Lines in a Plane
Lines and Equations
A line in a plane represents the solution of a linear equation. The linear equation may be in several different forms outlined below. The most common form for a linear equation is the Slope-Intercept form or where is the slope and is the -intercept.
For any value you choose there is a corresponding value where the equation is true. You could choose a set of values and form a pair of numbers for each value, . These pairs become a set of points in the plane where the value is the horizontal distance from the origin and the y value is the vertical distance. Another way to say this is to move from the origin to the right and upward to place each point. All of these points would form a straight line.
The following illustrates a line with a slope and a -intercept of . Both values can be changed with the sliders. Note how the line changes when you change either value.
Activities
Answer the following questions.
What happens to the line as the slider is changed?
What happens to the line as the slider is changed?
Pick an value and substitute it into the equation shown under the sliders. How does this compare to the value of the line at the chosen value?
Slope of a Line
A line can also be defined by two points. The following shows a line defined by two points. Both points can be moved. A key feature of a line is the slope. The slope can be calculated from the two points with the formula . The slope is how fast the line is rising as you move from left to right. A brown triangle 1 unit wide is shown to represent the slope.
The equation for the line in Slope-Intercept form is also shown.
Slope from Two Points
Activities
Note what happens to the slope as you move the points.
What happens to the slope if the points are both the same height ( value)?
What happens to the slope if the points are directly above each other ( same value )?
What happens to the line if the points are at the same location?
Line Definitions
The following shows a line and some values and points associated with a line.
The value shown are:
- Two points on the line, and . A line can be defined by two points.
- A brown triangle illustrating the slope.
- The -intercept, the point on the line where .
- The -intercept, the point where the line crosses the x axis at .
Line Definitions
Slope Intercept
The equation for finding the slope from two points was shown above. The following shows how to find the -intercept from a point and the slope. For illustration slope and intercept sliders can modify the line.
In addition a point can be moved along the -axis. This will define a point on the line, . A brown line segment shows that the run from the -axis is . The definition of the slope can be rearranged to give the Rise , shown as a green line segment. Projecting and onto the axis shows that the -intercept is . This then shows how to convert a slope and a point into the Slope-Intercept form of an equation for a line.
Point-Slope Form
Another form of an equation for a line is the Point-Slope form. This form works well if you know a point and a slope. It starts with the definition of the slope, which is true for any point 2, since the slope of a linear line is constant. Therefor you can substitute an arbitrary point on the line for point 2, giving . Multiplying both sides by and switching sides then gives the Point Slope form of the equation for a line, .
Activity
Compare the Point-Slope form and the Slope-Intercept form shown above of linear equations.
Two Points to Equations
The following illustrates how to obtain three forms of equations for a line from two points. The two points can be moved. The formulas for all terms are shown on the left as well as the values obtained from the two point line.
The inclination is the angle of the line with the axis. It can be calculated directly from the slope.
An added form is the General or Standard form.
What happens to the General form for vertical or horizontal lines?
Parallel and Perpendicular
The equation for a parallel or perpendicular line through another point can also be found. The following illustrates Parallel or Perpendicular lines. The two Points A and B define a blue line and a parallel or perpendicular green dash-dot line through a third point C is drawn. Checking the Show Slope box will show blue and green slope triangles for both lines.
How do the slopes compare for parallel lines?
How do the slopes compare for perpendicular lines?
For perpendicular lines place points A and C at the same location.
What do you notice about the slopes?
It turns out that all parallel lines have the same slope and perpendicular lines have negative inverse slopes. The equation for the slope of a line perpendicular to a line with a slope of is . Thus you can find the slope and if given a point you can use the methods above to find the Slope-Intercept of Point-Slope forms of the equation for the line.
Two Linear Equations
A line represents all of the solutions for an equation with two variables. If you have two lines that intersect at a point, the point of intersection is a solution to both lines equations. In the illustration below a single or two equations can be entered and the solutions to the equations are shown. Where the lines intersect is simultaneously the solution to both equations.
For two lines there are three possibilities for the intersection.
Activities
The initial setting show the solution of two equations and where they intersect with the equations
and .
What happens if the first equation is changed to ?
What happens if the first equation is changed to ?
What are the three possibilities for intersection of two lines?