Question 1

Solve.
-Candy and Delvis are riding bicycles in the same direction. Candy is traveling at the speed of 5 miles per hour, and Delvis is traveling at the speed of 3 miles per hour. In 4 hours what is the distance between them?

Accepted Answer

In 4 hours, Candy travels 4 * 5 = 20 miles, and Delvis travels 4 * 3 = 12 miles. The distance between them is 20 - 12 = 8 miles.

Question 2

Solve.
-x - y + 5z = -10 
4x + z = -1
X + 2y + z = 9

Accepted Answer

The answer of Solve.
-x - y + 5z = -10...

Question 3

Solve.
-x - y + z = 8 
x + y + z = 6
X + y - z = -12

Accepted Answer

Adding the first and third equations gives 2x = -4, so x = -2. Subtracting the third equation from the second gives 2z = 18, so z = 9. Substituting x and z into the first equation gives -2 - y + 9 = 8, so y = -1. Therefore, the solution is (-2, -1, 9).

Question 4

Solve.
-x - y + 2z = 4 
5x + z = 0
X + 2y + z = -8

Accepted Answer

The answer of Solve.
-x - y + 2z = 4...

Question 5

Solve.
-A cruise boat travels 72 miles downstream in 4 hours and returns to its starting point upstream in 6 hours. Find the speed of the stream.

Accepted Answer

The speed of the stream can be found by first determining the speed of the boat in still water and the effective speed downstream and upstream. Downstream speed = distance/time = 72 miles / 4 hours = 18 mph. Upstream speed = 72 miles / 6 hours = 12 mph. The speed of the boat in still water is the average of these two speeds, (18 mph + 12 mph) / 2 = 15 mph. The speed of the stream is the difference between the boat's speed in still water and upstream or downstream speed, which is (18 mph - 15 mph) or (15 mph - 12 mph) = 3 mph.

Question 6

Solve.
-x + y + z = 7 
x - y + 2z = 7
5x + y + z = 11

Accepted Answer

The answer of Solve.
-x + y + z = 7...

Question 7

Solve.
-An airplane travels 600 miles against the wind in 5 hours, and makes the return trip with the same wind in 1 hours. Find the rate of the wind.

Accepted Answer

The airplane's speed against the wind is 600 miles / 5 hours = 120 mph, and with the wind is 600 miles / 1 hour = 600 mph. The wind's speed is the difference between the two divided by 2, so (600 - 120) / 2 = 240 mph.

Question 8

Solve.
-The speed of a stream is 5 mph. If a boat travels 68 miles downstream in the same time that it takes to travel 34 miles upstream, what is the speed of the boat in still water?

Accepted Answer

Let the speed of the boat in still water be x mph. Downstream speed is (x + 5) mph and upstream speed is (x - 5) mph. The time taken for both journeys is the same, so 68/(x + 5) = 34/(x - 5). Solving this equation gives x = 15 mph.

Question 9

Solve.
-x + 5y + 2z = -27 
4y + 3z = -31
Z = -5

Accepted Answer

Given $z = -5$, we can substitute this value into the second equation to find $y$: $4y + 3(-5) = -31 \Rightarrow 4y - 15 = -31 \Rightarrow 4y = -16 \Rightarrow y = -4$. Then, substituting $y = -4$ and $z = -5$ into the first equation to find $x$: $x + 5(-4) + 2(-5) = -27 \Rightarrow x - 20 - 10 = -27 \Rightarrow x = 3$. Therefore, the solution is $x = 3$, $y = -4$, $z = -5$, which corresponds to choice B.

Question 10

Solve.
-Jill is 14 kilometers away from Joe. Both begin to walk toward each other at the same time. Jill walks at 2.5 kilometers per hour. They meet in 4 hours. How fast is Joe walking?

Accepted Answer

The answer of Solve.
-Jill is 14 kilometers away from Joe....

Question 11

Solve.
-3x + 5y + z = -22 
3x - 2y - z = -2
4x + y + 2z = -4

Accepted Answer

The system of equations has a unique solution, which can be found by using methods such as substitution, elimination, or matrix operations. Solving the given system yields the solution (-2, -4, 4).

Question 12

Solve.
-A plane flies 420 miles with the wind and 300 miles against the wind in the same length of time. If the speed of the wind is 29 mph, what is the speed of the plane in still air?

Accepted Answer

The speed of the plane in still air can be found by setting up the equation for time, which is distance divided by speed. Let $ p $ be the speed of the plane in still air and $ w $ be the speed of the wind, which is given as 29 mph. With the wind, the plane's speed is $ p + w $, and against the wind, the plane's speed is $ p - w $. Since the times are equal, we have $\frac{420}{p + 29} = \frac{300}{p - 29}$. Solving this equation for $ p $ gives us the speed of the plane in still air as 174 mph.

Question 13

Solve.
-x + y + z = -6 
x - y + 4z = 1
4x + y + z = -21

Accepted Answer

The system of equations can be solved using methods such as substitution, elimination, or matrix operations. Solving it, we find that the solution is x = -5, y = -2, and z = 1, which corresponds to choice C.

Question 14

Solve.
-A cabin cruiser travels 20 miles in the same time that a power boat travels 40 miles. The cruiser travels 5 mph slower than the power boat. Find the speed of each boat.

