Question 1

Solve.
-x + 5y + 2z = 1 3y + 4z = -2
Z = -2

Accepted Answer

The system of equations given is underdetermined (meaning there are more variables than equations), so typically, there would be an infinite number of solutions depending on the values of the free variables. However, since a specific value for $z$ is given ($z = -2$), we can substitute this value into the second equation to find $y$, and then use both $y$ and $z$ values to find $x$ from the first equation.Substituting $z = -2$ into the second equation:$$3y + 4(-2) = -2$$$$3y - 8 = -2$$$$3y = 6$$$$y = 2$$Now, substituting $y = 2$ and $z = -2$ into the first equation:$$x + 5(2) + 2(-2) = 1$$$$x + 10 - 4 = 1$$$$x + 6 = 1$$$$x = -5$$Therefore, the solution is $x = -5$, $y = 2$, and $z = -2$, which corresponds to choice B $(-5, 2, -2)$.

Question 2

Solve.
-Jill is 10.5 kilometers away from Joe. Both begin to walk toward each other at the same time. Jill walks at 1.5 kilometers per hour. They meet in 3 hours. How fast is Joe walking?

Accepted Answer

Jill walks 1.5 km/h for 3 hours, covering 4.5 km. The total distance is 10.5 km, so Joe covers the remaining 6 km in 3 hours, walking at 2 km/h.

Question 3

Solve.
-4x + 4y + z = 15 2x - 5y - z = 17
2x + y + 3z = 17

Accepted Answer

Rearranging the equations in matrix form, we have:
\begin{bmatrix} 4 & 4 & 1 \ 2 & -5 & -1 \ 2 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 15 \ 17 \ 17 \end{bmatrix}
The determinant of the coefficient matrix is -63. Therefore, the system has a unique solution given by: 
\begin{align*} x &= \frac{1}{-63} \begin{vmatrix} 15 & 4 & 1 \ 17 & -5 & -1 \ 17 & 1 & 3 \end{vmatrix} = 5 \ y &= \frac{1}{-63} \begin{vmatrix} 4 & 15 & 1 \ -5 & 17 & -1 \ 1 & 17 & 3 \end{vmatrix} = -2 \ z &= \frac{1}{-63} \begin{vmatrix} 4 & 4 & 15 \ -5 & -5 & 17 \ 1 & 1 & 17 \end{vmatrix} = 3 \end{align*}
Therefore, the solution is (x,y,z) = (5,-2,3).

Question 4

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

Accepted Answer

We can solve this system using substitution or elimination. Here, we will use elimination. Adding the first two equations together eliminates y:
2x + 3z = 14
Multiplying the first equation by 5 and adding it to the third equation eliminates y again:
6x + 6z = 46
Now we have two equations in terms of only x and z. Solving this system gives us:
x = 1
z = 2
Substituting these values back into any of the three original equations gives us y = 4. Therefore, the solution is (1, 2, 4), which corresponds to answer choice (B). We can solve this system by using elimination or substitution method. Here is the solution using elimination:
First, we can add the first two equations to eliminate y:
x + y + z = 7
x - y + 2z = 7
-------------
2x + 3z = 14
Next, we can multiply the second equation by 5 and subtract the first equation to eliminate y again:
5x - 5y + 10z = 35
- (x + y + z = 7)
-------------
4x + 9z = 28
Now we have two equations with two variables:
2x + 3z = 14
4x + 9z = 28
We can multiply the first equation by 2 and subtract it from the second equation to eliminate x:
4x + 9z = 28
- (4x + 6z = 28)
-----------
3z = 0
z = 0
Substitute z = 0 into the first equation to solve for x:
2x + 3z = 14
2x + 3(0) = 14
2x = 14
x = 7
Finally, substitute x = 7 and z = 0 into any of the three equations to solve for y:
x + y + z = 7
7 + y + 0 = 7
y = 0
Therefore, the solution is (x, y, z) = (7, 0, 0), which corresponds to choice B.

Question 5

Solve.
-An airplane travels 800 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 answer of Solve.
-An airplane travels 800 miles against the...

Question 6

Solve.
-A cruise boat travels 96 miles downstream in 3 hours and returns to its starting point upstream in 6 hours. Find the speed of the stream.

Accepted Answer

Let's assume the speed of the boat in still water is 'B' and the speed of the stream is 'S'.

When the boat travels downstream, the effective speed of the boat will be (B + S), and when it travels upstream, the effective speed of the boat will be (B - S).

According to the problem,

Distance downstream = 96 miles 
Time taken downstream = 3 hours 
Therefore, (B + S) = 96/3 = 32

Distance upstream = 96 miles 
Time taken upstream = 6 hours 
Therefore, (B - S) = 96/6 = 16

Now, we can solve these two equations simultaneously to find the values of 'B' and 'S'.

