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:  [latex]y = 3966x+74,400[/latex]

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

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.

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:

 [latex]y = -1.75x+16[/latex]

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.

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 0 in for x.

Hawaii

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

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.

Mississippi

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

The y-intercept is (0, 25,200).  This means that in 1950 the value of a house in Mississippi was $25,200. Remember that x represents the number of years since 1950, so if [latex]x=0[/latex] the year is 1950.

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.

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

The y-intercept is [latex](0,16)[/latex].  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.

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? [latex]2035 - 1950 = 85[/latex]

This is our x-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 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:

[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 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?

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.

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

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.

[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, x is representing the number of years since the start of the data—which was 2011, 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 0 smokers in high school?

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, [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 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 [latex]\frac{\Delta{y}}{\Delta{x}}[/latex], and y is in dollars and x is in months. The slope will be [latex]\frac{55\text{ dollars }}{1\text{ month }}[/latex]. [latex]m=\frac{55}{1}=55[/latex]

Y-Intercept: the y-intercept is defined as a point [latex]\left(0,b\right)[/latex].  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. [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 24 months?

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

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

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:

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