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Quiz 5: Systems of Linear Equations

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Question 1

 
 
Which of the following equations is/are linear in the variables x, y, z?
a) x2 + 2y + 7z = 3   b) √- 2x + ey + z = 0
c) 3x-y -z = 0   d) x + 78z = 31
e) sinx + 3cosy - tanz = 0   f) x = 2y

 

There is at least one mistake.
For example, choice (a) should be false.
In a linear equation the variables cannot occur in the form of squares. When x occurs it must be in the form kx for some constant k (and the other terms must also be in this form).
There is at least one mistake.
For example, choice (b) should be false.
Terms such as ey are not permitted - the general form of a linear equation in x, y, z is ax + by + cz = d for some constants a, b, c, d.
There is at least one mistake.
For example, choice (c) should be true.
This equation is linear in x, y, z because it is in the form ax + by + cz = d for some constants a, b, c, d.
There is at least one mistake.
For example, choice (d) should be true.
This is a linear equation in x, y, z since we can think of it as x + 0y + 78z = 31.
There is at least one mistake.
For example, choice (e) should be false.
This equation is linear in the variables u = sinx, v = cosy and w = tanz but is not linear in x, y, z.
There is at least one mistake.
For example, choice (f) should be true.
This is a linear equation in x, y, z since we can think of it as x - 2y + 0z = 0.
Your answers are correct
  1. False. In a linear equation the variables cannot occur in the form of squares. When x occurs it must be in the form kx for some constant k (and the other terms must also be in this form).
  2. False. Terms such as ey are not permitted - the general form of a linear equation in x, y, z is ax + by + cz = d for some constants a, b, c, d.
  3. True. This equation is linear in x, y, z because it is in the form ax + by + cz = d for some constants a, b, c, d.
  4. True. This is a linear equation in x, y, z since we can think of it as x + 0y + 78z = 31.
  5. False. This equation is linear in the variables u = sinx, v = cosy and w = tanz but is not linear in x, y, z.
  6. True. This is a linear equation in x, y, z since we can think of it as x - 2y + 0z = 0.
 

Question 2

 
 
Which matrix is the augmented matrix of the linear system
3x + y + z - t = 2
x + y = 1
y - 2t = 7
in the variables x, y, z, t?
a) ⌊ | ⌋ 3 1 1 - 1|2 ⌈ 1 1 0 0 |1 ⌉ 0 1 0 - 2 7   b) ⌊ | ⌋ 3 1 1 - 1|2 ⌈ 0 1 0 1 |1 ⌉ 0 1 0 - 2 7
c) ⌊ 3 1 1 - 1|2 ⌋ ⌈ 1 1 0 0 |1 ⌉ 1 - 2 0 0 |7   d) ⌊ 3 1 1 - 1 |2 ⌋ ⌈ 1 1 0 0 |1 ⌉ 0 1 - 2 0 |7

 

Your answer is correct.
Not correct. Choice (b) is false.
The second row corresponds to the equation y + t = 1, which is not the correct second equation.
Not correct. Choice (c) is false.
The third row corresponds to the equation x - 2y = 7, which is not the correct third equation.
Not correct. Choice (d) is false.
The third row corresponds to the equation y - 2z = 7, which is not the correct third equation.
 

Question 3

 
 
Which system of equations corresponds to the following augmented matrix?
⌊ | ⌋ 1 2 0 |3 ⌈ 0 1 3 |6 ⌉ 0 - 1 - 1 4
a)
a + 2b = 3
b + 3c = 6
b + c = 4
  b)
x + 2z = 3
x + 3y = 6
- x - y = -4
c)
u + 2v = 3w + 3
v + 3w = 6
v + w = -4
  d)
r + 2s = 3
s + 3t = 6
- s - t = 4

 

Not correct. Choice (a) is false.
Check the last row of the augmented matrix, in particular the signs!
Not correct. Choice (b) is false.
Check the second and third rows of the augmented matrix.
Not correct. Choice (c) is false.
Check the first row of the augmented matrix.
Your answer is correct.
 

Question 4

 
 
Which two linear systems have exactly the same (unique) solution?
a)
x + 2y = 5
2x - 3y = -4
  b)
3x - 4y = 2
x - 3y = -1
c)
5x + y = 7
6x - 2y = 2

 

There is at least one mistake.
For example, choice (a) should be true.
There is at least one mistake.
For example, choice (b) should be false.
The solution to this system is x = 2, y = 1 which is not a solution to either of the other two systems.
There is at least one mistake.
For example, choice (c) should be true.
Your answers are correct
  1. True.
  2. False. The solution to this system is x = 2, y = 1 which is not a solution to either of the other two systems.
  3. True.
 

Question 5

 
 
Find the solution of the system
x + 5y + 7z = -13
2y - z = -1
3z = 9
by back substitution.
a) x = 13, y = 1, z = 3.   b) x = 8, y = -1, z = 3.
c) x = 23, y = 1, z = -3.   d) x = 3, y = -1, z = 3.
e) x = -39, y = 1, z = 3.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Your answer is correct.
 

