Systems And Linear Equations Assessment

1) The length of a rectangle is twice its width, while the perimeter of the rectangle is 36 units. Which system of equations can be used to find the length "L" and the width "W" of the rectangle?

2L = W2L + 2W = 36

L = 2W2L + 2W = 36

2L = WL + W = 18

L = 2W + 236 - W = L

The first equation, 2L = W, represents the relationship between the length and width of the rectangle, where the length is twice the width. The second equation, 2L + 2W = 36, represents the perimeter of the rectangle, which is the sum of the lengths of all sides. Therefore, by solving this system of equations, we can find the values of L (length) and W (width) of the rectangle.

Explanation

The first equation, 2L = W, represents the relationship between the length and width of the rectangle, where the length is twice the width. The second equation, 2L + 2W = 36, represents the perimeter of the rectangle, which is the sum of the lengths of all sides. Therefore, by solving this system of equations, we can find the values of L (length) and W (width) of the rectangle.

2) The sum of two factors of 48 is nineteen. The larger factor, x, is one more than five times the smaller factor, y. Which system of equations can be used to find the numbers?

X = 19 - yx = 5y + 1

X = 19 + yy = 5x + 1

X = y + 19y = 5x - 1

X = y - 19x = 5y + 1

The question states that the sum of two factors of 48 is nineteen. This can be represented by the equation x + y = 19. It also states that the larger factor, x, is one more than five times the smaller factor, y. This can be represented by the equation x = 5y + 1. Therefore, the system of equations that can be used to find the numbers is x = 19 - y and x = 5y + 1.

Explanation

The question states that the sum of two factors of 48 is nineteen. This can be represented by the equation x + y = 19. It also states that the larger factor, x, is one more than five times the smaller factor, y. This can be represented by the equation x = 5y + 1. Therefore, the system of equations that can be used to find the numbers is x = 19 - y and x = 5y + 1.

3) The drama department sold 300 tickets for their last show. Adult tickets cost $10 and student tickets cost $5. If they sold $2750 worth of tickets, what is a reasonable conclusion that can be made about the number of tickets sold?

Only adults bought tickets

More adults than students bought tickets

More students than adults bought tickets

The same number of adults and students bought tickets

Based on the information given, we can conclude that more adults than students bought tickets. This is because adult tickets cost $10, which is more expensive than student tickets that cost $5. Therefore, in order for the drama department to have sold $2750 worth of tickets, they would need to sell more adult tickets than student tickets.

Explanation

Based on the information given, we can conclude that more adults than students bought tickets. This is because adult tickets cost $10, which is more expensive than student tickets that cost $5. Therefore, in order for the drama department to have sold $2750 worth of tickets, they would need to sell more adult tickets than student tickets.

4) Dustin wants to buy 9 tickets to the circus. The lower balcony tickets cost $20.95 while the upper balcony tickets cost only $12.50. How many of each type of ticket did Dustin buy if he spent $146.30?

2 lower balcony tickets, 7 upper balcony tickets

4 lower balcony tickets, 5 upper balcony tickets

5 lower balcony tickets, 4 upper balcony tickets

6 lower balcony tickets, 3 upper balcony tickets

Dustin spent a total of $146.30 on 9 tickets. Let's assume he bought x lower balcony tickets and y upper balcony tickets. The cost of x lower balcony tickets would be 20.95x, and the cost of y upper balcony tickets would be 12.50y. Since he bought a total of 9 tickets, we can write the equation x + y = 9. Also, the total cost of the tickets is $146.30, so we can write the equation 20.95x + 12.50y = 146.30. Solving these two equations simultaneously, we find that x = 4 and y = 5. Therefore, Dustin bought 4 lower balcony tickets and 5 upper balcony tickets.

Explanation

Dustin spent a total of $146.30 on 9 tickets. Let's assume he bought x lower balcony tickets and y upper balcony tickets. The cost of x lower balcony tickets would be 20.95x, and the cost of y upper balcony tickets would be 12.50y. Since he bought a total of 9 tickets, we can write the equation x + y = 9. Also, the total cost of the tickets is $146.30, so we can write the equation 20.95x + 12.50y = 146.30. Solving these two equations simultaneously, we find that x = 4 and y = 5. Therefore, Dustin bought 4 lower balcony tickets and 5 upper balcony tickets.

5)

(-1, -2)

(1, 2)

(2, 1)

(-1, 1)

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Explanation

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6)

Which of the following is closest to the solution to this system of linear equations?

