Category Archives: Equations

2.2 Lines, 2.4 Parallel and Perpendicular Lines, Finding the Equation of a Line

Finding the equation of a line perpendicular to another line

Example: Write the equation of a line in point-slope form passing through the point (-3,9) and perpendicular to the line whose equation is y={6/5}x+9/5.

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)

Although the slope of the line is not given, the slope can be deducted from the line being perpendicular to y={6/5}x+9/5.

Perpendicular lines have negative reciprocal slopes.  Since the slope of the given line is 6/5, the slope of the perpendicular line -5/6.

m=-{5/6} and  (-3,9)

Substitute the values into the point-slope formula.

y-9 = {-5/6}(x-(-3))

The point-slope form of the line is as follows.

y-9 = {-5/6}(x+3)

 

Example: Find the equation of a line perpendicular to another line and passing through a specific point. (The other line in slope intercept form)

Example: Find the equation of a line perpendicular to another line and passing through a specific point. (The other line in slope standard form)

Example: Find the equation of a line perpendicular to the x-axis.

Example: Find the equation of a line perpendicular to the x-axis and perpendicular to the y-axis.

 

2.2 Lines, 2.4 Parallel and Perpendicular Lines, Finding the Equation of a Line

Finding the Equation of a Line parallel to another line

Example: Find the equation of the line parallel to another line and passing through a specific point. (parallel equation in slope intercept form)

Example: Find the equation of the line parallel to another line and passing though a specific point. (parallel line in standard form)

Example: Find the equation of the line parallel to the x-axis or y-axis and passing through a specific point.

Example: What is an equation parallel to the y-axis?

Example: What is an equation parallel to the x-axis?

 

2.2 Lines, 2.3 Lines, Finding the Equation of a Line

Finding the Equation of a line given a fractional slope and a point

Example:  Find the equation of a line in slope intercept form given the slope of the line is -{2/3} and the line passes through the point (-4,7)

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=-{2/3} and  (-4,7)

Substitute the values into the formula.

 y-7 = -{2/3}(x-(-4))

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.  I will also use the clearing fractions method to avoid having to add fractions.

 y-7 = -{2/3}(x-(-4))

3(y-7) = 3[-{2/3}(x+4)] (Multiply by LCM)

3(y-7) = -2(x+4) (Cancel Denominator)

3y-21 = -2x-8

3y-21+21 = -2x-8+21

3y = -2x+13

{3y}/3 = {-2x}/3+{13}/3

y = -{2/3}x+{13}/3

The equation of a line in slope intercept form with a slope of -{2/3} and  passing through the point (-4,7) is   y = -{2/3}x+{13}/3

2.2 Lines, 2.3 Lines, Finding the Equation of a Line

Finding the Equation of a Line given two points on the line

Example:  Find the equation of a line in slope intercept form given the line passes through the two points (5,-3) and (6,-1).

Solution:

First find the slope of the line.

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, -3)  to be   ( x_1, y_1)  and choose   ( 6, -1) to be   ( x_2, y_2).

Substitute these values into the slope formula and simplify.

  m= {y_2-y_1} / {x_2-x_1} ={-1-(-3)}/{6-5}={-1+3}/{1} =2/1=2

The slope of the line containing the points   ( 5, -3) and   ( 6, -1)  is m= 2.

Then, use the point-slope formula of the line to start building the line.  m represents the slope of the line and you can use (x_1,y_1) or (x_2,y_2) as the point on the line.

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

m=2 and  (5,-3)

Substitute the values into the formula.

 y-(-3) = 2(x-5)

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) = 2(x-5)

 y+3 = 2x-10

 y+3-3 = 5x-10-3

 y = 5x-13

The equation of a line in slope intercept form passing through the two points (5,-3) and (6,-1) is  y = 5x-13.

2.2 Lines, 2.3 Lines, Finding the Equation of a Line

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.

2.2 Lines, 2.3 Lines, Slope

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.

 

2.2 Lines, 2.3 Lines, 4.1 Linear Functions and Their Properties, Slope

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.