Welcome to lesson in Writing Linear Equations
in Slope-Intercept Form to Model Situations.
Before we begin, remember, an equation
in slope-intercept form is in the form, y equal m x plus b,
or if we have a function, f of x equal m x plus b,
where m is the slope of the line,
but for an application problem,
the slope is often given as the rate of change,
and b is equal to the vertical intercept,
and again for an application problem,
it's often referred to as
the initial value or starting value.
Remember as an ordered pair,
the vertical intercept would be the point zero comma b.
So for this example, you have just bought a new
Sony 55 inch 3D television for $2,300.
The TV's value decreases at a rate of $250 per year.
We're asked to construct a linear function
to represent this situation.
Let's highlight the important information.
The cost of the new TV is $2,300,
and it decreases at a rate of $250 per year.
Let's define our variables,
and what we'll do is write an equation,
and then write the equation as a function when we finish,
because notice how the question
does ask for a linear function.
So let's let V equal to the value of the TV in dollars,
and let's let t equal time in years,
because the TV is decreasing at a rate of $250 per year.
And now because the TV cost $2,300,
again this is the starting value, initial value,
so let's go ahead and write this down.
The starting value is equal to $2,300,
so this is the value at time t equal zero,
and therefore the starting value would be
the ordered pair, zero comma 2,300.
So this is the vertical intercept,
and therefore we know that b equal $2,300 in our equation.
And then finally the TV is decreasing
at a rate of $250 per year, so the rate of change
is equal to negative $250 per year,
which means that slope m equals negative 250.
So our equation in slope-intercept form
would be in the form of V equals m t plus b,
notice how we replaced y with V and x with t.
So our linear equation is going to be
V equals negative 250 t plus 2,300.
Now we are asked to write the equation as a function,
notice how V is a function of t, so let's write this as
V of t equals negative 250 t plus 2,300.
Before we look at example two,
let's look at this another way,
let's think of it more logically.
We can think to ourselves that the value of the TV,
V of t is going to be equal to the starting value of 2,300,
and then because it decreases at a rate of $250 per year,
we would subtract $250 for every year,
so it's minus 250 times t.
Notice how these two terms are equivalent,
we just have the terms on the right side
are in different order.
Both of these would be acceptable.
Let's look at example two.
Before we find the equation for example two, remember,
if we're not given the slope and the vertical intercept,
we can still find the equation or linear function,
linear equation, if we're given a point
in the slope, or two points.
So for example two, in 1998, the cost of tuition
at a large Midwestern university was $144 per credit hour.
In 2008, tuition had risen to $238 per credit hour.
We're asked to determine a linear equation this time
to represent the cost, C, of the tuition
as a function of x, the number of years since 1990.
So notice here, we are given our variables,
where C equals the cost in dollars,
and x is equal to the number of years since 1990.
So the input would be the number of years since 1990,
and the output is the cost in dollars.
Let's highlight the important information.
We're told in 1998, the cost of tuition
was $144 per credit hour, and then in 2008,
tuition had risen to $238 per credit hour.
Let's write this information down below,
so in 1998, the cost was $144 per credit hour,
and then in 2008, the cost was $238 per credit hour.
This information represents two ordered pairs,
that would satisfy our linear equation,
we need to be careful here though.
We know the input is x, but x is the number of years
since 1990, so x is not 1998, and x is not 2008,
x would be the number of years since 1990.
So here, x is equal to the given year
minus the base year of 1990, so x equals eight,
which means that this information would be
the ordered pair eight, the number of years since 1990,
and the output would be the cost per credit hour, 144.
And then here, x is equal to 2008 minus 1990
which equals 18, so the ordered pair here
would be 18 comma 238.
Now that we have two ordered pairs that we know
must satisfy our linear equation,
we can define the slope using the slope formula,
then perform substitution to find the vertical intercept.
So let's take these two ordered pairs
onto the next slide and find the slope.
I know our variables are aren't x and y,
but for the purpose of our slope formula,
let's call this ordered pair x sub one, y sub one,
and this ordered pair x sub two, y sub two.
So using our slope formula, we know m is equal to
y sub two minus y sub one, or 238 minus 144,
divided by x sub two minus x sub one, that's 18 minus eight,
so this simplifies to 94 divided by 10, which equals 9.4.
To understand what this means, remember,
this is the ratio of the change in the cost
to the change in time in years after 1990,
so if we have 9.4 over one, this is telling us that the
credit hour cost is increasing
at a rate of $9.40 per year.
And for the purpose of our equation, we know m is now 9.4,
and therefore our equation,
because we're using the variables C and x,
so far would be C equals 9.4 x plus b,
where again b is the vertical intercept.
We don't know the vertical intercept,
but now we can perform substitution
with one of these ordered pairs to find b,
so let's use this point here,
this ordered pair, so using eight comma 144,
we'll substitute eight for x, and 144 for C.
So we would have 144 equals 9.4 times eight plus b,
9.4 times eight is equal to 75.2.
So we have 144 equals 75.2 plus b,
subtracting 75.2 on both sides, simplifying.
The difference on the left is equal to 68.8.
On the right we just have b.
Now that we have the slope and the vertical intercept,
we now can write the equation, modeling the situation.
We would have C equals 9.4 x plus 68.8.
I hope you found this helpful.