Question

which equation represents a line that passes through (2, -1/2) and has a slope of 3?
○ y - 2 = 3(x + 1/2)
○ y - 3 = 2(x + 1/2)
○ y + 1/2 = 3(x - 2)
○ y + 1/2 = 2(x - 3)

Explanation:

Step1: Recall point - slope form

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

Step2: Identify the given values

We are given that the line passes through the point $(x_1,y_1)=(2,-\frac{1}{2})$ and has a slope $m = 3$.

Step3: Substitute into point - slope form

Substitute $x_1 = 2$, $y_1=-\frac{1}{2}$ and $m = 3$ into the point - slope form $y - y_1=m(x - x_1)$.
We get $y-(-\frac{1}{2})=3(x - 2)$, which simplifies to $y+\frac{1}{2}=3(x - 2)$.
We can also check the slope of each option:

• For the first option $y - 2=3(x+\frac{1}{2})$, the slope is 3, but the point on the line would be $(-\frac{1}{2},2)$ (by comparing with $y - y_1=m(x - x_1)$), which is not the given point.
• For the second option $y - 3=2(x+\frac{1}{2})$, the slope is 2, not 3, so we can eliminate it.
• For the fourth option $y+\frac{1}{2}=2(x - 3)$, the slope is 2, not 3, so we can eliminate it.

Answer:

$y+\frac{1}{2}=3(x - 2)$ (the third option)