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In this unit you will learn how to write the equation of a line given specific information such as the slope and a point on the line or two points that lie ...
Typology: Exercises
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In this unit you will learn how to write the equation of a line given specific information such as the slope and a point on the line or two points that lie on the line. You will also investigate the graphs and equations of parallel and perpendicular lines.
Linear Equations in Two Variables Parallel and Perpendicular Lines
Point-slope form : y โ y 1 (^) = m x ( โ x 1 ) x 1 and y 1 represent the coordinates of the given point.
To write a linear equation:
Given the slope ( m ) and a point ( x 1 (^) , y 1 ) 1.) Use the point-slope form: y โ y 1 (^) = m x ( โ x 1 ). 2.) Substitute the given slope for m and the given point for x 1 and y 1. 3.) Solve for y.
Example #1 : Find the equation of a line whose slope is 4 and goes through the point (2, 3). 1.) Use the point slope form: y โ y 1 (^) = m x ( โ x 1 ). 2.) Replace m with the slope 4, y 1 with 3, and x 1 with 2. 3.) Solve for y. 3 4( 2) 3 4 8 4 5
y x y x y x
The equation of the line that has a slope of 4 and passes through the point (2, 3) is y = 4 x โ 5. Example #2 : Find the equation of a line with slope = 34 and passes through the point (โ12, 2). 1 (^1 ) 2 3 ( ( 12)) 4 2 3 9 4 (^3 ) 4
y y m x x y x
y x
y x
Parallel lines have the same slope.
Perpendicular lines have opposite reciprocal slopes.
Example #1 :^23 and^ โ 23 would be opposite reciprocals.
To Write Equations of Lines Parallel to Given Equations
Parallel to a given equation through a given point:
y y m x x y x y x y x y x
*Thus y = โ2 x + 1 is the equation of a line that is parallel to y = โ2 x + 4 going through the point (โ1, 3).
Letโs take a look at the graphs of the equations and make sure they are parallel.
The graphs of the two equations confirm that the two are parallel. Example #3 : Write the equation of a line that is parallel to y = โ3 x + 4 going through the point (0, โ1). 1 (^1 ) 4 ( 1) 3( 0 ) point 0 , 1
y y m x x y x y x y x
m
y x
To Write Equations Perpendicular to Given Equations
Perpendicular to a given equation going through a given point: 1.) Find the slope of the given equation y = mx + b and determine the opposite reciprocal of that number. 2.) Use the point slope form y โ y 1 (^) = m x ( โ x 1 ) to replace m (slope) and the ( x 1 (^) , y 1 )with the given point. 3.) Solve for y.
y = โ2 x + 4 y = โ2 x + 1
y =^12 x + 3
The equation of the line that is perpendicular to 2 x + y = 6 passing through the point (4, 5) is y =^12 x + 3.
Letโs take a look at the graphs of the two equations to confirm they are perpendicular. We will look at a graph that has not been made on the calculator because the pixels on the calculator distort the graphs.
The graphs of the equations confirm that the two are perpendicular.
y = โ 2 x + 6
y = 2 x +
