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 =?
Further suppose that the number of one bedroom apartments must equal the total number of bigger apartments. Then we get the restriction:
x +(?)y +(?)z = 0.
Finally, find the solution to these equations: x =?, y =?, and z =?
New age? this has been standard fare for a hundred years, and millions before you have figured it out. The subject my generation suffered through was candy mixtures.
25x+35y+54z=8277000
x+y+z=236
x+-y-z=0 (in the sentence, it was stated that x=y+z)
have fun, I am not going to do the solution for you.
To find the cost equation for the builder, we need to multiply the cost of each type of apartment by the respective number of apartments and add them together.
Given that the costs for the three types of apartments are $25,000, $35,000, and $54,000 respectively, and the number of one-bedroom, two-bedroom, and three-bedroom apartments is x, y, and z respectively, the cost equation will be:
25,000x + 35,000y + 54,000z = ?
Next, let's consider the equation that describes the total number of apartments to be built, which is 236 in this case. The equation will be:
x + y + z = ?
Now, let's move on to the restriction that the number of one-bedroom apartments must be equal to the total number of bigger apartments. This means that the sum of the number of two-bedroom and three-bedroom apartments must be equal to x. So the equation becomes:
x - y - z = 0
Now, let's solve these equations to find the values of x, y, and z.
To find x, y, and z, we need to solve the system of equations:
25,000x + 35,000y + 54,000z = 8,277,000 (cost equation)
x + y + z = 236 (total number of apartments)
x - y - z = 0 (restriction)
We can use a variety of methods to solve this system, such as substitution or elimination. Let's use the elimination method.
First, let's multiply the third equation by -1 to make the coefficients of y and z the same but with opposite signs. The equations become:
25,000x + 35,000y + 54,000z = 8,277,000
x + y + z = 236
-x + y + z = 0
Now, let's add the first and second equations to eliminate x:
25,000x + 35,000y + 54,000z + x + y + z = 8,277,000 + 236
26,000x + 36,000y + 55,000z = 8,277,236
Next, let's add the second and third equations to eliminate x:
x + y + z + -x + y + z = 236 + 0
2y + 2z = 236
Now we have two equations:
26,000x + 36,000y + 55,000z = 8,277,236
2y + 2z = 236
To further simplify the equations, divide the second equation by 2:
y + z = 118
Now we have a system of two equations:
26,000x + 36,000y + 55,000z = 8,277,236
y + z = 118
Solving this system will give us the values of x, y, and z.