2. Slope-Intercept Form of a Line
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Chapter 7
2.
Slope-Intercept Form of a Line
This lesson focuses on the slope-intercept form of a linear equation, a common way to represent straight lines on a graph. It explains how to interpret the slope and y-intercept, graph linear equations using this form, and write equations for lines based on graphs or two points. The concepts are essential for analyzing and solving problems involving linear relationships, making them useful in academic and practical applications like predicting trends or understanding rates of change.
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Lesson Settings & Tools
| | 17 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Slope-Intercept Form of a Line
Slide of 17
The slope and y-intercept are key characteristics of a line. They can be used to write the equation of the line and are very useful when graphing the line. This lesson will focus on writing equations in slope-intercept form and graphing them.
Catch-Up and Review
Here is a recommended reading before getting started with this lesson.
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Challenge
Deriving the Equation for a Nonproportional Relationship
In a nonproportional relationship, the line passes through the point (0,b), or the y-intercept. Use the Slope Formula to derive the equation for a nonproportional relationship. m=y_2-y_1/x_2-x_1
Assume that the other point the line passes through is (x,y).
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Explore
Analyzing Graphs of Equations
The applet shows the graphs of different equations. Analyze these lines.
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Discussion
Points Where a Line Crosses the Axes
The intercepts of a line provide information about the position of the line in the coordinate plane. They can be used to identify the equation of a line from known points or to graph a line from its equation.
Concept
Intercept
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Pop Quiz
Finding the Intercepts of a Line
The applet shows the graphs of different lines. Identify the intercepts of these lines.
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Discussion
Linear Equation
A linear equation is an equation with at least one linear term and any number of constants. No other types of terms may be included. Linear equations in one variable have the following form, where a and b are real numbers and a≠ 0.
ax+b=0 or ax = b
Linear equations in two variables have the form below, where a, b, and c are real numbers and a≠ 0 and b≠ 0.
ax+by+c=0 or ax+by = c
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Discussion
Slope-Intercept Form
A linear equation can be written in the following form called the slope-intercept form.
y=mx+b
In this form, m is the slope and b is the y-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.
This line has a slope of 2 and a y-intercept of 1. The equation of the line can be written in slope-intercept form using these values.
y= mx+ b ⇓ y= 2x+ 1
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Pop Quiz
Identifying Slope-Intercept Form
The following applet generates linear equations. Those equations represent the relation between two variables x and y. Determine whether the generated equation is in slope-intercept form.
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Discussion
Graphing a Linear Equation in Slope-Intercept Form
A linear equation in slope-intercept form has the following form. y=mx+b The slope m and y-intercept b are used to graph the equation. Consider the following function. y=2x-3 There are three steps to follow to graph it.
1
Plot the y-Intercept
expand_more The y-intercept b can be used to find the the first point the line passes through. y=2x-3 ⇔ y= 2x+( - 3)
The y-intercept b is - 3. Plot the point (0, - 3) on a coordinate plane.
2
Use the Slope to Plot the Second Point
expand_more 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 m. Based on the equation, the slope is 2. y= 2x+( - 3)
This means that the rise is 2 and the run is 1. m=rise/run ⇔ 2=2/1
From the first point (0,- 3), move 1 unit right and 2 units up to plot the second point.
3
Draw a Line Through the Points
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Example
Following the Fish Population
Jordan is investigating the fish population in Grassy Lake. The number of fish in the lake was counted as 60 in 2016.
She finds out that the population increases by 120 after each year. The following equation represents the fish population in the lake. y=120x+60 In this equation, y is the number of fish in the lake after x years.
a Graph the equation on a coordinate plane.
b Interpret what the slope and the y-intercept represent.
Answer
a 
b Slope: Increase in the fish population per year
y-intercept: Number of fish in the year 2016
Hint
a Begin by identifying the slope and y-intercept.
b Interpret the given information based on the slope and y-intercept found in Part A.
