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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.**

Challenge

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=x_{2}−x_{1}y_{2}−y_{1} $

Assume that the other point the line passes through is $(x,y).$Explore

The applet shows the graphs of different equations. Analyze these lines.

How do the coefficient of $x$ and the constant term affect position of the line? Try to find a relation between the line and its equation.

Discussion

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

Pop Quiz

The applet shows the graphs of different lines. Identify the intercepts of these lines.

Discussion

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=0orax=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=0orax+by=c $

Discussion

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 $

Pop Quiz

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.

Discussion

A linear equation 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

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.

a

b **Slope:** Increase in the fish population per year

$y-$**intercept:** Number of fish in the year $2016$

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.

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.

$Slopey-intercept =120=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 $2016.$
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

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.

a $y=-800x+100000$

b **Slope:** Decrease in the cheetah population per year

$y-$**intercept:** Number of cheetahs in the year $1900$

c $2025$

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.

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 $(0,100000).$ This means that the $y-$intercept of the line is $100000.$ Substitute $100000$ for $b$ into the slope-intercept form of an equation.
$y=mx+b⇓y=mx+100000 $

The slope of a line is the ratio of the rise and run of the line. $m=runrise $

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 $(50,60000)$ for simplicity.
The rise is $-40000$ and the run is $50.$ Substitute these values into the ratio and calculate the slope.
$m=runrise $

SubstituteII

$rise=-40000$, $run=50$

$m=50-40000 $

MoveNegNumToFrac

Put minus sign in front of fraction

$m=-5040000 $

CalcQuot

Calculate quotient

$m=-800$

$y=mx+100000⇓y=-800x+100000 $

b Remember the equation from Part A.

$y=-800x+100000 $

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 $100000$ 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.

$y=-800x+100000$

Substitute

$y=0$

$0=-800x+100000$

AddEqn

$LHS+800x=RHS+800x$

$800x=100000$

DivEqn

$LHS/800=RHS/800$

$800800x =800100000 $

CalcQuot

Calculate quotient

$x=125$

$1900+125=2025 $

This means that if the current trend continues, cheetahs will go extinct in $2025.$
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

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 $y$ | $147$ | $95$ | $17$ |

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Write an equation for the given situation in slope-intercept form using the Slope Formula.

The given equations are in slope-intercept form.

The slope of the relationship can be found by using the Slope Formula.
The slope of the relationship is $-13.$
The $y-$intercept of the relationship is $212.$ Finally, the equation can be completed.

$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 $(x,y)$ | $(5,147)$ | $(9,95)$ | $(15,17)$ |

$m=x_{2}−x_{1}y_{2}−y_{1} $

Use any two of the points to calculate the slope!
$m=x_{2}−x_{1}y_{2}−y_{1} $

SubstitutePoints

Substitute $(5,147)$ & $(9,95)$

$m=9−595−147 $

SubTerms

Subtract terms

$m=4-52 $

MoveNegNumToFrac

Put minus sign in front of fraction

$m=-452 $

CalcQuot

Calculate quotient

$m=-13$

$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

$b=212$

$y=-13x+b⇓y=-13x+212 $

This equation can also be rearranged as follows.
$y=-13x+212⇔y=212−13x $

Pop Quiz

Write an equation in slope-intercept form using the given line, two points, or equation.

Pop Quiz

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!

Closure

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=x_{2}−x_{1}y_{2}−y_{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=x_{2}−x_{1}y_{2}−y_{1} $

SubstitutePoints

Substitute $(0,b)$ & $(x,y)$

$m=x−0y−b $

SubTerm

Subtract term

$m=xy−b $

MultEqn

$LHS⋅x=RHS⋅x$

$mx=xy−b ⋅x$