This is the (average) sale price for single family property in Seattle and Port Townsend
Year Seattle Port Townsend
1970 $38000 $8400
1990 $175000 $168400
(a) When will the average sales price in seattle and port t. be equal and what is this price?
(b)When will the average sales price in port t. be $15000 less than seattle sales price? What are the two sales prices at this time?
I need help setting up the equations for these two questions.
i found the linear model relating to this question
seattle is :
y=6850(x-1970)+38000
port t:
y=7980(x-1970)+8400
Thanks for helping. What you explained to me makes sense. But i checked the back of the book and it said for (a) they will be equal in 1995 at $214,313.
so is (b) it said its 1982.70 Sea=$124,965. weird...
I going to try these problems out! and see what answers I get! thanks again.
(a) To find when the average sales price in Seattle and Port Townsend will be equal, we can set their linear models equal to each other and solve for x:
6850(x - 1970) + 38000 = 7980(x - 1970) + 8400
Let's simplify this equation:
6850x - 6850*1970 + 38000 = 7980x - 7980*1970 + 8400
6850x - 13415550 + 38000 = 7980x - 15756600 + 8400
6850x - 13377550 = 7980x - 15748200
Now let's rearrange the equation to isolate x:
6850x - 7980x = -15748200 + 13377550
-1130x = -2370650
x = (-2370650) / (-1130)
x ≈ 2096.46
Therefore, the average sales price in Seattle and Port Townsend will be equal in the year 2096.46 (rounded to the nearest year).
To find the price, substitute this value of x into either equation:
y = 6850(x - 1970) + 38000
y ≈ 6850(2096.46 - 1970) + 38000
y ≈ 6850(126.46) + 38000
y ≈ 866729 + 38000
y ≈ $904,729
So, the average sales price when Seattle and Port Townsend will be equal is approximately $904,729.
(b) To find when the average sales price in Port Townsend will be $15,000 less than Seattle, we can set up another equation:
7980(x - 1970) + 8400 = 6850(x - 1970) + 38000 - 15000
Let's simplify this equation:
7980x - 7980*1970 + 8400 = 6850x - 6850*1970 + 38000 - 15000
7980x - 15756600 + 8400 = 6850x - 13415550 + 23000
7980x - 15748200 + 8400 = 6850x - 13406550
7980x - 7980x = -15748200 + 13406550 + 8400
0 = -23332850 + 8400
0 = -23324450
Since the equation above leads to an inconsistent statement, it means that the average sales price in Port Townsend will never be $15,000 less than Seattle. Therefore, there is no specific year or sales price for this condition.
To set up the equations for these questions, let's define the variables as follows:
Let x be the number of years since 1970.
Let y be the average sales price.
(a) When will the average sales price in Seattle and Port Townsend be equal, and what is this price?
For Seattle, the linear equation is given as:
y1 = 6850(x - 1970) + 38000.
For Port Townsend, the linear equation is given as:
y2 = 7980(x - 1970) + 8400.
To find when the average sales prices will be equal, we need to solve the equation: y1 = y2.
Substituting the equations into y1 = y2:
6850(x - 1970) + 38000 = 7980(x - 1970) + 8400.
Simplifying the equation:
6850x - 6850*1970 + 38000 = 7980x - 7980*1970 + 8400.
Now, you can solve this equation for x to determine the year when the average sales prices in Seattle and Port Townsend will be equal.
(b) When will the average sales price in Port Townsend be $15000 less than the Seattle sales price? What are the two sales prices at this time?
To find when the average sales price in Port Townsend is $15000 less than the Seattle sales price, we can set up the equation: y2 = y1 - 15000.
Substituting the equations for y1 and y2 respectively:
7980(x - 1970) + 8400 = 6850(x - 1970) + 38000 - 15000.
Simplifying the equation:
7980x - 7980*1970 + 8400 = 6850x - 6850*1970 + 38000 - 15000.
Now, you can solve this equation for x to determine the year when the average sales price in Port Townsend is $15000 less than the Seattle sales price.
To solve these questions, we need to set up equations based on the given linear models for Seattle and Port Townsend. Let's go step by step:
(a) To find when the average sales price in Seattle and Port Townsend will be equal and what this price is, we can set up an equation by equating the two linear models:
6850(x - 1970) + 38000 = 7980(x - 1970) + 8400
Simplifying this equation will give us the value of x (the year) when the average prices are equal:
6850x - 13869500 + 38000 = 7980x - 15780600 + 8400
Solving this equation, we find that x ≈ 2002. Therefore, the average sales price in Seattle and Port Townsend will be equal in the year 2002.
To find the price at this time, substitute the value of x = 2002 into either equation. Let's use the Seattle equation to find the price:
y = 6850(2002 - 1970) + 38000 ≈ $377,900
So, the average sales price at that time will be approximately $377,900.
(b) To find when the average sales price in Port Townsend will be $15,000 less than the Seattle sales price, we can set up another equation:
6850(x - 1970) + 38000 = 7980(x - 1970) + 8400 - 15000
Simplifying this equation will give us the value of x (the year) when the Port Townsend price is $15,000 less than the Seattle price:
6850x - 13869500 + 38000 = 7980x - 15780600 - 6600
Solving this equation will give us the value of x ≈ 1991.8. Therefore, the average sales price in Port Townsend will be $15,000 less than the Seattle price in the year 1991.8.
To find the prices at this time, substitute the value of x = 1991.8 into both equations:
For Seattle:
y = 6850(1991.8 - 1970) + 38000 ≈ $199,770
For Port Townsend:
y = 7980(1991.8 - 1970) + 8400 ≈ $184,770
So, the average sales price in Seattle at that time will be approximately $199,770, and in Port Townsend, it will be approximately $184,770.
The linear equations are:
seattle: y=6580(x-1970)+38000
port t: y=8000(x-1970)+8400 <---you made an arithmetic error in calculating the slope for this one
1. To find when they will be equal just set the to equations equal to each other:
6580(x-1970)+38000=8000(x-1970)+8400
6580x-12962600+3800=8000x-15760000+8400
2792800=1420x
x=1966.76
1966
2. Here you just take the port t equation, add 15000, and set that equal to the seattle equation
so
8000(x-1970)+8400+15000=6580(x-1970)+38000
do the algebra and you get: 1980.28
so 1980
the second part ask for the two sales prices during this time, so just plug our answer, 1980, back into the equations:
seattle: y=6580([1980.28-1970)+38000, y= $105,642.4
port t: y=y=8000(1980.28-1970)+8400, y= $90640
we can then check our answer by doing 105,642-90640=15002 which confirms that seattle at this time is 15000 more than port t. (the number is a tiny bit off because 1980.28 is rounded)