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Solving Systems of Equations by Elimination
1. Solve the system of equations using elimination: \( 2x + 3y = 12 \) \( 4x - 3y = 6 \)
A. \( (3, 0) \)
B. \( (2, 2) \)
C. \( (0, 4) \)
D. \( (1, 3) \)
Multiply the first equation by 2: \( 4x + 6y = 24 \). Subtract the second equation: \( (4x + 6y) - (4x - 3y) = 24 - 6 \) gives \( 9y = 18 \) so \( y = 2 \). Substitute back to find \( x = 3 \).
2. Solve the system of equations using elimination: \( x - 2y = 3 \) \( 3x + 4y = 10 \)
A. \( (1, -1) \)
B. \( (2, -1) \)
C. \( (3, 0) \)
D. \( (0, 2) \)
Multiply the first equation by 3: \( 3x - 6y = 9 \). Subtract from the second equation: \( (3x + 4y) - (3x - 6y) = 10 - 9 \) gives \( 10y = 1 \) so \( y = -1 \). Substitute back to find \( x = 2 \).
3. Solve the system of equations using elimination: \( 5x + 2y = 11 \) \( 3x - 4y = -1 \)
A. \( (1, 3) \)
B. \( (2, 1) \)
C. \( (0, 5) \)
D. \( (3, -1) \)
Multiply the first equation by 2: \( 10x + 4y = 22 \). Add to the second equation: \( (10x + 4y) + (3x - 4y) = 22 - 1 \) gives \( 13x = 21 \), so \( x = \frac{21}{13} \). Substitute back to find \( y \).
4. Solve the system of equations using elimination: \( 2x + 5y = 20 \) \( 3x - 2y = 8 \)
A. \( (4, 2) \)
B. \( (1, 3) \)
C. \( (0, 4) \)
D. \( (2, 4) \)
Multiply the first equation by 3 and the second by 2: \( 6x + 15y = 60 \) and \( 6x - 4y = 16 \). Subtract the second from the first to get \( 19y = 44 \), so \( y = 2 \). Substitute back to find \( x = 4 \).
5. Solve the system of equations using elimination: \( 7x - 2y = 5 \) \( 4x + 3y = 21 \)
A. \( (1, 2) \)
B. \( (3, 1) \)
C. \( (2, 5) \)
D. \( (0, 3) \)
Multiply the first equation by 3 and the second by 2: \( 21x - 6y = 15 \) and \( 8x + 6y = 42 \). Add the two equations to get \( 29x = 57 \), so \( x = 3 \). Substitute back to find \( y = 1 \).
6. Solve the system of equations using elimination: \( 6x + 4y = 12 \) \( 3x - 2y = 2 \)
A. \( (0, 3) \)
B. \( (2, 0) \)
C. \( (4, 0) \)
D. \( (3, 1) \)
Multiply the second equation by 2: \( 6x - 4y = 4 \). Subtract the second equation from the first: \( (6x + 4y) - (6x - 4y) = 12 - 4 \) gives \( 8y = 8 \), so \( y = 1 \). Substitute back to find \( x = 2 \).
Solving Systems of Equations by Substitution
7. Solve the system of equations using substitution: \( y = 2x + 3 \) \( 3x - y = 4 \)
A. \( (1, 5) \)
B. \( (0, 3) \)
C. \( (2, 3) \)
D. \( (4, 3) \)
Substituting \( y = 2x + 3 \) into \( 3x - y = 4 \): \( 3x - (2x + 3) = 4 \) simplifies to \( x = 7 \). Substitute back to find \( y = 5 \).
8. Solve the system of equations using substitution: \( 2x + y = 8 \) \( y = x - 2 \)
A. \( (2, 4) \)
B. \( (0, 2) \)
C. \( (1, 5) \)
D. \( (3, 3) \)
Substituting \( y = x - 2 \) into \( 2x + y = 8 \): \( 2x + (x - 2) = 8 \) gives \( 3x = 10 \), so \( x = \frac{10}{3} \), \( y = \frac{4}{3} \).
9. Solve the system of equations using substitution: \( x + 2y = 12 \) \( y = 3 - x \)
A. \( (0, 6) \)
B. \( (3, 3) \)
C. \( (2, 4) \)
D. \( (4, 0) \)
Substituting \( y = 3 - x \) into \( x + 2y = 12 \): \( x + 2(3 - x) = 12 \) gives \( x = 3 \), so \( y = 3 \).
10. Solve the system of equations using substitution: \( 3x + 4y = 20 \) \( y = 2 - x \)
A. \( (2, 0) \)
B. \( (1, 1) \)
C. \( (0, 2) \)
D. \( (3, 0) \)
Substituting \( y = 2 - x \) into \( 3x + 4y = 20 \): \( 3x + 4(2 - x) = 20 \) gives \( x = 2 \), so \( y = 0 \).
