The market value (in thousands of dollars)of a certain home is a linear function of t, the time in years since 1990. In 1992 the market value was 68 thousand dollars, while in 1997 the market value was 79 thousand dollars

-Find the slope of V as a linear function of t. Explain in practical terms the meaning of the slope.
-Find a formula for V as a function of t.

Well, well, well, talking about home values always puts a roof over my head! Let's dive into your questions, shall we?

First, to find the slope of V as a linear function of t, we need to use the two given data points. We know that in 1992, t=2 and V=68, and in 1997, t=7 and V=79. So, let's calculate the slope using the good ol' slope formula:

slope = (change in V) / (change in t)

Slope = (79 - 68) / (7 - 2)
Slope = 11 / 5
Slope = 2.2

Now, let's get practical! The slope tells us how much the market value of the home increases (or decreases) per year. So, in this case, for every additional year you wait, the value of the home is expected to increase by $2,200 (since we're talking about thousands of dollars). It's like having your very own money-making property!

Next up, finding a formula for V as a function of t. Since we know the slope and have two data points, we can use the point-slope form of a linear equation:

V - V1 = m(t - t1)

Let's pick a data point, say 1992: (t1, V1) = (2, 68)

V - 68 = 2.2(t - 2)

Now, we can rearrange this equation to get V on one side:

V = 2.2t - 4.4 + 68
V = 2.2t + 63.6

And there you have it, a formula for V as a function of t! Just plug in any year since 1990 into this equation, and you'll get an estimate of the market value of the home in thousands of dollars. It's like a fortune teller for home prices!

Hope that brings a smile to your face, now go forth and calculate those home values!

To find the slope of V as a linear function of t, we need to use the formula for the slope of a linear function.

Let's label the market value in thousands of dollars as V and the time in years since 1990 as t.

Given that the market value in 1992 was 68 thousand dollars and in 1997 was 79 thousand dollars, we can create two points on the line: (2, 68) and (7, 79).

Using the formula for slope, which is (change in vertical)/(change in horizontal), we can calculate the slope:

slope = (79 - 68) / (7 - 2)
= 11 / 5
= 2.2

The slope of V as a linear function of t is 2.2.

In practical terms, the slope represents the rate at which the market value of the home is changing with respect to time. In this case, the slope of 2.2 means that for every additional year since 1990, the market value of the home increases by $2,200.

To find a formula for V as a function of t, we can use the point-slope form of a linear function, which is:

V - V1 = m(t - t1),

where V1 is the market value in the given year, t1 is the corresponding time since 1990, and m is the slope.

Using the point (2, 68) and the slope of 2.2:

V - 68 = 2.2(t - 2).

Simplifying:

V - 68 = 2.2t - 4.4,
V = 2.2t - 4.4 + 68,
V = 2.2t + 63.6.

Therefore, the formula for V as a function of t is V = 2.2t + 63.6.

To find the slope of V as a linear function of t, we can use the formula for calculating the slope of a line. It is given by:

slope = (change in y) / (change in x)

In this case, the y-values represent the market values in thousands of dollars, and the x-values represent the time in years since 1990. So, the formula becomes:

slope = (change in market value) / (change in time)

To calculate the change in market value, we subtract the initial value from the final value:

change in market value = final value - initial value

Substituting the given values, we have:

change in market value = 79 - 68 = 11 (thousand dollars)

Similarly, the change in time is 1997 (year) - 1992 (year) = 5 (years).

Now, we can calculate the slope:

slope = change in market value / change in time = 11 / 5 = 2.2 (thousand dollars per year)

In practical terms, the slope represents the rate at which the market value of the home is changing with respect to time. In this case, the slope of 2.2 (thousand dollars per year) means that for each additional year since 1990, the market value of the home has increased by an average of $2,200.

Next, to find a formula for V as a function of t, we can use the point-slope form of a linear equation:

V - V₁ = m(t - t₁)

Here, V represents the market value, V₁ is the initial market value (68 thousand dollars), m is the slope (2.2 thousand dollars per year), t is the time in years since 1990, and t₁ is the initial time (1992).

Substituting the values into the formula, we get:

V - 68 = 2.2(t - 1992)

Simplifying further:

V = 2.2(t - 1992) + 68

Thus, the formula for V as a function of t is:

V = 2.2t - 4384 + 68

Simplifying again:

V = 2.2t - 4316 (in thousands of dollars)

So, this formula represents the market value of the home in thousands of dollars, as a function of the time in years since 1990.