A building contractor is planning to build an apartment complex with one, two or three bedroom apartments. Let x,y,z respectively denote the number of apartments of each type to be built. Suppose that the builder will spend a total of $8,277,000, and that the costs for the three types of apartments are $25,000, $35,000, and $54,000 respectively. Then the 'cost equation' for the builder is:
?x + ?y + 54 z =?
Suppose he plans to build a total of 236 apartments, then complete the equation in x, y, and z which describes this:
x +(?)y +(?)z = 0.
Finally, find the solution to these equations: x =?, y =?, and z =?
If we express the cost in thousands of dollars, the costs are 25, 35 and 54 K$.
So the budget is 8277 k$.
The cost equation is then:
25x + 35y + 54z = 8277 .....(1)
The total number of apartments built
x+y+z = 236 ....(1)
Solve the system of equations (1) and (2) by eliminating x:
25x+35y+54z=8277 ....(1)
25x+25y+25z =5900 ....25*(2)
Subtract:
10y+29z=2377....(3)
Therefore, let z=t, then
x=236-y-t.....(1a)
y=(2377-29t)/10.....(2a)
z=t.......(3a)
where t is an integer such that x,y,z∈ℤ+.
with the only restriction that x,y,z are integers where x≥0, y≥0, and z≥0.
Equation (3) can be solved in integers y and z and 2377 with the help of the Euclid Algorithm applied to Diophantine equations to give
y=7131+29n
z=-2377-10n
Solving for n with 236≥y≥0 and 236≥z≥0, and such that 236≥x≥0, we get
8 possible answers for (x,y,z) as follows:
(4,229,3)
(23,200,13)
(42,171,23)
(61,142,33)
(80,113,43)
(99,84,53)
(118,55,63)
(137,26,73)
Check that each solution satisfies all the given conditions (cost and number of units).
For example:
25*137+35*26+54*73=8277
137+26+73=236.
If you have not done Diophantine equations before, then the answer is provided by 1a to 3a, with an appropriate choice of t to give x,y and z as integers, and satisfies x,y,z≥0.
Obvious choices for t are 10k+3 (because the last digit will end in a 7, which makes y an integer.
Thanks for trying. But what you just wrote does not help at all. Can someone else try?
Sorry that it is not helping.
If no one answers in the next while, I suggest you repost your question, preferably indicating your subject, such as algebra, number theory, diophantine euqations, etc.
However, if you had read through the first six lines, you would have found the answer to the first two of the three questions. Correction:
x+y+z = 236 ....(1)
Above equation should have read equation (2), which is the answer to the second question.
The rest is at college or university level, but unfortunately I do not know if you are doing algebra or number theory. The solution (third question) is long because there are only two equations for three unknowns, and there are eight possible answers, for example x=4, y=229, z=3, etc.
Good luck!
To find the cost equation for the builder, we need to multiply the number of each type of apartment by its respective cost and sum them up.
The cost equation for the builder is: 25,000x + 35,000y + 54,000z = 8,277,000.
To complete the equation in x, y, and z for a total of 236 apartments, we can use the fact that the total number of apartments should be equal to the sum of individual types of apartments.
So, the equation becomes: x + y + z = 236.
To find the solution to these equations, we can use a method called substitution or elimination. Let's solve it using the substitution method.
1. Solving the cost equation and equation for total apartments simultaneously:
From the cost equation: 25,000x + 35,000y + 54,000z = 8,277,000.
From the equation for total apartments: x + y + z = 236.
We can solve for one variable in one equation and substitute it into the other equation.
Let's solve for x in terms of y and z from the equation for total apartments:
x = 236 - y - z.
Substituting this value of x in the cost equation:
25,000(236 - y - z) + 35,000y + 54,000z = 8,277,000.
2. Simplifying and rearranging the equation:
5,900,000 - 25,000y - 25,000z + 35,000y + 54,000z = 8,277,000.
Combining like terms:
-10,000y + 29,000z = 2,377,000.
3. Now, we have one equation with two variables. To find a unique solution, we need another equation. Let's use the equation for total apartments.
x + y + z = 236.
Substituting the value of x from earlier:
236 - y - z + y + z = 236.
Simplifying the equation:
236 = 236.
4. The equation 236 = 236 indicates that this equation is true for all values of y and z. Therefore, y and z can have any values.
5. To find the solution to these equations, we can choose any values for y and z and calculate the corresponding values for x.
For example, let's assume y = 0 and z = 0:
Using the equation x = 236 - y - z, we get:
x = 236 - 0 - 0 = 236.
Therefore, the solution to the equations is x = 236, y can be any value, and z can be any value.