## Interpreting the y-Intercept and Making Predictions

### Learning Outcomes

• Interpret the characteristics of a linear equation and use that equation to make predictions

## Interpret the y-intercept of a linear equation

Earlier in this module, we learned how to write the equation of a line given the slope and $y$-intercept.  Often, when the line in question represents a set of data or observations, the $y$-intercept can be interpreted as a starting point.  We will continue to use the examples for house value in Mississippi and Hawaii and high school smokers to interpret the meaning of the $y$-intercept in those equations.

### Example

Recall the equations and data for house value:

Linear equations describing the change in median home values between $1950$ and $2000$ in Mississippi and Hawaii are as follows:

Hawaii:  $y = 3966x+74,400$

Mississippi:  $y = 924x+25,200$

The equations are based on the following dataset.

$x$ = the number of years since $1950$, and y = the median value of a house in the given state.

Year (x) Mississippi House Value (y) Hawaii House Value (y)
$0$ $25,200$ $74,400$
$50$ $71,400$ $272,700$

And the equations and data for high school smokers:

A linear equation describing the change in the number of high school students who smoke, in a group of $100$, between $2011$ and $2015$ is given as:

$y = -1.75x+16$

And is based on the data from this table, provided by the Centers for Disease Control.

$x$ = the number of years since $2011$, and $y$ = the number of high school smokers per $100$ students.

 Year Number of  High School Students Smoking Cigarettes (per 100) $0$ $16$ $4$ $9$

Also recall that the equation of a line in slope-intercept form is as follows:

$y = mx + b$

$\begin{array}{l}\,\,\,\,\,m\,\,\,\,=\,\,\,\text{slope}\\(x,y)=\,\,\,\text{a point on the line}\\\,\,\,\,\,\,\,b\,\,\,\,=\,\,\,\text{the y value of the y-intercept}\end{array}$

The examples that follow show how to interpret the y-intercept of the equations used to model house value and the number of high school smokers. Additionally, you will see how to use the equations to make predictions about house value and the number of smokers in future years.

### Example

Interpret the y-intercepts of the equations that represent the change in house value for Hawaii and Mississippi.

Hawaii:  $y = 3966x+74,400$

Mississippi:  $y = 924x+25,200$

The y-intercept of a two-variable linear equation can be found by substituting $0$ in for $x$.

#### Hawaii

$y = 3966x+74,400\\y = 3966(0)+74,400\\y = 74,400$

The $y$-intercept is a point, so we write it as (0, 74,400).  Remember that $y$-values represent dollars and $x$ values represent years.  When the year is $0$—in this case $0$ because that is the first date we have in the dataset—the price of a house in Hawaii was $74,400$.  (Remember that $x$ represents the number of years since $1950$, so if $x=0$ the year is $1950$.)

#### Mississippi

$y = 924x+25,200\\y = 924(0)+25,200\\y = 25,200$

The $y$-intercept is $(0, 25,200)$.  This means that in $1950$ the value of a house in Mississippi was $25,200$.

### Example

Interpret the $y$-intercept of the equation that represents the change in the number of high school students who smoke out of $100$.

Substitute $0$ in for $x$.

$y = -1.75x+16\\y = -1.75(0)+16\\y = 16$

The $y$-intercept is $(0,16)$.  The data starts at $2011$, so we represent that year as $0$. We can interpret the $y$-intercept as follows:

In the year $2011$, $16$ out of every $100$ high school students smoked.

In the following video you will see an example of how to interpret the $y$– intercept given a linear equation that represents a set of data.

## Use a linear equation to make a prediction

Another useful outcome we gain from writing equations from data is the ability to make predictions about what may happen in the future. We will continue our analysis of the house price and high school smokers. In the following examples you will be shown how to predict future outcomes based on the linear equations that model current behavior.

### Example

Use the equations for house value in Hawaii and Mississippi to predict house value in $2035$.

We are asked to find house value, y, when the year, $x$, is $2035$. Since the equations we have represent house value increase since $1950$, we have to be careful. We can’t just plug in $2035$ for $x$, because $x$ represents the years since $1950$.

How many years are between $1950$ and $2035$? $2035 - 1950 = 85$

This is our $x$-value.

For Hawaii:

$y = 3966x+74,400\\y = 3966(85)+74,400\\y = 337110+74,400 = 411,510$

Holy cow! The average price for a house in Hawaii in $2035$ is predicted to be $411,510$ according to this model. See if you can find the current average value of a house in Hawaii. Does the model measure up?

