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There are several forms in which linear equations can be written. A type called the slope-intercept form will be presented and analyzed in this lesson. It will also be shown how to graph an equation written in such form.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Analyzing the Graphs of Linear Functions

Analyze the graphs of different linear functions. Do these lines have anything in common?

Randomly generated graphs of different linear functions

Explore

Finding Similarities Between Lines

Examine the graphs of different linear functions. Are there any similarities between these graphs?
Randomly generated graphs of different linear functions with the same slope
What does it mean for the equations of these functions? Do they have a common part?

Discussion

Slope-Intercept Form of a Linear Equation

When trying to find similarities between lines, the first group of lines all have the same intercept, while the second group of lines have the same slope. These two characteristics can be used to write an equation of any line.

Concept

Slope-Intercept Form

A linear equation or linear function can be written in the following form called the slope-intercept form.

In this form, is the slope and is the intercept. These are the general characteristics or parameters of the line. They determine the steepness and the position of the line on the coordinate plane. Consider the following graph.

The graph of the linear function y=2*x+1 with a slope of 2 (2 rise, 1 run) and a y-intercept at (0, 1)

This line has a slope of and a intercept of The equation of the line can be written in slope-intercept form using these values.

Pop Quiz

Identifying Slope-Intercept Form

Consider the given linear equation that represents the relation between two variables and Determine whether the equation is written in slope-intercept form.

Random generator creates an equation written in different forms

Discussion

Graphing a Linear Function in Slope-Intercept Form

A linear equation or linear function in slope-intercept form has the following form.
The slope and intercept are used to graph the equation. Consider the following function.
There are three steps to follow to graph it.
1
Plot the Intercept
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The intercept can be used to find the the first point the line passes through.
The intercept is Plot the point on a coordinate plane.
The point (0,-3) plotted on a coordinate plane
2
Use the Slope to Plot the Second Point
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There should be at least two points to draw a line. The second point can be plotted on the coordinate plane by using the slope Based on the equation, the slope is
This means that the rise is and the run is
From the first point move unit right and units up to plot the second point.
Two points (0,-3) and (1,-1) on the coordinate plane
3
Draw a Line Through the Points
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Finally, use a straightedge to draw a line through both points.

A line with the equation y=2x-3 passing through the points (1, -1) and (0, -3).

This line is the graph of

Example

Graphing an Equation and Using the Graph to Solve Problems

Tiffaniqua is driving from her home in New York to visit her sister, who lives in Springfield, Missouri. On the first day of the trip, she traveled miles to Washington DC to pick up her friend Maya.

The route of Tiffaniqua
Together they continued traveling miles per day. The number of miles that Tiffaniqua traveled is represented by the following linear equation.
Here, is the total number of miles traveled and is the number of days after Tiffaniqua picked up her friend.
a Graph the equation on a coordinate plane.
b Use the graph to find on which day Tiffaniqua traveled a total of miles.

Answer

a
Graph of the given equation
b On the second day of traveling together, the third day of traveling for Tiffaniqua, Tiffaniqua traveled a total of 1200 miles.

Hint

a Determine the slope and the intercept using the given equation and use their values to plot two points on a coordinate plane.
b Find the coordinate of the point on the line with the coordinate of

Solution

a The equation that represents the situation is given in slope-intercept form. Therefore, the slope and the intercept can be used to graph it. To begin, identify their values by comparing the equation with the general equation of a line written in slope-intercept form.
As seen above, the slope is and the intercept is Now, the intercept can be used to find the first point. Since the value of the intercept is plot on a coordinate plane.
A point on a coordinate plane

Next, by using the slope, the second point on the line can be determined. Since the slope is move unit right and units up from the first point, then plot the new point.

Second point is plotted on the coordinate plane

Finally, by drawing the line through the two plotted points, the graph of the equation can be completed.

Graph of the equation

Note that since and represent the number of miles and days, respectively, they cannot have negative values. This is why the line is only drawn in the first quadrant and is actually a ray.

b To determine on which day Tiffaniqua passed the mark of traveled miles, find that value on the axis and then identify the corresponding point on the line.
Graph of the equation

Next, determine the coordinate of that point on the line.

Graph of the equation

From the graph it can be concluded that Tiffaniqua passed the mark of miles on the second day of traveling together with Maya. Since on the first day Tiffaniqua traveled miles alone to pick up Maya, the second day of traveling together with Maya is her third day of travel in total.

Example

Finding and Graphing an Equation of a Real-Life Problem

Tiffaniqua's car broke down right after she arrived at her sister's house, so Tiffaniqua decided to rent a new car while visiting her sister. The rental company charges a one-time insurance fee of and an additional per hour.