11. Solve the system of equations using substitution: \( 4x - y = 8 \) \( y = 2x + 4 \)
A. \( (3, 10) \)
B. \( (2, 8) \)
C. \( (1, 6) \)
D. \( (0, 4) \)
Substituting \( y = 2x + 4 \) into \( 4x - y = 8 \): \( 4x - (2x + 4) = 8 \) gives \( 2x = 12 \), so \( x = 6 \), \( y = 16 \).
12. Solve the system of equations using substitution: \( 2x + 3y = 9 \) \( y = 3 - x \)
A. \( (0, 3) \)
B. \( (3, 0) \)
C. \( (1, 2) \)
D. \( (4, 2) \)
Substituting \( y = 3 - x \) into \( 2x + 3y = 9 \): \( 2x + 3(3 - x) = 9 \) gives \( 2x + 9 - 3x = 9 \), so \( x = 0 \) and \( y = 3 \).
Word Problems on Systems of Equations
13. A bookstore sells fiction and non-fiction books. The total number of books sold was 50. The fiction books sold were 10 more than twice the non-fiction books sold. How many fiction books were sold?
A. 34
B. 20
C. 25
D. 15
Let \( x \) be the number of non-fiction books sold. Then \( 50 - x \) is the number of fiction books sold. The equation is \( 50 - x = 2x + 10 \). Solve for \( x \) to find \( x = 13 \), so fiction books sold = \( 34 \).
14. Two trains leave a station at the same time. Train A travels at 60 mph, while Train B travels at 90 mph. How far apart will they be after 2 hours?
A. 300 miles
B. 150 miles
C. 240 miles
D. 180 miles
In 2 hours, Train A travels \( 60 \times 2 = 120 \) miles and Train B travels \( 90 \times 2 = 180 \) miles. Thus, the distance apart is \( 180 + 120 = 300 \) miles.
15. A total of $240 is to be divided between Alice and Bob. Alice should receive $30 more than twice what Bob receives. How much does Alice receive?
A. $180
B. $120
C. $90
D. $30
Let \( x \) be the amount Bob receives. Then \( 240 - x \) is the amount Alice receives. The equation is \( 240 - x = 2x + 30 \). Solve for \( x \) to find \( x = 70 \), so Alice receives \( 240 - 70 = 180 \).
16. A farmer has chickens and cows. The total number of animals is 20, and there are 4 more chickens than cows. How many chickens does the farmer have?
A. 12
B. 10
C. 8
D. 6
Let \( x \) be the number of cows. Then \( x + 4 \) is the number of chickens. The equation is \( x + (x + 4) = 20 \). Solving gives \( 2x + 4 = 20 \), so \( x = 8 \) (cows) and \( 12 \) (chickens).
17. A restaurant sells pasta and salads. If the total revenue from both dishes is $250, and the pasta sales are $50 more than double the salad sales, how much revenue is from pasta?
A. $150
B. $200
C. $100
D. $50
Let \( x \) be the revenue from salads. Then \( 2x + 50 \) is the revenue from pasta. The equation is \( x + (2x + 50) = 250 \). Solve for \( x \) to find \( x = 66.67 \), so pasta revenue is \( 200 \).
18. The sum of two numbers is 40. If one number is three times the other, what are the two numbers?
A. 10 and 30
B. 30 and 10
C. 5 and 35
D. 20 and 20
Let \( x \) be the smaller number. Then \( 40 - x \) is the larger number. The equation is \( x + 3x = 40 \). Solving gives \( 4x = 40 \), so \( x = 10 \), and the numbers are \( 10 \) and \( 30 \).
19. If a car travels 60 miles in 1 hour and 90 miles in 1.5 hours, what is the average speed of the car for the total distance?
A. 72 mph
B. 75 mph
C. 80 mph
D. 90 mph
The total distance traveled is \( 60 + 90 = 150 \) miles. The total time is \( 1 + 1.5 = 2.5 \) hours. Average speed is \( \frac{150}{2.5} = 60 \) mph.
20. A painter has two colors of paint. If the painter uses 3 times as much of color A as color B and the total amount of paint used is 40 liters, how many liters of color A did the painter use?
A. 30 liters
B. 20 liters
C. 15 liters
D. 10 liters
Let \( x \) be the amount of color B. Then \( 3x \) is the amount of color A. The equation is \( x + 3x = 40 \). Solve for \( x \) to find \( x = 10 \), so color A used = \( 30 \) liters.
21. Solve the following system of equations by elimination: \[ 2x + 3y = 12 \] and \[ 4x - y = 2 \].
A. \( (3, 0) \)
B. \( (0, 4) \)
C. \( (2, 2) \)
D. No solution
To eliminate \( y \), multiply the second equation by 3: \[ 12x - 3y = 6 \]. Now solve:
Subtract the first equation from this: \[ (12x - 3y) - (2x + 3y) = 6 - 12 \] gives \( 10x - 6y = -6 \), leading to \( x = 3 \) and \( y = 0 \).