For Mississippi:

$y = 924x+25,200\\y = 924(85)+25,200\\y = 78540+25,200 = 103,740$

The average price for a home in Mississippi in $2035$ is predicted to be $103,740$ according to the model. See if you can find the current average value of a house in Mississippi. Does the model measure up?

In the following video, you will see the example of how to make a prediction with the home value data.

### Example

Use the equation for the number of high school smokers per $100$ to predict the year when there will be $0$ smokers per $100$.

$y = -1.75x+16$

This question takes a little more thinking.  In terms of $x$ and $y$, what does it mean to have $0$ smokers?  Since y represents the number of smokers and $x$ represent the year, we are being asked when $y$ will be $0$.

Substitute $0$ for $y$.

$y = -1.75x+16$

$0 = -1.75x+16$

$-16 = -1.75x$

$\frac{-16}{-1.75} = x$

$x = 9.14$ years

Again, like the last example, $x$ is representing the number of years since the start of the data—which was $2011$, based on the table:

 Year Number of  High School Students Smoking Cigarettes (per 100) $0$ $16$ $4$ $9$

So we are predicting that there will be no smokers in high school by $2011+9.14=2020$. How accurate do you think this model is? Do you think there will ever be $0$ smokers in high school?

The following video gives a thorough explanation of making a prediction given a linear equation.

## Bringing it Together

The last example we will show will include all of the concepts that we have been building up throughout this module.  We will interpret a word problem, write a linear equation from it, graph the equation, interpret the $y$-intercept and make a prediction. Hopefully this example will help you to make connections between the concepts we have presented.

### Example

It costs $600$ to purchase an iphone, plus $55$ per month for unlimited use and data.

Write a linear equation that represents the cost, $y$,  of owning and using the iPhone for $x$ amount of months. When you have written your equation, answer the following questions:

1. What is the total cost you’ve paid after owning and using your phone for $24$ months?
2. If you have spent $2,580$ since you purchased your phone, how many months have you used your phone? iPhone

Read and Understand: We need to write a linear equation that represents the cost of owning and using an iPhone for any number of months.  We are to use $y$ to represent cost, and $x$ to represent the number of months we have used the phone.

Define and Translate: We will use the slope-intercept form of a line, $y=mx+b$, because we are given a starting cost and a monthly cost for use.  We will need to find the slope and the $y$-intercept.

Slope: in this case we don’t know two points, but we are given a rate in dollars for monthly use of the phone.  Our units are dollars per month because slope is $\frac{\Delta{y}}{\Delta{x}}$, and $y$ is in dollars and $x$ is in months. The slope will be $\frac{55\text{ dollars }}{1\text{ month }}$:  $m=\frac{55}{1}=55$

$Y$-Intercept: the $y$-intercept is defined as a point $\left(0,b\right)$.  We want to know how much money we have spent, $y$, after $0$ months.  We haven’t paid for service yet, but we have paid $600$ for the phone. The $y$-intercept in this case is called an initial cost. $b=600$

Write and Solve: Substitute the slope and intercept you defined into the slope-intercept equation.

$\begin{array}{c}y=mx+b\\y=55x+600\end{array}$

Now we will answer the following questions:

1. What is the total cost you’ve paid after owning and using your phone for $24$ months?

Since $x$ represents the number of months you have used the phone, we can substitute $x=24$ into our equation.

$\begin{array}{c}y=55x+600\\y=55\left(24\right)+600\\y=1320+600\\y=1920\end{array}$

$Y$ represents the cost after $x$ number of months, so in this scenario, after $24$ months, you have spent $1920$ to own and use an iPhone.

1. If you have spent $2,580$ since you purchased your phone, how many months have you used your phone?

We know that $y$ represents cost, and we are given a cost and asked to find the number of months related to having spent that much. We will substitute $y=2,580$ into the equation, then use what we know about solving linear equations to isolate $x$:

$\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y=55x+600\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2580=55x+600\\\text{ subtract 600 from each side}\,\,\,\,\,\,\,\underline{-600}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-600}\\\text{}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1980=55x\\\text{}\\\text{ divide each side by 55 }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1980}{55}=\frac{55x}{55}\\\text{}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,36=x\end{array}$

If you have spent $2,580$ then you have been using your iPhone for $36$ months, or $3$ years.

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