A board with prices of the car renting company
a Write an equation in slope-intercept form for the total cost of renting a car for hours.
b Graph the equation.
c Using the graph, determine how much Tiffaniqua would have to pay for hours of the rental.

Answer

a
b Graph:
Graph of the equation
c

Hint

a Recall what an equation in slope-intercept form looks like. What is the total cost of renting a car for hours?
b First, use the intercept and the slope to locate two points that lie on the line.
c Locate the point on the line where the coordinate is What is its coordinate?

Solution

a Start by recalling what an equation in slope-intercept form looks like.
Here, is the slope and is the intercept. The hourly rate of the renting company is so by multiplying that value by the number of hours the total hourly amount of renting the car for hours can be calculated.
Furthermore, by adding a one-time insurance fee of the total cost of renting the car can be found.
b In order to graph the equation found, the slope and the intercept will be used. First, plot the intercept on a coordinate plane. The intercept is which corresponds to the point
One point on a coordinate plane

Next, the slope will be used to locate a second point. In order to plot this point, move unit right and units up. Points on the line can also be plotted by using multiples of the slope — in this case, for example, units right and units up.

Second point plotted on the coordinate plane

Finally, draw a line through these two points. Note that since and represent the cost and number of hours the car is rented, respectively, they can only have non-negative values. This means that the line should only be graphed in the first quadrant.

Drawing a line through the two plotted points
c To find the cost of renting a car for hours, find on the axis and then move vertically to the corresponding point on the line.
A vertical arrow from the x-axis to the line is drawn and the point on the line is located

Now move horizontally to the axis to identify the coordinate of this point.

A horizontal arrow from the point on the line to the y-axis is drawn

The coordinate is which means that when renting a car for hours, Tiffaniqua will have to pay a total of

Discussion

Writing the Equation of a Line in Slope-Intercept Form From a Graph

The intercept and the slope of a line must be found to write the equation of the graph of the line in slope-intercept form .
Consider the line shown as an example.
A line on a coordinate plane
There are four steps to writing the equation of this line.
1
Find the Intercept
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The intercept is the coordinate of the point where the line crosses the axis. This line intercepts the axis at which means that the intercept is

The y-intercept of the line is identified on the graph
2
Replace With the -Intercept
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The intercept can be substituted into the slope-intercept form equation for
3
Find the Slope
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Next, the slope of the line must be determined. The slope of a line is the ratio of the rise and run of the line.
The rise is the vertical distance between two points and the run is the horizontal distance. Any two points on the line can be used to find the slope.
Rise and run between the two chosen points are determined
For this line, the is and the is Substitute these values into the formula to calculate the slope of the line.
4
Replace With the Slope
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Finally, substitute in to the equation from Step to complete the equation.
The equation of the line in slope-intercept form is now complete.

Example

Using a Graph to Write an Equation in Slope-Intercept Form

The evening after Tiffaniqua and Maya arrived at Tiffaniqua's sister's house, the girls decided to pass the evening by putting together puzzles. The number of the remaining puzzle pieces as the girls complete the puzzle is shown in the following graph.

The graph of the presented situation

On the diagram, represents the number of puzzle pieces and represents time spent completing the puzzle in minutes.

a Write the equation of the line in slope-intercept form.
b How many pieces are in the puzzle?
c How many pieces per minute did the girls put together?

Hint

a Identify the intercept of the line. Then choose a second point on the line and find the slope by analyzing the rise and run between the two points.
b Which point on the line describes the moment, when the girls have not yet started completing the puzzle?
c The speed at which the girls put the puzzle together is represented by the slope.

Solution

a The equation of the line should be written in slope-intercept form, which means that it should have the following form.
However, in this case, instead of and the variables will be and respectively. Use the given graph to determine the values of the slope and the intercept To find the intercept, locate the point where the line intercepts the axis.
The y-intercept is located on the graph
The coordinates of the intercept are so is equal to This value can be substituted for in the general equation.
To determine the value of the slope, locate a second point on the line and analyze the rise and run between the two points.
The run and rise between the y-intercept and the second chosen point
As can be seen, for each units to the right, the line goes units down. By dividing both values by it is obtained that for each unit to the right, the line goes units down.
The line goes down as it moves to the right, so it has a negative slope. Therefore, is By substituting this value for the equation of the line can be completed.
b To determine how many pieces the puzzle has, think of what the graph represents. The line shows the number of remaining pieces as the girls spend more time putting the puzzle together. Therefore, the point where represents the total number of puzzle pieces, as the girls have not yet begun putting the puzzle together.
The y-intercept of the line on the graph
Note that the point where is also the intercept. In Part A, the intercept was found to be so there are puzzles in the game.
c The speed at which the girls put the puzzle together is represented by the slope of the line. In Part A, the slope was found to be This means that the number of remaining puzzle pieces decreased by every minute the girls worked on the puzzle. In other words, the girls completed pieces per minute.

