Example

Recall the equations and data for house value:

Linear equations describing the change in median home values between [latex]1950[/latex] and [latex]2000[/latex] in Mississippi and Hawaii are as follows:

Hawaii:  [latex]y = 3966x+74,400[/latex]

Mississippi:  [latex]y = 924x+25,200[/latex]

The equations are based on the following dataset.

[latex]x[/latex] = the number of years since [latex]1950[/latex], and y = the median value of a house in the given state.

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 [latex]100[/latex], between [latex]2011[/latex] and [latex]2015[/latex] is given as:

 [latex]y = -1.75x+16[/latex]

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

[latex]x[/latex] = the number of years since [latex]2011[/latex], and [latex]y[/latex] = the number of high school smokers per [latex]100[/latex] students.

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

[latex]y = mx + b[/latex]

[latex]\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}[/latex]

Example

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

Hawaii:  [latex]y = 3966x+74,400[/latex]

Mississippi:  [latex]y = 924x+25,200[/latex]

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

Hawaii

[latex]y = 3966x+74,400\\y = 3966(0)+74,400\\y = 74,400[/latex]

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

Mississippi

[latex]y = 924x+25,200\\y = 924(0)+25,200\\y = 25,200[/latex]

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

Example

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

Substitute [latex]0[/latex] in for [latex]x[/latex].

[latex]y = -1.75x+16\\y = -1.75(0)+16\\y = 16[/latex]

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

In the year [latex]2011[/latex], [latex]16[/latex] out of every [latex]100[/latex] high school students smoked.

Example

Use the equations for house value in Hawaii and Mississippi to predict house value in [latex]2035[/latex].

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

How many years are between [latex]1950[/latex] and [latex]2035[/latex]? [latex]2035 - 1950 = 85[/latex]

This is our [latex]x[/latex]-value.

For Hawaii:

[latex]y = 3966x+74,400\\y = 3966(85)+74,400\\y = 337110+74,400 = 411,510[/latex]

Holy cow! The average price for a house in Hawaii in [latex]2035[/latex] is predicted to be [latex]$411,510[/latex] 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:

[latex]y = 924x+25,200\\y = 924(85)+25,200\\y = 78540+25,200 = 103,740[/latex]

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

Example

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

[latex]y = -1.75x+16[/latex]

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

Substitute [latex]0[/latex] for [latex]y[/latex].

[latex]y = -1.75x+16[/latex]

[latex]0 = -1.75x+16[/latex]

[latex]-16 = -1.75x[/latex]

[latex]\frac{-16}{-1.75} = x[/latex]

[latex]x = 9.14[/latex] years

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

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

Example

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

Write a linear equation that represents the cost, [latex]y[/latex],  of owning and using the iPhone for [latex]x[/latex] 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 [latex]24[/latex] months?
2. If you have spent [latex]$2,580[/latex] 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 [latex]y[/latex] to represent cost, and [latex]x[/latex] to represent the number of months we have used the phone.

Define and Translate: We will use the slope-intercept form of a line, [latex]y=mx+b[/latex], because we are given a starting cost and a monthly cost for use.  We will need to find the slope and the [latex]y[/latex]-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 [latex]\frac{\Delta{y}}{\Delta{x}}[/latex], and [latex]y[/latex] is in dollars and [latex]x[/latex] is in months. The slope will be [latex]\frac{55\text{ dollars }}{1\text{ month }}[/latex]:  [latex]m=\frac{55}{1}=55[/latex]

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

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

[latex]\begin{array}{c}y=mx+b\\y=55x+600\end{array}[/latex]

Now we will answer the following questions:

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

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

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

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

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

We know that [latex]y[/latex] 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 [latex]y=$2,580[/latex] into the equation, then use what we know about solving linear equations to isolate [latex]x[/latex]:

 [latex]\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}[/latex]

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