Solution
a The equation that represents the fish population in the lake is in slope-intercept form. In this form, the slope and y-intercept are the important characteristics of the line of the equation. They determine the steepness and the position of the line on a coordinate plane. Begin by identifying these characteristics.
y= mx+ b ⇓ y= 120x+ 60 The slope is 120 and the y-intercept is 60. This means that the line crosses the y-axis at (0, 60).
Next, move 1 unit to the right and 120 units up from the y-intercept to plot another point.
Finally, the graph of the equation can be competed by drawing a line passing through these two points.
Note that the number of years passed or the number of fish cannot be negative. Therefore, only the first quadrant of the coordinate plane is shown.
b Recall the slope and the y-intercept of the line.
Slope &= 120 y-intercept &= 60 In this context, the slope 120 represents the increase in the fish population per year. The y-intercept 60 represents the number of fish in 2016.
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Discussion
Writing the Equation of a Line in Slope-Intercept Form From a Graph
The y-intercept b and the slope m of a line must be found to write the equation of the graph of the line in slope-intercept form . y=mx+b Consider the line shown as an example.
There are four steps to writing the equation of this line.
1
Find the y-Intercept
expand_more The y-intercept is the y-coordinate of the point where the line crosses the y-axis. This line intercepts the y-axis at (0,- 4), which means that the y-intercept is - 4.
2
Replace b With the y-Intercept
expand_more The y-intercept can be substituted into the slope-intercept form equation for b.
y=mx+ b ⇓ y=mx - 4
3
Find the Slope
expand_more 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.
m=rise/run
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.
For this line, the rise is 6 and the run is 2. Substitute these values into the formula to calculate the slope of the line. m=6/2 ⇒ m=3
4
Replace m With the Slope
expand_more Finally, substitute m= 3 in to the equation from Step 2 to complete the equation.
y= mx-4 ⇓ y= 3x-4
The equation of the line in slope-intercept form is now complete.
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Example
Chasing the Big Cats
Jordan turns her attention to the cheetah population over years.
She finds a graph on the Internet that shows the number of cheetahs over years since 1900.
Jordan wants to represent the graph algebraically to interpret it further.
a Write the equation of the line in slope-intercept form.
b Interpret what the slope and the y-intercept represent.
c Use the equation from Part A to estimate in which year cheetahs will go extinct if the current trend continues.
Answer
a y=-800x+100 000
b Slope: Decrease in the cheetah population per year
y-intercept: Number of cheetahs in the year 1900
c 2025
Hint
a Begin by determining the slope and y-intercept of the line.
b How many cheetahs were there in 1900? How does the number of cheetahs change over the years?
c In the equation, y represents the number of cheetahs after x years.
Solution
a The y-intercept and the slope of the line need to be determined to write the equation of the line in slope-intercept form. Begin by remembering slope-intercept form of a line.
y= mx+ b In this form, m is the slope and b is the y-intercept. The y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis.
In the given graph, the line intercepts the y-axis at (0,100 000). This means that the y-intercept of the line is 100 000. Substitute 100 000 for b into the slope-intercept form of an equation. y= mx+ b ⇓ y= mx+ 100 000 The slope of a line is the ratio of the rise and run of the line. m=rise/run Any two points on the line can be chosen to determine the rise and run. One point can be the one where the line intercepts the y-axis. The other point can be (50,60 000) for simplicity.
The rise is -40 000 and the run is 50. Substitute these values into the ratio and calculate the slope.
m=rise/run
SubstituteII
rise= -40 000, run= 50
m=-40 000/50
MoveNegNumToFrac
Put minus sign in front of fraction
m=-40 000/50
CalcQuot
Calculate quotient
m=-800
The slope of the line is -800. Finally, the equation of the line can be completed by substituting -800 for m. y= mx+ 100 000 ⇓ y= -800x+ 100 000
b Remember the equation from Part A.
y= -800x+ 100 000 The slope of the line is negative, which suggests that the number of cheetahs decreases by 800 per year. The y-intercept represents that there were 100 000 cheetahs in 1900.
c In the equation, y represents the number of cheetahs after x years. Substitute 0 for y into the equation and solve for x to find after how many years cheetahs will go extinct, according to the graph.
y=-800x+100 000
Substitute
y= 0
0=-800x+100 000
AddEqn
LHS+800x=RHS+800x
800x=100 000
DivEqn
.LHS /800.=.RHS /800.
