Category Archives: MAC1105

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 Circles, 2.2 Lines, 2.3 Lines, Intercpets

X-intercepts and Y-intercepts

An x-intercept is where the graph touches or crosses the x-axis.

A y-intercept is where the graph touches of crosses the y-axis.

In this picture, the graph crosses the x-axis at the ordered pair (2, 0).  Since every ordered pair on the x-axis has a y coordinate of zero we can let y=0 to find x-intercepts.

To find an x-intercept: Let y=0 and solve for x.

In this picture, the graph crosses the y-axis at the ordered pair (0, 6).  Since every ordered pair on the y-axis has a x coordinate of zero we can let x=0 to find y-intercepts.

To find an y-intercept: Let x=0 and solve for y.

2.2 Lines, 2.3 Lines, Graphing

Graphing Linear Equations by Finding Intercepts

Steps for Graphing with the Intercept Method

  1. Find the x intercept and the y-intercept.
    • To find an x-intercept let y=0 and solve for x.
    • To find a y-intercept let x=0 and solve for y.
  2. Plot the x-intercept and y-intercept.
  3. Draw the line that connects the intercepts.

Example:   Graph the linear equation 2x-3y = 12

Solution:

 1. Find the x-intercept and the y-intercept.

To find an x-intercept: Let y=0 and solve for x.

2x-3y = 12

2x-3(0) = 12

2x-0 = 12

2x = 12

{2x}/2 = {12}/2

x = 6

The x-intercept of this equation is (6,0)

To find a y-intercept: Let x=0 and solve for y.

2x-3y = 12

2(0)-3y = 12

0-3y = 12

-3y = 12

{-3y}/{-3} = {12}/{-3}

y = -4

The y-intercept of this equation is (0,-4)

2. Plot the x-intercept and the y-intercept.

3. Draw the line that connects the intercepts.

 

Example:   Graph the linear equation y=3x

Solution:

 1. Find the x-intercept and the y-intercept.

To find an x-intercept: Let y=0 and solve for x.

y=3x

0=3x

{0}/3 = {3x}/3

0 = x

The x-intercept of this equation is (0,0)

To find a y-intercept: Let x=0 and solve for y.

y=3x

y=3(0)

y=0

The y-intercept of this equation is (0,0)

Since the x-intercept and the y-intercept are the same point and we need two distinct points to graph a line, we must find another ordered pair that is a solution to the equation.

Let x=1 and find the associated y value. (I chose x=1 but you could choose a different value)

y=3x

y=3(1)

y=3

Another ordered pair on the graph is (1,3)

2. Plot the x-intercept and the y-intercept.

3. Draw the line that connects the intercepts.

Example: Graphing a linear equation with intercepts.

Example: Graphing a linear equation with intercepts.

Example: Graphing a linear equation with intercepts.

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.

https://www.youtube.com/watch?v=1Cm7hjMUsrQ

Example: Finding the slope with the formula.

Example: Finding the slope from the graph.