Question 6

 
 
Which row operation should be performed next, in order to efficiently solve (by back substitution) the system of equations corresponding to the following augmented matrix?
⌊ 1 1 5|3 ⌋ ⌈ 0 1 3|4 ⌉ 0 2 1|1
a) An operation on row 1: R1 - R2, that is, subtract row 2 from row 1.   b) An operation on row 2: R2 -1 2R3, that is, subtract half of row 3 from row 2.
c) An operation on row 3: R3 - 2R2, that is, subtract twice row 2 from row 3.   d) An operation on row 3: R3 - 2R1, that is, subtract twice row 1 from row 3.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
This is the best move because it will produce a new row 3 with entries [0 0 - 5  | - 7] from which we immendiately see that the third variable is equal to 7∕5. The remaining variables can now be found by back substitution.
Not correct. Choice (d) is false.
 

Question 7

 
 
Which option is the complete set of solutions of the equation x + 2y - 4z = 13, written in parametric form with s and t as the parameters?
a) [s, - 2s + 13, 4t + 13]   b) [t, s, -134 + s4 + t2]
c) [s, t, 134- -s4 -t2]   d) [s, t, -134 + s4 + t2]

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Let x = s and y = t. Then since z = 14(x + 2y - 13), we must have z = 14(s + 2t - 13) = -143 + s4 + t2.
 

Question 8

 
 
There is a unique solution of the following linear system. What is it?
x1 + x2 - 3x3 = -6
- 7x2 + 7x3 = 7
3x3 = 9
a) x1 = 1, x2 = 2, x3 = 3.   b) x1 = -1, x2 = 2, x3 = 3.
c) x1 = -1, x2 = -2, x3 = 3.   d) x1 = 1, x2 = 2, x3 = -3.

 

Your answer is correct.
These values satisfy all three equations.
Not correct. Choice (b) is false.
The first equation is not satisfied.
Not correct. Choice (c) is false.
The first and second equations are not satisfied.
Not correct. Choice (d) is false.
None of the equation is satisfied.
 

Question 9

 
 
The following augmented matrices represent different linear systems in x, y, z. Which of the linear systems has a unique solution?
⌊ | ⌋ ⌊ | ⌋ ⌊ | ⌋ ⌈ 3 30 90 |120 ⌉ ⌈ 12 1 5 |15 ⌉ ⌈ 1 2 5 |3 ⌉ A. 1 10 30 | 40 B. 11 1 3 |14 C. 0 1 3 |4 2 20 60 80 1 0 2 1 0 0 1 1
a) A only.   b) B only.
c) C only.   d) None has a unique solution.

 

Not correct. Choice (a) is false.
System A must have infinitely many solutions. Each row of the augmented matrix is a multiple of the same row, namely [1  10  30 | 40]. So we have essentially just one equation linking the three variable in system A. This means that two parameters are needed to specify a complete set of solutions; that is A does not have a unique solution.
Not correct. Choice (b) is false.
This can’t be the right answer because in system B, the first row is the sum of the second and third rows. This means that the row operation on R1 which subtracts R2 from R1 will produce a new first row identical to R3. This means that instead of having three independent equations linking the three unknown variables, we have only two. In fact, system B has infinitely many solutions.
Your answer is correct.
The unique solution is x = -4, y = 1, z = 1.
Not correct. Choice (d) is false.
 

Question 10

 
 
Two of the following augmented matrices correspond to linear systems with no solution (inconsistent systems). Which two are they?
a) ⌊ 1 1 2 |4 ⌋ ⌈ 3 1 0 |1 ⌉ 3 1 0 |2   b) ⌊ 1 1 5|15 ⌋ ⌈ 0 1 3|14 ⌉ 0 0 2| 4
c) ⌊ 2 1 5 |0 ⌋ ⌈ 1 1 3 |- 1 ⌉ 1 0 2 |1   d) | ⌊ 1 3 2 |2 ⌋ ⌈ 4 12 8 |7 ⌉ 2 4 1 1

 

There is at least one mistake.
For example, choice (a) should be true.
This represents an inconsistent system. The second row corresponds to the equation 3x + y = 1. However the third row corresponds to 3x + y = 2. Clearly these are contradictory statements and therefore the system has no solution.
There is at least one mistake.
For example, choice (b) should be false.
This system has the unique solution z = 2, y = 8, x = -3, found by back substitution.
There is at least one mistake.
For example, choice (c) should be false.
This system has infinitely many solutions, as the first row is the sum of the second and third rows. There are only two independent equations involving the three unknowns, leading to infinitely many solutions.
There is at least one mistake.
For example, choice (d) should be true.
This represents an inconsistent system. If we multiply the first row by 4 we obtain [4  12  8 | 8] which corresponds to the equation 4x + 12y + 8z = 8. However the second row corresponds to 4x + 12y + 8z = 7. Clearly these are contradictory statements and therefore the system has no solution.
Your answers are correct
  1. True. This represents an inconsistent system. The second row corresponds to the equation 3x + y = 1. However the third row corresponds to 3x + y = 2. Clearly these are contradictory statements and therefore the system has no solution.
  2. False. This system has the unique solution z = 2, y = 8, x = -3, found by back substitution.
  3. False. This system has infinitely many solutions, as the first row is the sum of the second and third rows. There are only two independent equations involving the three unknowns, leading to infinitely many solutions.
  4. True. This represents an inconsistent system. If we multiply the first row by 4 we obtain [4  12  8 | 8] which corresponds to the equation 4x + 12y + 8z = 8. However the second row corresponds to 4x + 12y + 8z = 7. Clearly these are contradictory statements and therefore the system has no solution.