800x/800=100 000/800
CalcQuot
Calculate quotient
x=125
The equation suggests that cheetahs will go extinct after 125 years. The specific year can be found by adding 125 to the starting year 1900. 1900+125=2025 This means that if the current trend continues, cheetahs will go extinct in 2025.
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Discussion
Writing the Equation of a Line in Slope-Intercept Form Using Two Points
The slope m and the y-intercept b of a line must be known to write a linear equation in slope-intercept form. y=mx+b 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 (- 4,1) and (8,4) will be written.
1
Find the Slope
expand_more Given two points on a line, the slope of the line can be found by using the Slope Formula. In this case, the coordinates ( - 4, 1) and ( 8, 4) will be substituted in place of (x_1,y_1) and (x_2,y_2), respectively.
m = y_2-y_1/x_2-x_1
SubstitutePoints
Substitute ( - 4, 1) & ( 8,4)
m=4- 1/8-( - 4)
SubNeg
a-(- b)=a+b
m=4-1/8+4
AddSubTerms
Add and subtract terms
m=3/12
CalcQuot
Calculate quotient
m=0.25
The slope m of the line passing through the two points is 0.25.
2
Replace m With the Slope
expand_more Now that the value of the slope is known, it can be substituted for m in the slope-intercept form of an equation.
y= mx+b ⇓ y= 0.25x+b
3
Find b Using a Point
expand_more Next, the y-intercept can be found by substituting either of the given points into the equation and solving for b. In the considered example, (8,4) can be used. Substitute its coordinates into the equation from Step 2 and solve for b.
y=0.25x+b
SubstituteII
x= 8, y= 4
4=0.25( 8)+b
Multiply
Multiply
4=2+b
SubEqn
LHS-2=RHS-2
2=b
RearrangeEqn
Rearrange equation
b=2
Therefore, the y-intercept is 2.
4
Write the Equation
expand_more Lastly, the complete equation in slope-intercept form can be written by substituting the y-intercept into the equation from Step 2. y=0.25x+ b ⇓ y=0.25x+ 2
The equation of the line in slope-intercept form is now complete.
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Example
Reading the Research Rag
Jordan is interested in endangered animals. She reads a magazine to learn what can be done to save these animals.
Suppose Jordan reads the same number of pages each day. The following table shows the number of pages y left in the magazine after x days.
| Number of Days x | 5 | 9 | 15 |
|---|---|---|---|
| Number of Pages y | 147 | 95 | 17 |
Which of the following equations represents the information in the table?
{"type":"choice","form":{"alts":["y=212-13x","y=13x+212","y=13x-212","y=-212-13x"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
Hint
Write an equation for the given situation in slope-intercept form using the Slope Formula.
Solution
The given equations are in slope-intercept form. y= mx+ b
In this form, m is the slope and b is the y-intercept. These two characteristics need to be determined to write an equation in slope-intercept form. Begin by representing given information in terms of points (x,y).
| Number of Days x | 5 | 9 | 15 |
|---|---|---|---|
| Number of Pages y | 147 | 95 | 17 |
| Point (x,y) | ( 5, 147) | ( 9, 95) | ( 15, 17) |
The slope of the relationship can be found by using the Slope Formula. m=y_2- y_1/x_2- x_1 Use any two of the points to calculate the slope!
m=y_2-y_1/x_2-x_1
SubstitutePoints
Substitute ( 5, 147) & ( 9, 95)
m=95- 147/9- 5
SubTerms
Subtract terms
m=- 52/4
MoveNegNumToFrac
Put minus sign in front of fraction
m=- 52/4
CalcQuot
Calculate quotient
m= - 13
The slope of the relationship is - 13. y= mx+ b ⇓ y= - 13x+ b Next, any of the points can be substituted into the above equation to find the y-intercept of the relation.
