1)Choose the correct description of the system of equations:2x+3y=10 and 4x+6y=20. I got consistent/dependent
2)The first equation of the system is multiplied by 2. By what number would you multiply the second equation to eliminate the x variable by adding?
6x-5y=21
4x+7y=15 I got 3
3)The first equation is multiplied by 4.By what number would you multiply the second equation to eliminate the y variable by adding?
2x+5y=16
8x-4y=10 I got 2
4)Find the coordinates of the vertices of the figure formed by the system of inequalities:
x>_-1,y>_-2,and 2x+y<_6
I got (0,0)(3,0)(0,6)
5)Use the systems of inequalities:
x>_2,y-x>_-3,and x+y<_5
Find the coordinates of the vertices of the feasible region.
I got (2,-1)(2,3)(4,1)
6)What is the value of z in the solution of the system of equations?
2x+3y=9
x-2y-z=4
x-3y+2z=-3
I got -2
7)An office building containing 96,000 square ft of spaceis to be made into apartments.There at most 15 one bedroom units,each with 800 square ft of space. The remaining units each with 1200 square ft of space,will have two bedrooms. Rent for each one bedroom is $650 and the two bedrooms are $900.How many two bedroom apts should be built to maximize revenue? 70,15,80 or 120? I got 70
8)At a university, 1200 students are enrolled in engineering. There are twice as many in electrical engineering(e)as in mechanical engineering(m),and three times as many in chemical engineering(c)as in mechanical engineering(m). Which system of equations represents the number of students in each program?
a)c+m+e=1200,2m=e,3m=c
b)c+m+e=1200,3m=e,2m=c
c)c+m+e=1200,2e=m,3c=m
d)c+m+e=1200,2m=e,3m=2e I got A
9)How many students are enrolled in the mechanical engineering program?
I got 200
1, 2, 8 and 9 are correct. Your answer to 3 is incorrect. I don't have time to check the others for you right now. For quicker reswponse, it is better to ask fewer questions in a single post.
No problem, let's go through the questions and explanations one by one:
1) Choose the correct description of the system of equations: 2x + 3y = 10 and 4x + 6y = 20.
To determine the nature of the system of equations, we can check if the two equations are consistent or inconsistent and if they have a unique solution or infinitely many solutions. One way to do this is by comparing the slopes of the two equations, which can be done by putting them in slope-intercept form (y = mx + b).
For the first equation, 2x + 3y = 10, we can rewrite it as y = (-2/3)x + (10/3). The slope of this equation is -2/3.
For the second equation, 4x + 6y = 20, we can simplify it to y = (-2/3)x + (10/3) as well. The slope of this equation is also -2/3.
Since the slopes are the same, we can conclude that the equations are consistent and dependent, meaning they represent the same line and have infinitely many solutions.
2) The first equation of the system is multiplied by 2. By what number would you multiply the second equation to eliminate the x variable by adding?
The goal here is to multiply one of the equations by a suitable factor so that when you add or subtract the two equations, one of the variables is eliminated.
The given equations are:
6x - 5y = 21
4x + 7y = 15
To eliminate the x variable, we need to find the common multiple of the coefficients of x in both equations. In this case, the common multiple is 12. So, you would multiply the first equation by 2 and the second equation by 3 to get:
12x - 10y = 42
12x + 21y = 45
Now, when you subtract these two equations, the x variable will be eliminated.
3) The first equation is multiplied by 4. By what number would you multiply the second equation to eliminate the y variable by adding?
Similar to the previous question, the goal here is to multiply one of the equations by a suitable factor so that when you add or subtract the two equations, one of the variables is eliminated.
The given equations are:
2x + 5y = 16
8x - 4y = 10
To eliminate the y variable, we need to find the common multiple of the coefficients of y in both equations. In this case, the common multiple is 20. So, you would multiply the first equation by 4 and the second equation by 5 to get:
8x + 20y = 64
40x - 20y = 50
Now, when you add these two equations, the y variable will be eliminated.
4) Find the coordinates of the vertices of the figure formed by the system of inequalities:
x > -1, y > -2, and 2x + y < 6
To find the vertices, we need to find the points where the inequalities intersect.
The vertex points can be found by solving the system of equations formed by the equalities:
x = -1, y = -2, and 2x + y = 6
Solving these equations, we get the vertex points: (-1, -2), (0, -2), (0, 6).