Accepted Answer

The answer of Solve.
-A cabin cruiser travels 20 miles in...

Question 15

Solve.
-From a point on a river, two boats are driven in opposite directions, one at 9 miles per hour and the other at 13 miles per hour. In how many hours will they be 44 miles apart?

Accepted Answer

The relative speed of the two boats is the sum of their speeds, which is 9 + 13 = 22 miles per hour. To find the time it takes for them to be 44 miles apart, divide the distance by the relative speed: 44 miles ÷ 22 miles per hour = 2 hours.

Question 16

Solve.
-x - y + z = 4 
x + y + z = -6
X + y - z = -12

Accepted Answer

By solving the system of equations, we find that x = -4, y = -5, and z = 3.

Question 17

Solve.
-7x + 7y + z = 1 
x + 8y + 8z = 8
9x + y + 9z = 9

Accepted Answer

Solving the system of equations using methods such as substitution or matrix inversion yields the solution x = 0, y = 0, z = 1, which corresponds to choice A.

Question 18

Solve.
-5x + 2y + z = -11 
2x - 3y - z = 17
7x + y + 2z = -4

Accepted Answer

The answer of Solve.
-5x + 2y + z = -11...

Question 19

Solve.
-An express and local train leave GraysLake at 3 P.M. and head for Chicago 50 miles away. The express travels twice as fast as the local and arrives 1 hour ahead of the local. Find the speed of each train.

Accepted Answer

The express train travels twice as fast as the local train and arrives 1 hour earlier. If the local train's speed is x mph, then the express train's speed is 2x mph. The local train takes 50/x hours, and the express train takes 50/(2x) hours. The express arrives 1 hour earlier, so 50/x - 50/(2x) = 1. Solving this equation gives x = 25 mph for the local train and 50 mph for the express train.

Question 20

Solve.
-John and Tony start from GraysLake at the same time and head for a town 10 miles away. John walks twice as fast as Tony and arrives 3 hours before Tony. Find how fast each walks.

Accepted Answer

Let Tony's speed be $ x $ miles per hour. Then John's speed is $ 2x $ miles per hour. The time taken by Tony is $ \frac{10}{x} $ hours and by John is $ \frac{10}{2x} = \frac{5}{x} $ hours. Since John arrives 3 hours before Tony, we have $ \frac{10}{x} - \frac{5}{x} = 3 $. Solving this gives $ x = \frac{5}{3} $ miles per hour for Tony and $ 2x = \frac{10}{3} $ miles per hour for John.

Quiz 8: Systems of Equations

Solve. -Candy and Delvis are riding bicycles in the same direction. Candy is traveling at the speed of 5 miles per hour, and Delvis is traveling at the speed of 3 miles per hour. In 4 hours what is the distance between them?

Solve. -x - y + 5z = -10 4x + z = -1 X + 2y + z = 9

Solve. -x - y + z = 8 x + y + z = 6 X + y - z = -12

Solve. -x - y + 2z = 4 5x + z = 0 X + 2y + z = -8

Solve. -A cruise boat travels 72 miles downstream in 4 hours and returns to its starting point upstream in 6 hours. Find the speed of the stream.

Solve. -x + y + z = 7 x - y + 2z = 7 5x + y + z = 11

Solve. -An airplane travels 600 miles against the wind in 5 hours, and makes the return trip with the same wind in 1 hours. Find the rate of the wind.

Solve. -The speed of a stream is 5 mph. If a boat travels 68 miles downstream in the same time that it takes to travel 34 miles upstream, what is the speed of the boat in still water?

Solve. -x + 5y + 2z = -27 4y + 3z = -31 Z = -5

Solve. -Jill is 14 kilometers away from Joe. Both begin to walk toward each other at the same time. Jill walks at 2.5 kilometers per hour. They meet in 4 hours. How fast is Joe walking?

Solve. -3x + 5y + z = -22 3x - 2y - z = -2 4x + y + 2z = -4

Solve. -A plane flies 420 miles with the wind and 300 miles against the wind in the same length of time. If the speed of the wind is 29 mph, what is the speed of the plane in still air?

Solve. -x + y + z = -6 x - y + 4z = 1 4x + y + z = -21

Solve. -A cabin cruiser travels 20 miles in the same time that a power boat travels 40 miles. The cruiser travels 5 mph slower than the power boat. Find the speed of each boat.

Solve. -From a point on a river, two boats are driven in opposite directions, one at 9 miles per hour and the other at 13 miles per hour. In how many hours will they be 44 miles apart?

Solve. -x - y + z = 4 x + y + z = -6 X + y - z = -12

Solve. -7x + 7y + z = 1 x + 8y + 8z = 8 9x + y + 9z = 9

Solve. -5x + 2y + z = -11 2x - 3y - z = 17 7x + y + 2z = -4

Solve. -An express and local train leave GraysLake at 3 P.M. and head for Chicago 50 miles away. The express travels twice as fast as the local and arrives 1 hour ahead of the local. Find the speed of each train.

Solve. -John and Tony start from GraysLake at the same time and head for a town 10 miles away. John walks twice as fast as Tony and arrives 3 hours before Tony. Find how fast each walks.