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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?
Explore
Finding Similarities Between Lines
Examine the graphs of different linear functions. Are there any similarities between these graphs? 
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 y-intercept, while the second group of lines have the same slope. These two characteristics can be used to write an equation of any line.
Pop Quiz
Identifying Slope-Intercept Form
Consider the given linear equation that represents the relation between two variables x and y. Determine whether the equation is written in slope-intercept form.
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 250 miles to Washington DC to pick up her friend Maya.
m=350d+250
Here, m is the total number of miles traveled and d 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 1200 miles.
Answer
a 
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 y-intercept using the given equation and use their values to plot two points on a coordinate plane.
b Find the x-coordinate of the point on the line with the y-coordinate of 1200.
Solution
a The equation that represents the situation is given in slope-intercept form. Therefore, the slope and the y-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.
y=mx+bm=350d+250
As seen above, the slope is 350 and the y-intercept is 250. Now, the y-intercept can be used to find the first point. Since the value of the y-intercept is 250, plot (0,250) on a coordinate plane. 
Next, by using the slope, the second point on the line can be determined. Since the slope is 350, move 1 unit right and 350 units up from the first point, then plot the new point.
Finally, by drawing the line through the two plotted points, the graph of the equation can be completed.
Note that since m and d 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 1200 traveled miles, find that value on the y-axis and then identify the corresponding point on the line.
Next, determine the x-coordinate of that point on the line.
From the graph it can be concluded that Tiffaniqua passed the mark of 1200 miles on the second day of traveling together with Maya. Since on the first day Tiffaniqua traveled 250 miles alone to pick up Maya, the second day of traveling together with Maya is her third day of travel in total.
Discussion
Writing the Equation of a Line in Slope-Intercept Form From a Graph
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.
On the diagram, p represents the number of puzzle pieces and t represents time spent completing the puzzle in minutes.
a Write the equation of the line in slope-intercept form.
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b How many pieces are in the puzzle?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"pieces","answer":{"text":["500"]}}
c How many pieces per minute did the girls put together?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"pieces","answer":{"text":["10"]}}
Hint
a Identify the y-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.
y=mx+b
However, in this case, instead of x and y, the variables will be t and p, respectively. Use the given graph to determine the values of the slope m and the y-intercept b. To find the y-intercept, locate the point where the line intercepts the y-axis. 
pp=mt+b ⇓=mt+500
To determine the value of the slope, locate a second point on the line and analyze the rise and run between the two points. 
10 units right and 100 units down⇕1 unit right and 10 units down
The line goes down as it moves to the right, so it has a negative slope. Therefore, m is -10. By substituting this value for m, the equation of the line can be completed.
pp=mt+500 ⇓=-10t+500
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 t=0 represents the total number of puzzle pieces, as the girls have not yet begun putting the puzzle together. 
Note that the point where t=0 is also the y-intercept. In Part A, the y-intercept was found to be 500, so there are 500 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 -10. This means that the number of remaining puzzle pieces decreased by 10 every minute the girls worked on the puzzle. In other words, the girls completed 10 pieces per minute.
Discussion
Writing the Equation of a Line in Slope-Intercept Form Using Two Points
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.
(4,3) and (6,4.5)
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 y=0.75x
b Graph:
c From the diagram, it can be seen that the points (4,3) and (6,4.5) lie on 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 y-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 y-intercept of the line should be known. First, find the slope by substituting the coordinates of the two given points into the Slope Formula.
y=mx+b⇓y=0.75x+b
Next, by substituting either of the two given points into the partly completed equation, the y-intercept b can be found. For example, substitute (4,3) and solve for b.
The y-intercept of the line is 0. Now the equation can be completed.
y=0.75x+0⇕y=0.75x
b In order to graph an equation in slope-intercept form, the slope and the y-intercept can be used. First, plot the y-intercept, which in this case is 0, on a coordinate plane.
Next, use the slope of 0.75 to plot the second point that lies on the line. Note that 0.75 can be rewritten as 43. Therefore, by moving 4 units right and 3 units up, the second point can be located.
Finally, draw a line through the two points to obtain the graph of the equation.
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 x.
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 (4,3) and (6,4.5).
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.
Pop Quiz
Identifying the Slope and Y-Intercept of a Line
Determine the slope or y-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.
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?
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.