Category Archives: Finding the Equation of a Line

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.