Question

A line passes through the points (-21, -22) and (-14, -16). Find this line's equation in point-slope form. Using the point (-21, -22), this line's point-slope form equation is: Using the point (-14, -16), this line's point-slope form equation is:the point slope form has to be simplified

Answer 1

Using point (-21, -22): y = (6/7)x - 4

Using point (-14, -16): y = (6/7)x - 4

Step-by-step explanation:

The point-slope form of a line's equation is:

`y-y_1=m(x-x_1)`

Where (x₁, y₁) is a point in the line and m is the slope.

The slope of a line can be calculated as:

`m=(y_2-y_1)/(x_2-x_1)`

Where (x₁, y₁) and (x₂, y₂) are two points in the line. So, replacing (x₁, y₁) by (-21, -22) and (x₂, y₂) by (-14, -16), we get that the slope of the line is:

`m=(-16-(-22))/(-14-(-21))=(-16+22)/(-14+21)=(6)/(7)`

Now, using the point (-21, -22), we get that the equation of the line is:

`\begin{gathered} y-(-22)=(6)/(7)(x-(-21)) \\ y+22=(6)/(7)(x+21) \end{gathered}`

Then, we can simplify the equation as:

`\begin{gathered} y+22=(6)/(7)(x)+(6)/(7)(21) \\ y+22=(6)/(7)x+18 \\ y+22-22=(6)/(7)x+18-22 \\ y=(6)/(7)x-4 \end{gathered}`

On the other hand, using the point (-14, -16), the equation of the line is:

`\begin{gathered} y-(-16)=(6)/(7)(x-(-14)) \\ y+16=(6)/(7)(x+14) \end{gathered}`

Simplifying, we get:

`\begin{gathered} y+16=(6)/(7)x+(6)/(7)(14) \\ y+16=(6)/(7)x+12 \\ y+16-16=(6)/(7)x+12-16 \\ y=(6)/(7)x-4 \end{gathered}`

So, the answers are:

Using point (-21, -22): y = (6/7)x - 4

Using point (-14, -16): y = (6/7)x - 4