y=- 13x+b
SubstituteII
x= 15, y= 17
17=- 13( 15)+b
MultNegPos
(- a)b = - ab
17=-195+b
AddEqn
LHS+195=RHS+195
212=b
RearrangeEqn
Rearrange equation
b=212
The y-intercept of the relationship is 212. Finally, the equation can be completed. y= -13x+ b ⇓ y= - 13x+ 212 This equation can also be rearranged as follows. y=- 13x+212 ⇔ y=212-13x
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Pop Quiz
Writing Equations in Slope-Intercept Form
Write an equation in slope-intercept form using the given line, two points, or equation.
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Pop Quiz
Determining the Slope and y-Intercept of a Line
Determine the slope or y-intercept the given line based on its graph, points, or its equation. Be careful — the equation may or may not be written in slope-intercept form!
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Closure
Equation of a Nonproportional Relationship
In a nonproportional relationship, the line passes through the point (0,b). The y-coordinate of this point is the y-intercept of the line. The Slope Formula will be used to derive the equation for a nonproportional relationship. m=y_2- y_1/x_2- x_1 Two points are needed to use the slope formula. If one of the points is ( 0, b), let the other point be ( x, y). This point could be any point that the line passes through. Then, substitute these points into the formula and solve for y.
m=y_2-y_1/x_2-x_1
SubstitutePoints
Substitute ( 0, b) & ( x, y)
m=y- b/x- 0
SubTerm
Subtract term
m=y-b/x
MultEqn
LHS * x=RHS* x
mx=y-b/x* x
FracMultDenomToNumber
a/x* x = a
mx=y-b
AddEqn
LHS+b=RHS+b
mx+b=y
RearrangeEqn
Rearrange equation
y=mx+b
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Slope-Intercept Form of a Line
Exercise 2.1
The table below shows the relation between degrees Celsius and degrees Fahrenheit.
| ^(∘) C | -20 | -10 | 0 | 5 | 15 |
|---|---|---|---|---|---|
| ^(∘) F | -4 | 14 | 32 | 41 | 59 |
Write an equation in slope-intercept form for this table.
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We want to write an equation in slope-intercept form for the given table. y= mx+ b In this form, m is the slope and b is the y-intercept. Let x represent the degrees Celsius and y represent the degrees Fahrenheit. We can now choose any two ordered pairs from the table to find the slope of the equation.
| ^(∘) C | -20 | -10 | 0 | 5 | 15 |
|---|---|---|---|---|---|
| ^(∘) F | -4 | 14 | 32 | 41 | 59 |
Let's use ( 5, 41) and ( 15, 59). We can substitute these points into the slope formula and simplify to find the slope of the line. Let's do it!
The slope of the equation is 95. Next, we will find the y-intercept of the equation. To make this easier, let's find an ordered pair with an x-coordinate 0.
| ^(∘) C | -20 | -10 | 0 | 5 | 15 |
|---|---|---|---|---|---|
| ^(∘) F | -4 | 14 | 32 | 41 | 59 |
The line of the equation intercepts the y-axis at (0,32). Therefore, the y-intercept is 32. Let's substitute the values of the slope and y-intercept into the slope-intercept form to write our equation. y= mx+ b ⇒ y= 9/5x+ 32 The equation in slope-intercept form that represents the values in the table is y= 95x+32.
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Solution
Jordan says the equation of the line given below is y= 13x-2. Tadeo says that the equation of the line is y=- 13x+2.
Who is correct?
{"type":"choice","form":{"alts":["Jordan","Tadeo","Both","Neither"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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We want to decide who is correct about the equation of the given line. Notice that both equations are in slope-intercept form. Jordan:& y= 1/3x-2 ⇔ y= 1/3x+( -2) [0.8em] Tadeo:& y=-1/3x+2 ⇔ y= -1/3x+ 2 Let's also write an equation of the line in slope-intercept form. We will first find the slope of the line. We can choose two points on the line, then use them to determine the rise and run.
