-2$ and $y_1 = 5$ into $y - y_1=m(x - x_1)$:
$y-5 = 4(x+2)$

Step3: Simplify the equation

Expand the right - hand side: $y - 5=4x + 8$
Add 5 to both sides: $y=4x+8 + 5=4x + 13$

Step1: Find the slope of the line

The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. For the points $(2,4)$ and $(3,-1)$, $x_1 = 2,y_1 = 4,x_2=3,y_2=-1$.
So, $m=\frac{-1 - 4}{3 - 2}=\frac{-5}{1}=-5$.
The equation of the line using point - slope form with point $(2,4)$ is $y - 4=-5(x - 2)$.
Simplify: $y-4=-5x + 10$, so $y=-5x+14$.

Step2: Analyze each option

• Option A: To find where the line crosses the $x$ - axis, set $y = 0$: $0=-5x + 14\Rightarrow5x=14\Rightarrow x=\frac{14}{5}=2.8$. So the line crosses the $x$ - axis.
• Option B: Substitute $x = 0,y = 0$ into $y=-5x + 14$: $0

eq14$, so the line does not pass through $(0,0)$.

• Option C: When $x = 3$, $y=-1$, which is below the $x$ - axis, so the line does not stay above the $x$ - axis at all times.
• Option D: A line parallel to the $y$ - axis has an undefined slope, but our slope is - 5, so the line is not parallel to the $y$ - axis.

Step1: Find the slope

For the points $(2,-6)$ and $(-4,3)$, the slope $m=\frac{3-(-6)}{-4 - 2}=\frac{3 + 6}{-6}=\frac{9}{-6}=-\frac{3}{2}$.

Step2: Use point - slope form

Using the point $(2,-6)$ and $m =-\frac{3}{2}$ in $y - y_1=m(x - x_1)$:
$y+6=-\frac{3}{2}(x - 2)$

Step3: Simplify the equation

Expand: $y+6=-\frac{3}{2}x+3$
Subtract 6 from both sides: $y=-\frac{3}{2}x+3 - 6=-\frac{3}{2}x-3$

Answer:

D. $y = 4x+13$

Question 52