Adding the two equations:
(B+S) + (B-S) = 32 + 16
2B = 48
B = 24 mph

Substituting the value of 'B' in one of the equations:
(B+S) = 32
24 + S = 32
S = 8 mph

Therefore, the speed of the stream is 8 mph. So, the correct answer is (A). Let's assume the speed of the boat is "B" and the speed of the stream is "S".
When the boat travels downstream, its effective speed is increased due to the stream. So its speed can be given by the formula,
Speed downstream = B + S
Similarly, when the boat travels upstream, its effective speed is decreased due to the stream. So its speed can be given by the formula,
Speed upstream = B - S

According to the problem,
Distance downstream = Distance upstream = 96 miles
Time taken downstream = 3 hours
Time taken upstream = 6 hours

Using the formula, Distance = Speed × Time,
We can write,
Speed downstream × Time taken downstream = Distance downstream
(B + S) × 3 = 96
Speed upstream × Time taken upstream = Distance upstream
(B - S) × 6 = 96

Solving these two equations, we get,
B + S = 32            ---(1)
B - S = 16            ---(2)

Adding equation 1 and equation 2, we get,
2B = 48
B = 24

Substituting B = 24 in equation 1, we get,
24 + S = 32
S = 8

Hence, the speed of the stream is 8 mph.

Question 7

Solve.
-The speed of a stream is 4 mph. If a boat travels 60 miles downstream in the same time that it takes to travel 30 miles upstream, what is the speed of the boat in still water?

Accepted Answer

The answer of Solve.
-The speed of a stream is 4...

Question 8

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

Accepted Answer

There are a few ways to solve a system of linear equations, but one common method is to use elimination. First, add the second and third equations to eliminate z:
2x - 3y - z + 7x + y + 2z = 17 - 4
9x - 2y = 13
Next, multiply the first equation by 2, and subtract the second equation to eliminate y:
10x + 4y + 2z - (2x - 3y - z) = -11 - 17
8x + 7y + 3z = -28
Now, we have two equations with two variables:

9x - 2y = 13
8x + 7y + 3z = -28

We can solve for y by multiplying the first equation by 7/2 and adding it to the second equation:

63/2x - 7y = 91/2
8x + 7y + 3z = -28
---------------------
71/2x + 3z = -119/2

Finally, we can solve for x by multiplying the first equation by 8/9 and substituting the expression for z:

x = (8/9)(13/2) + (8/9)(3z)
x = 52/9 + 8/3z

Now we can substitute these expressions for x and z into any of the previous equations to solve for y. Let's use the first equation:

9x - 2y = 13
9(52/9 + 8/3z) - 2y = 13
y = -6z + 1

So the solution is a set of three numbers: x, y, and z. We can use any of the equations to check our solution. For example:

5x + 2y + z = -11
5(52/9 + 8/3z) + 2(-6z + 1) + z = -11
z = 1

Now we can substitute z = 1 into our expressions for x and y:

x = 52/9 + 8/3(1) = -10/9
y = -6(1) + 1 = -5

Therefore, the solution is (-10/9, -5, 1). However, none of the answer choices match this solution exactly. We can round the decimal to the nearest whole number to see which answer choice is closest:

A) (3, 0, -4)   --> x off by 4, y off by 5, z off by 3
B) (0, 6, -1)   --> x off by 1, y off by 11, z off by 2
C) (0, -6, 1)   --> x off by 1, y off by 1, z off by 0   CLOSEST
D) (-3, 0, 4)   --> x off by 2, y off by 5, z off by 3

Therefore, the best choice is C, which has the closest values to the actual solution.

Question 9

Solve.
-x + y + z = -5 x - y + 3z = -11
5x + y + z = -21

Accepted Answer

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

Question 10

Solve.
-Candy and Delvis are riding bicycles in the same direction. Candy is traveling at the speed of 7 miles per hour, and Delvis is traveling at the speed of 5 miles per hour. In 5 hours what is the distance between them?

Accepted Answer

Over 5 hours, Candy will travel 35 miles (7 mph x 5 hours) and Delvis will travel 25 miles (5 mph x 5 hours). The distance between them can be found by subtracting Delvis' distance from Candy's: 35 miles - 25 miles = 10 miles. Therefore, the answer is C.

Question 11

Solve.
-A plane flies 420 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is 27 mph, what is the speed of the plane in still air?

Accepted Answer

The answer of Solve.
-A plane flies 420 miles with the...

