1.
Explain how you can tell if a system has no solution.
2.
Explain how you can tell is a system has infinitely many solutions.
3.
A rectangular park is three blocks longer than it is wide. The perimeter of the park is 26 blocks. If w represents the width, write an equation in terms of w for the perimeter of the park. Find the length and the width of the park.
4.
Suppose you and your friends form a band and you want to record a demo. Studio A rents for $75 plus $75 an hour and Studio B rents for $150 and $50 an hour. Let t = the number of hours and c = cost.- Write an equation to represent the cost at each studio.
- Solve the system
- Explain what the solution to the system means in terms of where your band will rent a studio.
5.
Solve the system:5x + 4y = 495x – 4y = 1
Correct Answer
A. (5,6)
Explanation
The correct answer is (5,6) because when we substitute x=5 and y=6 into the two equations, we get 5(5) + 4(6) = 49 and 5(5) - 4(6) = 1. Therefore, the solution satisfies both equations and is the correct answer.
6.
Solve the system:5x + y = 10x + 3y = -26
Correct Answer
B. (4,-10)
Explanation
The correct answer is (4,-10). This is the solution to the system of equations 5x + y = 10 and x + 3y = -2. By substituting x = 4 and y = -10 into both equations, we can see that both equations are satisfied. Therefore, (4,-10) is the solution to the system.
7.
Solve the system:3x + 5y = 62x + 5y = 4
Correct Answer
C. (2,0)
Explanation
The correct answer is (2,0) because when we substitute x=2 and y=0 into the equations, we get 3(2) + 5(0) = 6 and 2(2) + 5(0) = 4, which satisfies both equations. Therefore, (2,0) is a solution to the system of equations.
8.
If a + 3b = 13 and a + b = 5, the value of b is
Correct Answer
D. 4
Explanation
To find the value of b, we can subtract the equation a + b = 5 from the equation a + 3b = 13. This will eliminate the variable a and leave us with 2b = 8. Dividing both sides by 2, we get b = 4. Therefore, the value of b is 4.
9.
What is true of the graphs of the two lines 2x – 2y = 6 and 2x + 3y = 21?
Correct Answer
C. Intersect at (6,3)
Explanation
The two lines 2x - 2y = 6 and 2x + 3y = 21 intersect at the point (6,3). This means that the coordinates (x,y) of the point (6,3) satisfy both equations simultaneously, indicating that the lines intersect at this specific point.
10.
What is true of the graphs of the two lines 2x + 3y = 6 and -6x – 9x = -18?
Correct Answer
D. Infinitely many solutions
Explanation
The given answer, "Infinitely many solutions," is correct because the two equations represent the same line. By simplifying both equations, we can see that they are equivalent: 2x + 3y = 6 simplifies to y = -2/3x + 2, and -6x - 9y = -18 simplifies to y = -2/3x + 2. Since the two equations have the same slope and y-intercept, they represent the same line and therefore have infinitely many solutions.
11.
What is true of the graphs of the two lines -5x + 5y = -50 and 4x + 3y = 26?
Correct Answer
B. Intersect at (8,-2)
Explanation
The correct answer is "Intersect at (8,-2)". This means that the two lines represented by the equations -5x + 5y = -50 and 4x + 3y = 26 intersect at the point (8,-2). In other words, there is a single point where the two lines meet on the coordinate plane.
12.
What is the value of the y-coordinate of the solution to the system of equations 2x + y = 8 and x − 3y = −3?
Correct Answer
B. 2
Explanation
To find the value of the y-coordinate, we need to solve the system of equations. First, we can solve the second equation for x in terms of y: x = 3y - 3. Then, we substitute this expression for x into the first equation: 2(3y - 3) + y = 8. Simplifying this equation gives us 7y - 6 = 8. Solving for y, we find y = 2. Therefore, the value of the y-coordinate of the solution is 2.
13.
What is the solution of the system of equations c + 3d = 8 and c = 4d − 6?
Correct Answer
C. C = 2, d = 2
Explanation
The given system of equations can be solved by substitution. We can substitute the value of c from the second equation into the first equation. By substituting c = 4d - 6 into c + 3d = 8, we get 4d - 6 + 3d = 8. Simplifying this equation, we get 7d - 6 = 8. Solving for d, we find d = 2. Substituting this value of d back into the second equation, we get c = 4(2) - 6 = 2. Therefore, the solution to the system of equations is c = 2, d = 2.
14.
Alexandra purchases two doughnuts and three cookies at a doughnut shop and is charged $3.30. Briana purchases five doughnuts and two cookies at the same shop for $4.95. All the doughnuts have the same price and all the cookies have the same price. Find the cost of one doughnut.
Correct Answer
B. $0.75
Explanation
Let's assume the cost of one doughnut is x and the cost of one cookie is y. According to the given information, we can set up two equations:
2x + 3y = 3.30 (equation 1)
5x + 2y = 4.95 (equation 2)
To solve these equations, we can multiply equation 1 by 5 and equation 2 by 2 to eliminate y:
10x + 15y = 16.50 (equation 3)
10x + 4y = 9.90 (equation 4)
By subtracting equation 4 from equation 3, we get:
11y = 6.60
y = 0.60
Substituting the value of y back into equation 1, we can solve for x:
2x + 3(0.60) = 3.30
2x + 1.80 = 3.30
2x = 1.50
x = 0.75
Therefore, the cost of one doughnut is $0.75.