5) Use the system of inequalities:
x >= 2, y - x >= -3, x + y <= 5
To find the coordinates of the vertices of the feasible region, we need to find the points where the inequalities intersect.
The vertex points can be found by solving the system of equations formed by the equalities:
x = 2, y - x = -3, and x + y = 5
Solving these equations, we get the vertex points: (2, -1), (2, 3), (4, 1).
6) What is the value of z in the solution of the system of equations?
2x + 3y = 9
x - 2y - z = 4
x - 3y + 2z = -3
To solve this system, we can use the method of substitution or elimination. Let's use the elimination method to eliminate the x variable.
Multiply the first equation by 2 and the second equation by -1, then add the two equations together:
4x + 6y = 18
-x + 2y + z = -4
-----------
3y + z = 14
Now let's substitute this result into the third equation and solve for z:
x - 3y + 2z = -3
x = -3y + 14 - 2z
Substitute this value of x into the first equation:
2(-3y + 14 - 2z) + 3y = 9
-6y + 28 - 4z + 3y = 9
-3y - 4z = -19
3y + 4z = 19
Multiply the previous equation by 3 and add it to the new equation:
3(3y + 4z) = 3(19)
9y + 12z = 57
3y + 4z = 19
9y + 12z = 57
--------------
12y + 16z = 76
Divide the equation by 4:
3y + 4z = 19
12y + 16z = 76
--------------
3y + 4z = 19
12y + 16z = 76
--------------
9y + 12z = 57
Subtract the first equation from the second equation:
(12y + 16z) - (9y + 12z) = 76 - 57
12y - 9y + 16z - 12z = 19
3y + 4z = 19
Substitute the value of 3y + 4z from the previous equation into the first equation:
3y + 4z = 19
3y + 4z = 19
Since the two equations are identical, the value of z can vary. In this case, no specific value of z can be determined.
7) An office building containing 96,000 square ft of space is to be made into apartments. There are at most 15 one-bedroom units, each with 800 square ft of space. The remaining units, each with 1200 square ft of space, will have two bedrooms. Rent for each one bedroom is $650, and the two bedrooms are $900. How many two-bedroom apartments should be built to maximize revenue?
To maximize revenue, we need to find the number of two-bedroom apartments that will generate the most income.
Let's assume the number of two-bedroom apartments is x. The number of one-bedroom apartments would then be (15 - x).
The total revenue is given by:
Total Revenue = (Number of one-bedroom apartments * Rent for one-bedroom) + (Number of two-bedroom apartments * Rent for two-bedroom)
Total Revenue = ((15 - x) * 650) + (x * 900)
To maximize the revenue, we can find the value of x that maximizes this equation. We can do this by finding the vertex of the quadratic equation representing this revenue.
The vertex of a quadratic equation in the form ax^2 + bx + c can be found using the formula x = -b/2a. In this case, a = -250 and b = 2250.
x = -(2250) / (2 * -250)
x = -2250 / -500
x = 4.5
Since we can't have half of a two-bedroom apartment, we round down to 4. Therefore, the maximum revenue is obtained with 4 two-bedroom apartments.
8) At a university, 1200 students are enrolled in engineering. There are twice as many in electrical engineering (e) as in mechanical engineering (m), and three times as many in chemical engineering (c) as in mechanical engineering (m). Which system of equations represents the number of students in each program?
A) c + m + e = 1200, 2m = e, 3m = c
In this system of equations, we can write the relationship between the number of students in each program:
The number of students in electrical engineering (e) is twice the number in mechanical engineering (m), so 2m = e.
The number of students in chemical engineering (c) is three times the number in mechanical engineering (m), so 3m = c.
We can also add up the number of students in all three programs and set it equal to the total number of students in engineering, which is 1200, so c + m + e = 1200.
Therefore, the correct system of equations is:
c + m + e = 1200
2m = e
3m = c
9) How many students are enrolled in the mechanical engineering program?
To find the number of students enrolled in the mechanical engineering program, we can substitute the value of m into one of the other equations.
From the given system of equations:
2m = e
We can substitute 2m for e in the equation c + m + e = 1200:
c + m + 2m = 1200
c + 3m = 1200
Since we don't have a specific value for c, we can't determine the exact number of students in the mechanical engineering program.