The rise is 1 and the run is 3. Remember, the slope is the ratio of the rise to the run. m=rise/run ⇒ m=1/3 The slope of the line is 13. We can also see that the line intercepts the y-axis at (0,-2). This means that the y-intercept is -2. Let's substitute these values in slope-intercept form. y= mx+ b ⇓ y= 1/3x+( -2) The equation of the line is y= 13x-2. This means that Jordan is correct!
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Solution
A taxi fare can be determined by the following equation. y=0.5x + 3 In this equation, x is the number of miles traveled and y is the total charge in dollars. Graph the equation. What do the slope and the y-intercept represent in this context?
Choose the option that best describes the situation.
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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We want to graph and interpret the slope and the y-intercept of the following equation. y=0.5x + 3 We can see that the equation is written in slope-intercept form, so let's begin by identifying its slope and y-intercept. y= 0.5x + 3 The slope is 0.5 and the y-intercept is 3. This means that the line of this equation intercepts the y-axis at (0,3). Let's plot this point on the coordinate plane.
We need a second point to graph the line. We already identified that the slope of the line is 0.5, but we can convert this decimal to a fraction to make it easier to use to find another point on the line. If the slope is 0.5= 12, the rise is 1 and the run is 2. Therefore, we move 2 unit right and 1 unit up from the y-intercept to plot the second point.
We can now graph a line that passes through these two points. Note that the number of miles cannot be negative. Therefore, the minimum value of x is 0. This means that the line will be drawn only in the first quadrant of the coordinate plane.
We have the graph! Now we need to interpret the slope and the y-intercept. Remember that the y-intercept of an equation is its initial value. In this context, this can be interpreted as the initial fee.
We can also see that the total charge increases by $0.50 per mile.
This means that the slope of the line is the charge per mile. The best option that describes this situation is option C.
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Solution
The graph below shows the total earnings of salespeople in a company based on the sales they make.
Here are some statements about the total earnings of several salespeople. A.& Mr. Birch will earn $2500 if his & sales are $5000. B.& Ms. Underwood will not earn any & money if she has no sales. C.& Mr. Russo will earn $3000 if his & sales are $15 000. D.& Ms. Barnes earns $3300 if she sells & $17 500 worth of product. Which of these statements could be true?
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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We want to decide which of the given statements could be true for the given graph. Let's write an equation in slope-intercept form to verify the statements. y= mx+ b In this form, m is the slope and b is the y-intercept. We can see that the line intercepts the y-axis at (0,1500).
The y-intercept of the line is 1500. Next, we will find the slope of the line. Let's choose two points on the line.
We can calculate the slope by substituting these values into the Slope Formula.
The slope of the line is 110. Now that we have th slope and the y-intercept, let's write the equation of the line. y= mx+ b ⇒ y= 1/10x+ 1500 Our next step is to organize the given statements in a table.
| Statement | Sales ($) | Total Earnings ($) |
|---|---|---|
| A | 5000 | 2500 |
| B | 0 | 0 |
| C | 15 000 | 3000 |
| D | 17 500 | 3300 |
Finally, we will substitute these values into the equation of the line to see which statement is true.
| Statement | Sales ($) | Total Earnings ($) | Substitute | Verify |
|---|---|---|---|---|
| A | 5000 | 2500 | 2500? =1/10( 5000)+1500 | 2500 ≠ 2000 |
| B | 0 | 0 | 0? =1/10( 0)+1500 | 0 ≠ 1500 |
| C | 15 000 | 3000 | 3000? =1/10( 15 000)+1500 | 3000 = 3000 |
| D | 17 500 | 3300 | 3300? =1/10( 17 500)+1500 | 3300 ≠ 3250 |
We found that statement C is correct.
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Solution
Slope-Intercept Form of a Line
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