A.6 Point-Slope Form

point slope form

How to write an equation of a line given 2 points

Point-Slope form: y-y₁=m(x-x₁)

is another way to find the equation of a line given 2 points

Multiple Choice

What is point slope form?

• $$y=mx+b$$  

• $$m=\frac{y_2-y_1}{x_2-x_1}$$  

• $$y-y_1=m\left(x-x_1\right)$$  

• $$y-x=m\left(x_1-y_1\right)$$  

Remember to keep all the signs & simplify

Point-Slope Form

​y - y₁ = m(x - x₁)

The SLOPE is easy to identify.

The POINT is a little trickier.

​

First, the y-coordinate (y₁) is first.

And the x-coordinate (x₁) is second.

So we have to reverse the numbers

​AND the sign for (x,y) is opposite.

​

(easier to explain with an example)

EXAMPLE

Given the equation: y - 4 = -2(x - 3), identify the slope and point of the line.​

​y - y₁ = m(x - x₁)

Multiple Select

Given the equation: $$y+9=\frac{2}{5}(x-5)$$ , identify the slope and point of the line.

• Slope:  $$\frac{2}{5}$$  

• Point: (5,-9)

• Point: (-5,-9)

• Point: (-5,9)

• Point: (5,9)

y-y₁= m(x - x₁)

m= 2

(1, 6)

m is your slope

1 is your x value

6 is your y value

Plug in your values

y- 6= 2(x - 1)

Multiple Choice

m= 5

(2, 4)

• y - 2= 5(x - 4)

• y - 4= 5(x - 2)

• y- 5 = 4(x - 2)

• y - 2 = 4(x - 5)

Multiple Choice

m= 3

(1, 7)

• y - 7 = 3( x - 1)

• y - 1 = 3( x - 7)

• y - 3 = 3( x - 1)

• y - 1 = 7( x - 3)

Multiple Choice

Which of these represents point-slope form of a linear equation?

• y - y₁ = m(x - x₁)

• Ax + By = C

• y = mx + b

• y=kx

Multiple Choice

m=4

(2, 2)

• y- 2 = 4( x - 2)

• y- 4=4(x - 2)

• y- 2= 2( x - 2)

• y + 2 = 4(x + 2)

Multiple Choice

m=6

(2, 7)

• y - 7 = 2( x - 6)

• y - 7 = 6( x - 2)

• y - 7 = 6( x + 2)

• y - 7 = 3( x - 6)

y - y₁= m(x - x₁)

When your COORDINATE has a negative value, you must change it to positive.

m=5

(-1, 2)

y - 2 = 5(x + 1)

Notice how at the end it was +1 because it was -1 as the x value.

Multiple Choice

Write the equation in point-slope form of the line that passes through the given point and has the given slope:

(-2, -1); m= -3

• y + 1 = -3(x - 2)

• y + 1 = -3(x + 2)

• y - 1 = 3(x - 1)

• y + 2 = -3(x + 1)

Multiple Choice

Write the equation in point-slope form of the line that passes through the given point and has the given slope.

(4, -7); m = -1/4

• y + 7 = -1/4(x - 4)

• y - 4 = -1/4(x + 7)

• y + 7 = 4(x - 4)

• y - 7 = -1/4(x - 4)

Multiple Choice

Write the equation in point-slope form of the line that passes through the given point and has the given slope:
 (-2, -1); m= -3

• y + 1 = -3(x - 2)

• y + 1 = -3(x + 2)

• y - 1 = 3(x - 1)

• y + 2 = -3(x + 1)

Let's take a look the other way...

If I am given an equations can I identify the slope and point?

Multiple Choice

The line with the equation

y + 1 = -3(x - 5),

contains the point (5, -1)

and has a slope of _______.

• -1

• 1

• -3

• 5

Multiple Choice

For the equation
y - 3 = 2(x + 4),
which of the following is true?

• The line has m = 2 passing through the point (4, -3)

• The line has m = 2 passing through the point (3, -4)

• The line has m = 2 passing through the point (-4, 3)

• The line has m = 2 passing through the point (-3, 4)

Multiple Choice

What is the slope of the line through the points (4, 0) and (-1, 10)?

• -10/3

• 3/10

• 2

• -2

Rewriting into Slope-Intercept Form

Start with Point-Slope Form: y - y₁ = m(x - x₁)

And rewrite it in Slope-Intercept Form: y = mx + b

Multiple Choice

Convert to slope-intercept form.

y - 3 = -2(x + 4)

• y = 2x - 5

• y = -2x - 11

• y = -2x - 5

• y = -2x + 1

Multiple Choice

Convert to slope-intercept form.

$$y-5=\frac{1}{3}\left(x-18\right)$$  

• $$y=\frac{1}{3}x-11$$

• $$y=\frac{1}{3}x-6$$  

• $$y=\frac{1}{3}x-1$$

• $$y=\frac{1}{3}x-23$$

• $$y=\frac{1}{3}x-13$$

Multiple Choice

A line has a slope of -6 and passes through the point (-2,3). Write the equation for the line in Slope-Intercept form.