Question 12

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

Accepted Answer

Rearranging the equations in matrix form, we get:
$\begin{bmatrix}
7 & 7 & 1\
1 & 8 & 8\
9 & 1 & 9
\end{bmatrix} \begin{bmatrix}
x\
y\
z
\end{bmatrix} = \begin{bmatrix}
1\
8\
9
\end{bmatrix}$
Now, finding the inverse of the $3	imes3$ matrix on the left, we get:
$\begin{bmatrix}
\frac{7}{8} & -\frac{7}{64} & -\frac{1}{32}\
-\frac{1}{8} & \frac{9}{64} & \frac{1}{32}\
-\frac{1}{8} & -\frac{1}{64} & \frac{9}{32}
\end{bmatrix} \begin{bmatrix}
1\
8\
9
\end{bmatrix} = \begin{bmatrix}
0\
0\
1
\end{bmatrix}$
Therefore, the solution is $x=0,y=0,z=1$, which corresponds to answer choice D.

Question 13

Solve.
-x - y + 3z = -1 3x + z = 0
X + 5y + z = 5

Accepted Answer

The answer of Solve.
-x - y + 3z = -1...

Question 14

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's speed is twice the local train's speed, and it arrives 1 hour earlier. If the local train's speed is x, then the express train's speed is 2x. The time difference between their arrivals is 1 hour, which can be represented by the equation: $ \frac{50}{x} - \frac{50}{2x} = 1 $. Solving this equation gives x = 25 mph for the local train and 2x = 50 mph for the express train.

Question 15

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

Accepted Answer

To solve the system of equations, we can add the first and third equations to eliminate $z$, which gives us $2x = -4$, so $x = -2$. Then, substituting $x = -2$ into the second equation, we get $-2 + y + z = 6$, which simplifies to $y + z = 8$. Using the first equation, $-2 - y + z = 8$, simplifies to $-y + z = 10$. Solving the system of equations $y + z = 8$ and $-y + z = 10$ by adding them gives $2z = 18$, so $z = 9$. Substituting $z = 9$ into $y + z = 8$ gives $y = -1$. Therefore, the solution is $(-2, -1, 9)$.

Question 16

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

Accepted Answer

Using elimination method, adding the 1st and 2nd equation, we get 2x + 2z = -2 or x + z = -1. Subtracting the 1st and 3rd equation, we get 2y = 10 or y = 5. Substituting y = 5 and x + z = -1 in the 2nd equation, we get x + z + 5 = -4 or x + z = -9. Solving x + z = -1 and x + z = -9, we get x = -4 and z = 3. Therefore, the solution is (-4, 5, 3), which corresponds to option C.

Question 17

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 18

Solve.
-x - y + 2z = -7 2x + z = -5
X + 5y + z = -20

Accepted Answer

The answer of Solve.
-x - y + 2z = -7...

Question 19

Solve.
-From a point on a river, two boats are driven in opposite directions, one at 9 miles per hour and the other at 7 miles per hour. In how many hours will they be 80 miles apart?

Accepted Answer

Since the boats are moving in opposite directions, their speeds add up. So, their combined speed is 9 + 7 = 16 miles per hour. To find the time it takes for them to be 80 miles apart, divide the distance by their combined speed: 80 miles / 16 miles per hour = 5 hours.

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

The answer of Solve.
-John and Tony start from GraysLake at...

Quiz 3: Systems of Equations

Solve. -x + 5y + 2z = 1 3y + 4z = -2 Z = -2

Solve. -Jill is 10.5 kilometers away from Joe. Both begin to walk toward each other at the same time. Jill walks at 1.5 kilometers per hour. They meet in 3 hours. How fast is Joe walking?

Solve. -4x + 4y + z = 15 2x - 5y - z = 17 2x + y + 3z = 17

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

Solve. -An airplane travels 800 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. -A cruise boat travels 96 miles downstream in 3 hours and returns to its starting point upstream in 6 hours. Find the speed of the stream.

Solve. -The speed of a stream is 4 mph. If a boat travels 60 miles downstream in the same time that it takes to travel 30 miles upstream, what is the speed of the boat in still water?

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

Solve. -x + y + z = -5 x - y + 3z = -11 5x + y + z = -21

Solve. -Candy and Delvis are riding bicycles in the same direction. Candy is traveling at the speed of 7 miles per hour, and Delvis is traveling at the speed of 5 miles per hour. In 5 hours what is the distance between them?

Solve. -A plane flies 420 miles with the wind and 350 miles against the wind in the same length of time. If the speed of the wind is 27 mph, what is the speed of the plane in still air?

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

Solve. -x - y + 3z = -1 3x + z = 0 X + 5y + z = 5

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

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

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. -x - y + 2z = -7 2x + z = -5 X + 5y + z = -20

Solve. -From a point on a river, two boats are driven in opposite directions, one at 9 miles per hour and the other at 7 miles per hour. In how many hours will they be 80 miles apart?

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.