Category Archives: Equations

Finding the Equation of a Line given the slope and a point

Example:  Find the equation of a line in slope intercept form given the slope of the line is 7 and the line passes through the point (2,-3)

Solution:

Use the point-slope formula of the line to start building the line.  m represents the slope of the line and (x_1,y_1) is a point on the line.

Point-slope formula: y-y_1 = m(x-x_1)

m=7 and  (2,-3)

Substitute the values into the formula.

 y-(-3) = 7(x-2)

Since the instructions ask to write the equation in slope intercept form (y=mx+b) we will simplify and write the equation with y by itself on one side.

 y-(-3) = 7(x-2)

 y+3 = 7x-14

 y+3-3 = 7x-14-3

 y = 7x-17

The equation of a line in slope intercept form with a slope of 7 and  passing through the point (2,-3) is y = 7x-17.

Example: Find the equation of the line.

Calculating Slope

Given two points on the line   ( x_1, y_1) and   ( x_2, y_2), you can calculate the slope of a line by the following formula.

  m= {y_2-y_1} / {x_2-x_1}

  y_2-y_1 is also know as   Delta y or “the change in y.”

  x_2-x_1 is also know as   Delta x or “the change in x.”

 

  m= {y_2-y_1} / {x_2-x_1} ={Delta y} /{Delta x} ={rise}/{run} 

Example: Calculate the slope of the line containing the points   ( 5, 7) and   ( 9, 10).

Solution: Choose one of the points to be   ( x_1, y_1) and choose the other point to be   ( x_2, y_2).

I will choose   ( 5, 7)  to be   ( x_1, y_1)  and choose   ( 9, 10) to be   ( x_2, y_2).

Substitute these values into the slope formula and simplify.

  m= {y_2-y_1} / {x_2-x_1} ={10-7}/{9-5} =3/4

The slope of the line containing the points   ( 5, 7) and   ( 9, 10)  is m= 3/4.

Example: Calculate the slope of the line containing the points   ( -7, -2) and   ( 8, 8).

Solution: Choose one of the points to be   ( x_1, y_1) and choose the other point to be   ( x_2, y_2).

I will choose   ( -7, -2)  to be   ( x_1, y_1)  and choose   ( 8, 8) to be   ( x_2, y_2).

Substitute these values into the slope formula and simplify.

  m= {y_2-y_1} / {x_2-x_1} ={8-(-2)}/{8-(-7)}={8+2}/{8+7} =10/15=2/3

The slope of the line containing the points   ( -7, -2) and   ( 8, 8)  is m= 2/3.

Example: Finding the slope with the formula.

Example: Finding the slope with the formula.

Example: Finding the slope from the graph.

 

Interpretation of slope

The slope of a line is a number that indicates the “steepness” of a line.  Slope is usually denoted with the letter m.

If the slope of the line is positive, the line will be rising or increasing from left to right.

     

All three of the above graphs have a positive slope and the line is rising or increasing from left to right.  Notice as the value of the slope gets larger, the line is getting steeper.

If the slope of the line is negative, the line will be falling or decreasing from left to right.

   

All three of the above graphs have a negative slope and the line is falling or decreasing from left to right.  Notice as the value of the slope gets smaller, the line is getting steeper.

If the slope is zero, the line will be constant.  This results in a horizontal line.

     

All three of the the above graphs have a slope of zero.  The y values are constant.

A vertical line has a slope that is undefined.

   

All three of the vertical lines have undefined slope.