• y = -6x - 16

• y = -6x - 20

• y = -6x + 16

• y = -6x - 9

• y = -6x - 15

Slope-Intercept Form

Multiple Choice

For the line given by:

$$y=4x+5$$  

• $$m=5$$  and  $$\left(0,4\right)$$  is the y-intercept

• $$m=4$$  and  $$\left(0,5\right)$$  is the y-intercept

• $$m=4$$  and  $$\left(5,0\right)$$  is the y-intercept

• $$m=5$$  and  $$\left(4,0\right)$$  is the y-intercept

Multiple Choice

For the line given by:

$$y=-x+9$$  

• $$m=0$$  and  $$\left(0,9\right)$$  is the y-intercept

• $$m=-1$$  and  $$\left(9,0\right)$$  is the y-intercept

• $$m=9$$  and  $$\left(0,-1\right)$$  is the y-intercept

• $$m=-1$$  and  $$\left(0,9\right)$$  is the y-intercept

Multiple Choice

For the line given by:

$$y=x-5$$  

• $$m=0$$  and  $$\left(0,-5\right)$$  is the y-intercept

• $$m=0$$  and  $$\left(-5,0\right)$$  is the y-intercept

• $$m=1$$  and  $$\left(0,-5\right)$$  is the y-intercept

• $$m=1$$  and  $$\left(-5,0\right)$$  is the y-intercept

Multiple Choice

For the line given by:

$$y=4x$$  

• $$m=4$$  and  $$\left(0,4\right)$$  is the y-intercept

• $$m=0$$  and  $$\left(0,4\right)$$  is the y-intercept

• $$m=4$$  and  $$\left(4,0\right)$$  is the y-intercept

• $$m=4$$  and  

   

Point-Slope Form

Multiple Choice

For the line given by: $$y-3=4\left(x-5\right)$$  

• $$m=4$$ and $$\left(-3,-5\right)$$  is a point on the line

• $$m=4$$  and  $$\left(3,5\right)$$  is a point on the line

• $$m=4$$  and  $$\left(5,3\right)$$  is a point on the line

• $$m=4$$  and  $$\left(-5,-3\right)$$  is a point on the line

Multiple Choice

For the line given by:

$$y+2=6\left(x+7\right)$$  

• $$m=6$$  and  $$\left(-7,-2\right)$$  is a point on the line

• $$m=6$$  and  $$\left(7,2\right)$$  is a point on the line

• $$m=6$$  and  $$\left(2,7\right)$$  is a point on the line

• $$m=6$$  and  $$\left(-2,-7\right)$$  is a point on the line

Multiple Choice

For the line given by:

$$y-8=x+9$$  

• $$m=0$$  and  $$\left(-9,8\right)$$  is a point on the line

• $$m=1$$  and  $$\left(-9,8\right)$$  is a point on the line

• $$m=1$$  and  $$\left(8,-9\right)$$  is a point on the line

• $$m=1$$  and  $$\left(9,-8\right)$$  is a point on the line

Multiple Choice

Recall: Which of these formulas helps you calculate the SLOPE of a line?

What if...they don't give you the slope???

Open Ended

What if we just had two points, (0, -2) and (3, 4). Do you think could we still figure out the point-slope form equation? If so, how?

Multiple Choice

Given points (0,-2) and (3,4), what's the correct way to calculate the slope?

• $$\frac{4-2}{3-0}=\frac{2}{3}$$

• $$\frac{4--2}{3-0}=\frac{6}{3}=2$$

• $$\frac{3-0}{4--2}=\frac{3}{6}=\frac{1}{2}$$

• $$\frac{3-0}{4-2}=\frac{3}{2}$$

Now that we found the slope, we can substitute m with 2:

y - y₁ = 2 (x - x₁)

Multiple Select

We were given the points points (0,-2) and (3,4). Which one do you want to plug in?

• (0,-2)

• (3,4)

Two given points: (0, -2) and (3, 4), so there are two possible equations! Both are correct. 😃

• y - -2 = 2 (x - 0)

• y - 4 = 2 (x - 3)

Summary: If you have two points...

• Calculate the slope the slope formula. This will be your m.

• Choose either of the two points to plug in for x₁ and y₁.