Discussion

Writing the Equation of a Line in Slope-Intercept Form Using Two Points

The slope and the intercept of a line must be known to write a linear equation in slope-intercept form.
When only two points on the line are known, the following four-step method can be used. For example, the equation of the line that passes through the points and will be written.
1
Find the Slope
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Given two points on a line, the slope of the line can be found by using the Slope Formula. In this case, the coordinates and will be substituted in place of and respectively.
The slope of the line passing through the two points is
2
Replace With the Slope
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Now that the value of the slope is known, it can be substituted for in the slope-intercept form of an equation.
3
Find Using a Point
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Next, the intercept can be found by substituting either of the given points into the equation and solving for In the considered example, can be used. Substitute its coordinates into the equation from Step and solve for
Therefore, the intercept is
4
Write the Equation
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Lastly, the complete equation in slope-intercept form can be written by substituting the intercept into the equation from Step
The equation of the line in slope-intercept form is now complete.

Example

Using Two Points to Write an Equation and Graph a Line

Tiffaniqua really wants to make a dress that she has been dreaming about for a long time. In her sister's town, there is a very famous fabric store, so she decided to go there for the necessary fabric.

Tiffaniqua imagining a dress she wants to sew
After choosing the perfect fabric, she asked the shop assistant how much and yards of the fabric would cost. The shop assistant told her that the costs would be and respectively.
Tiffaniqua realizes that she may want to make a matching accessory, so she might need more fabric. The cost of the fabric will help her decide which, if any, accessory she wants to make.
a Help Tiffaniqua determine the price for however much fabric she wants to buy by finding the equation of the line. Write the equation in slope-intercept form.
b Graph the equation.
c Verify that the points written above lie on the line.

Answer

a
b Graph:
The graph of the equation
c From the diagram, it can be seen that the points and lie on the line.
Plotting the given points on the coordinate plane with the line

Hint

a Use the Slope Formula to find the slope of the line. Then substitute the coordinates of either of the given points into the equation with the slope.
b Plot the intercept on a coordinate plane. Then use the slope to plot another point on the line.
c Plot the given points and see if they lie on the line.

Solution

a To write the equation in slope-intercept form, the slope and the intercept of the line should be known. First, find the slope by substituting the coordinates of the two given points into the Slope Formula.
Simplify
Now that the slope is known, can be substituted for into the equation.
Next, by substituting either of the two given points into the partly completed equation, the intercept can be found. For example, substitute and solve for
Solve for
The intercept of the line is Now the equation can be completed.
b In order to graph an equation in slope-intercept form, the slope and the intercept can be used. First, plot the intercept, which in this case is on a coordinate plane.
The y-intercept of the line

Next, use the slope of to plot the second point that lies on the line. Note that can be rewritten as Therefore, by moving units right and units up, the second point can be located.

Using the slope to plot another point

Finally, draw a line through the two points to obtain the graph of the equation.

Drawing the line through the two points

Note that since it is not possible to buy a negative number of yards of fabric, the line should only be drawn for non-negative values of

c To verify that the given points lie on the line, plot them on the coordinate plane and see if they are on the line. Recall that the coordinates of the points are and
Plotting the given points on the coordinate plane with the line

As can be seen, the points indeed lie on the line.

Pop Quiz

Practice Writing Equations in Slope-Intercept Form

Given a line, two of its points, or its equation in either standard or point-slope form, write an equation in slope-intercept form.

Applet that generates a line, two of its points, or an equation

Pop Quiz

Identifying the Slope and Y-Intercept of a Line

Determine the slope or intercept of a line given its graph, two of its points, or its equation, which may or may not be written in slope-intercept form.

Applet that randomly generates a graph of a line, two its points, or an equation

Closure

Analyzing Two Lines With the Same Slope

In this lesson, writing and graphing linear equations written in slope-intercept form were studied. However, in every case a line or an equation was analyzed individually. Now, consider two lines with the same slope on one coordinate plane. What can be said about these pairs of lines?
Randomly generated two lines with the same slope
The lines with the same slope seem to be parallel. This observation is always true and can be proven by trying to solve the system of the lines' equations.