(3.5, -2.125)

(3.25, -1.25)

(2.5, -0.33)

(2.66, -1.75)

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Explanation

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7) At September Fest, the first dance of the year, the Student Council charged $3 for couples and $2 for singles. If 365 tickets were sold, and the total receipts were $925, which pair of equations would determine the number of each ticket sold?

X - y = 3653x + 2y = 925

X - y = 365x + y = 925

X + y = 3653x + 2y = 925

X + y = 9252x + 3y = 365

The given problem states that there were 365 tickets sold in total and the total receipts were $925. Since the cost of a couple ticket is $3 and the cost of a single ticket is $2, we can set up a system of equations to represent the situation. Let x represent the number of couple tickets sold and y represent the number of single tickets sold. The first equation x - y = 365 represents the total number of tickets sold. The second equation 3x + 2y = 925 represents the total amount of money collected from the ticket sales. Therefore, the pair of equations x + y = 365 and 3x + 2y = 925 would determine the number of each ticket sold.

Explanation

The given problem states that there were 365 tickets sold in total and the total receipts were $925. Since the cost of a couple ticket is $3 and the cost of a single ticket is $2, we can set up a system of equations to represent the situation. Let x represent the number of couple tickets sold and y represent the number of single tickets sold. The first equation x - y = 365 represents the total number of tickets sold. The second equation 3x + 2y = 925 represents the total amount of money collected from the ticket sales. Therefore, the pair of equations x + y = 365 and 3x + 2y = 925 would determine the number of each ticket sold.

8)

What is the solution to the system of equations represented by these tables?

(2,3)

(3,5)

(-1,1)

(5,11)

The solution to the system of equations represented by these tables is (5,11) because when we substitute the x and y values into the equations, they satisfy all the equations.

Explanation

The solution to the system of equations represented by these tables is (5,11) because when we substitute the x and y values into the equations, they satisfy all the equations.

9) Dominica has a total of 22 coins in her pocket, all of them nickels and dimes. If the value of the coins is $1.50, then how many nickels does Dominica have?

8 nickels

10 nickels

12 nickels

14 nickels

Dominica has a total of 22 coins in her pocket, all of them nickels and dimes. If the value of the coins is $1.50, we can set up the following equation: 0.05n + 0.10d = 1.50, where n represents the number of nickels and d represents the number of dimes. Since we know that the total number of coins is 22, we can also set up the equation n + d = 22. By solving these two equations simultaneously, we find that the number of nickels (n) is 14.

Explanation

Dominica has a total of 22 coins in her pocket, all of them nickels and dimes. If the value of the coins is $1.50, we can set up the following equation: 0.05n + 0.10d = 1.50, where n represents the number of nickels and d represents the number of dimes. Since we know that the total number of coins is 22, we can also set up the equation n + d = 22. By solving these two equations simultaneously, we find that the number of nickels (n) is 14.

10) The graph of p = 500 and the graph of p = 25s - 100 intersect at the point (24, 500). If "p" represents the profit and "s" represents sales, what does the intersection best represent?

The profit is always 500.

The sales are always 24.

When the sales are 500, the profit will be 24.

When the sales are 24, the profit will be 500.

The intersection of the graphs represents the point where the profit (p) is equal to 500 and the sales (s) is equal to 24. This means that when the sales are 24, the profit will be 500.

Explanation

The intersection of the graphs represents the point where the profit (p) is equal to 500 and the sales (s) is equal to 24. This means that when the sales are 24, the profit will be 500.

11) Bagels cost $6.99 per dozen and muffins cost $7.50 per dozen. If Rita has at most $75 to spend, which combination of bagels and muffins is not a reasonable purchase?

5 dozen bagels and 3 dozen muffins

2 dozen bagels and 8 dozen muffins

7 dozen bagels and 5 dozen muffins

4 dozen bagels and 4 dozen muffins

The combination of 7 dozen bagels and 5 dozen muffins is not a reasonable purchase because it would cost more than $75. To calculate the cost, we multiply the price per dozen by the number of dozens for each item. For bagels, the cost is $6.99 x 7 = $48.93. For muffins, the cost is $7.50 x 5 = $37.50. Adding these two costs together, we get $48.93 + $37.50 = $86.43, which is more than $75. Therefore, this combination is not reasonable within the given budget.

Explanation

The combination of 7 dozen bagels and 5 dozen muffins is not a reasonable purchase because it would cost more than $75. To calculate the cost, we multiply the price per dozen by the number of dozens for each item. For bagels, the cost is $6.99 x 7 = $48.93. For muffins, the cost is $7.50 x 5 = $37.50. Adding these two costs together, we get $48.93 + $37.50 = $86.43, which is more than $75. Therefore, this combination is not reasonable within the given budget.