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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

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

Explore

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

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.

Concept

A linear equation or linear function 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 $

Pop Quiz

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.

Discussion

A linear equation or linear function in slope-intercept form has the following form.
*expand_more*
*expand_more*
*expand_more*

$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

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

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=runrise ⇔2=12 $

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

Finally, use a straightedge to draw a line through both points.

This line is the graph of $y=2x−3.$

Example

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.

Together they continued traveling $350$ miles per day. The number of miles that Tiffaniqua traveled is represented by the following linear equation.$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.

a

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

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 $1200.$

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.

Example

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 $$20$ and an additional $$4$ per hour.

a Write an equation in slope-intercept form for the total cost $c$ of renting a car for $h$ hours.

b Graph the equation.

c Using the graph, determine how much Tiffaniqua would have to pay for $7$ hours of the rental.

a $c=4h+20$

b **Graph:**

c $$48$

a Recall what an equation in slope-intercept form looks like. What is the total cost $c$ of renting a car for $h$ hours?

b First, use the $y-$intercept and the slope to locate two points that lie on the line.

c Locate the point on the line where the $x-$coordinate is $7.$ What is its $y-$coordinate?

a Start by recalling what an equation in slope-intercept form looks like.

$y=mx+b $

Here, $m$ is the slope and $b$ is the $y-$intercept. The hourly rate of the renting company is $$4,$ so by multiplying that value by the number of hours $h,$ the total hourly amount of renting the car for $h$ hours can be calculated.
$4h $

Furthermore, by adding a one-time insurance fee of $$20,$ the total cost $c$ of renting the car can be found. $c=4h+20 $

b In order to graph the equation found, the slope and the $y-$intercept will be used. First, plot the $y-$intercept on a coordinate plane. The $y-$intercept is $20,$ which corresponds to the point $(0,20).$

Next, the slope $4$ will be used to locate a second point. In order to plot this point, move $1$ unit right and $4$ units up. Points on the line can also be plotted by using multiples of the slope — in this case, for example, $3$ units right and $3⋅4=12$ units up.

Finally, draw a line through these two points. Note that since $c$ and $h$ 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.

c To find the cost of renting a car for $7$ hours, find $7$ on the $x-$axis and then move vertically to the corresponding point on the line.

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

The $y-$coordinate is $48,$ which means that when renting a car for $7$ hours, Tiffaniqua will have to pay a total of $$48.$

Discussion

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 .
*expand_more*
*expand_more*
*expand_more*
*expand_more*

$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

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

The $y-$intercept can be substituted into the slope-intercept form equation for $b.$

$y=mx+b⇓y=mx−4 $

3

Find the Slope

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.
*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=runrise $

The $m=26 ⇒m=3 $

4

Replace $m$ With the Slope

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. Example

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"]}}

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.

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.
The coordinates of the $y-$intercept are $(0,500),$ so $b$ is equal to $500.$ This value can be substituted for $b$ in the general equation.
$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.
As can be seen, for each $10$ units to the right, the line goes $100$ units down. By dividing both values by $10,$ it is obtained that for each $1$ unit to the right, the line goes $10$ units down.
$10units right and100units down⇕1unit right and10units 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

The slope $m$ and the $y-$intercept $b$ of a line must be known to write a linear equation in slope-intercept form.
*expand_more*
*expand_more*
*expand_more*
*expand_more*

$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

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.
The slope $m$ of the line passing through the two points is $0.25.$

$m=x_{2}−x_{1}y_{2}−y_{1} $

SubstitutePoints

Substitute $(-4,1)$ & $(8,4)$

$m=8−(-4)4−1 $

SubNeg

$a−(-b)=a+b$

$m=8+44−1 $

AddSubTerms

Add and subtract terms

$m=123 $

CalcQuot

Calculate quotient

$m=0.25$

2

Replace $m$ With the Slope

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

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 $b.$
Therefore, the $y-$intercept is $2.$

$y=0.25x+b$

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

$b=2$

4

Write the Equation

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. Example

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.

After choosing the perfect fabric, she asked the shop assistant how much $4$ and $6$ yards of the fabric would cost. The shop assistant told her that the costs would be $$3$ and $$4.50,$ respectively.$(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.

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.

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.

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

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

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

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.

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