Question

the table represents a linear equation.

x	y
-4	-11
-2	-6
6	14
10	24

which equation correctly uses point (-2, -6) to write the equation of this line in the point - slope form?
options:

1. $y - 6 = \frac{5}{2}(x - 2)$
2. $y + 6 = \frac{2}{5}(x + 2)$
3. $y - 6 = \frac{2}{5}(x - 2)$
4. $y + 6 = \frac{5}{2}(x + 2)$

Explanation:

Step1: Recall point - slope formula

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 point

We are given the point $(x_1,y_1)=(-2,-6)$. So, $x_1=-2$ and $y_1 = - 6$. Substituting these values into the point - slope formula, we get $y-(-6)=m(x - (-2))$, which simplifies to $y + 6=m(x + 2)$.

Step3: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points from the table, say $(-4,-11)$ and $(-2,-6)$.
Using the slope formula: $m=\frac{-6-(-11)}{-2 - (-4)}=\frac{-6 + 11}{-2 + 4}=\frac{5}{2}$.

Step4: Substitute the slope into the point - slope form

We found that $m = \frac{5}{2}$ and the point is $(-2,-6)$. Substituting $m=\frac{5}{2}$, $x_1=-2$ and $y_1=-6$ into the point - slope formula $y - y_1=m(x - x_1)$ (which we already simplified to $y + 6=m(x + 2)$), we get $y+6=\frac{5}{2}(x + 2)$.

Answer:

$y + 6=\frac{5}{2}(x + 2)$