Question

What is the equation of the line that passes through the points (−2, 1) and (1, 10)?

Answer 1

Slope-Intercept form: y=3x+7

Standard form: 3x-y=-7

Point-slope form: y-1=3(x+2)

Explanation:

Slope-Intercept form:

First, find the slope, using the formula:
`m=(y_(2)-y_(1) )/(x_(2)-x_(1) )`

Our x₁ and y₁ will be the point (-2,1) and our x₂ and y₂ wwill be the point (1,10).

So let's write those in our equation to find slope:

`m=(10-1)/(1-(-2))=(9)/(3)=3`

Therefore, our slope is 3.

Now let's write our linear equation with what we have already in slope-intercept form:

y=3x+b

Well, we still need to find the y-intercept, or "b".

Plug in one of your points for the x and y values of the equation. We'll use the point (-2,1)

`y=3x+b\\1=3(-2)+b\\1=-6+b\\1+6=-6+6+b\\7=b`

This means our y-intercept is 7. Now we can write our equation in slope-intercept form completely:

y=3x+7

Standard form:

Now, let's find this equation is standard form.

Take your equation in slope-intercept form and write it out again:

`y=3x+7`

Now, standard form of a linear equation is ax+by=c, so subtract 3x from both sides:

`y-3x=3x-3x+7\\-3x+y=7`

The "a" coefficient in standard form cannot be negative, so divide the entire equation by -1:

`(-3x+y)/(-1)=(7)/(-1)\\3x-y=-7`

Therefore, your equation in standad form is:

3x-y=-7

Point-Slope form:

The formula for point-slope form is y-y₁=m(x-x₁). We already know that our x₁ and y₁ is the point (-2,1) and we know that our slope, m, is 3, so we just have to plug then in where the fit in the equation.

x₁ is -2 and y₁ is 1 and m is 3, so:

y-1=3(x-(-2)) or y-1=3(x+2)

That means our equation in point-slope form is:

y-1=3(x+2)