22. Solve the following system of equations by substitution: \[ y = 2x + 3 \] and \[ 3x - y = 4 \].
A. \( (1, 5) \)
B. \( (1, 5) \)
C. \( (2, 7) \)
D. No solution
Substituting \( y = 2x + 3 \) into the second equation: \[ 3x - (2x + 3) = 4 \]. This simplifies to \( x - 3 = 4 \), giving \( x = 7 \) and \( y = 17 \).
23. A total of 200 tickets were sold for a concert. Adult tickets cost $20 each and child tickets cost $10 each. If the total sales were $3500, how many adult tickets were sold?
A. 150
B. 100
C. 50
D. 200
Let \( x \) be the number of adult tickets. Then \( 200 - x \) is the number of child tickets. The equations are:
1. \( x + (200 - x) = 200 \)
2. \( 20x + 10(200 - x) = 3500 \)
Solving gives \( x = 150 \) (adult tickets sold).
24. Determine if the following system of equations has no solution, one solution, or infinitely many solutions: \[ 2x + 4y = 8 \] and \[ x + 2y = 4 \].
A. No solution
B. Infinitely many solutions
C. One solution
D. None of these
The second equation can be rewritten as \( 2x + 4y = 8 \), which is identical to the first. Therefore, the system has infinitely many solutions.
25. Use Cramer's Rule to solve the following system of equations: \[ 2x + 3y = 6 \] and \[ 4x - y = 5 \].
A. \( (0, 2) \)
B. \( (1, 0) \)
C. \( (2, 0) \)
D. No solution
The coefficient matrix is \( \begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix} \) and its determinant is \( (2)(-1) - (4)(3) = -14 \). For \( D_x \), replace the first column with constants, giving \( D_x = \begin{pmatrix} 6 & 3 \\ 5 & -1 \end{pmatrix} = -27 \). For \( D_y \), replace the second column giving \( D_y = \begin{pmatrix} 2 & 6 \\ 4 & 5 \end{pmatrix} = -14 \). Hence, \( x = \frac{-27}{-14} \) and \( y = \frac{-14}{-14} \). Thus, \( x = 1.93, y = 1 \).
26. A company produces two types of gadgets, A and B. Each gadget A costs $3 to produce and each gadget B costs $5. The company has a budget of $500 and wants to produce at least 50 gadgets in total. If they produce 100 gadgets, how many of each type should they produce?
A. 20 A, 80 B
B. 50 A, 50 B
C. 100 A, 0 B
D. 80 A, 20 B
Let \( x \) be the number of gadgets A and \( y \) be the number of gadgets B. The equations are:
1. \( x + y = 100 \)
2. \( 3x + 5y \leq 500 \)
Substituting from the first equation into the second gives the feasible solutions. The answer is \( (100, 0) \).
27. A boat travels upstream at a speed of 10 km/h and downstream at 15 km/h. How long does it take to travel 60 km upstream and then return downstream?
A. 4 hours
B. 6 hours
C. 8 hours
D. 2 hours
Time taken upstream = \( \frac{60}{10} = 6 \) hours.
Time taken downstream = \( \frac{60}{15} = 4 \) hours.
Total time = 6 + 4 = 10 hours.
28. Determine if the following system of equations has no solution, one solution, or infinitely many solutions: \[ x + 2y = 4 \] and \[ 2x + 4y = 8 \].
A. No solution
B. Infinitely many solutions
C. One solution
D. None of these
The second equation is simply a multiple of the first, indicating infinitely many solutions.
29. Use Cramer's Rule to solve the following system of equations: \[ x + 3y = 9 \] and \[ 2x - y = 5 \].
A. \( (0, 3) \)
B. \( (3, 0) \)
C. \( (4, 2) \)
D. No solution
The determinant \( D \) is \( 1 \cdot (-1) - 2 \cdot 3 = -7 \). The determinants for \( D_x \) and \( D_y \) give \( D_x = (9)(-1) - (3)(5) = -12 \) and \( D_y = (1)(9) - (3)(2) = 3 \). Hence, \( x = \frac{-12}{-7} = \frac{12}{7} \) and \( y = \frac{3}{-7} = -\frac{3}{7} \).
30. A store sells pens and pencils. A pen costs $2 and a pencil costs $1. If the total number of pens and pencils sold is 80 and the total revenue is $120, how many pens were sold?
A. 60 pens
B. 40 pens
C. 20 pens
D. 80 pens
Let \( x \) be the number of pens and \( y \) be the number of pencils. The equations are:
1. \( x + y = 80 \)
2. \( 2x + y = 120 \)
Subtracting gives \( x = 40 \) (number of pens sold).
31. Determine if the following system of equations has no solution, one solution, or infinitely many solutions: \[ 3x + 2y = 6 \] and \[ 3x + 2y = 9 \].
A. No solution
B. One solution
C. Infinitely many solutions
D. None of these
Both equations have the same left-hand side but different right-hand sides. This means they represent parallel lines, which do not intersect. Therefore, the system has no solution.
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