Master Linear Equations and Systems: Interactive Learning Quizzes

Question 1

A linear system is considered inconsistent if it has:

Accepted Answer

No solution.

Answer

Two distinct solutions.

Answer

Infinitely many solutions.

Answer

A unique solution.

Question 2

Which matrix represents the system of equations:

2x + 3y = 7

x - 2y = -1

Accepted Answer

[[2, 3], [1, -2]] * [[x], [y]] = [[7], [-1]]

Answer

[[4, 5], [2, -4]] * [[x], [y]] = [[7], [-1]]

Answer

[[2, 1], [3, -2]] * [[x], [y]] = [[7], [-1]]

Answer

[[7], [-1]] * [[2, 3], [1, -2]] = [[x], [y]]

Question 3

Find the solution to the following system of equations:

x + 2y = 5

3x - y = 1

Accepted Answer

x = 1, y = 2

Answer

x = 2, y = 1

Answer

x = 3, y = -1

Answer

x = -1, y = 3

Question 4

A system of linear equations with more unknowns than equations could have:

Accepted Answer

Either no solutions or infinitely many solutions.

Answer

A unique solution.

Answer

No solutions.

Answer

Exactly two solutions.

Question 5

The determinant of a 2x2 matrix [[a, b], [c, d]] is:

Accepted Answer

ad - bc

Answer

ab - cd

Answer

ac - bd

Answer

a + d - b - c

Question 6

Which technique is NOT used to solve a system of linear equations?

Accepted Answer

Newton's method

Answer

Gaussian elimination

Answer

Cramer's rule

Answer

Substitution method

Question 7

Solve the following system of linear equations:

x + y = 5
x - y = 1

Accepted Answer

(3, 1)

Answer

(4, 0)

Answer

(1, 3)

Answer

(2, 2)

Question 8

Find the value of x that satisfies the equation:

2x - 5 = 11

Accepted Answer

8

Answer

10

Answer

6

Answer

12

Question 9

Find the inverse of the matrix:

[2 1]
[3 2]

Accepted Answer

[2/7 -1/7]
[-3/7 4/7]

Answer

[1/4 -1/4]
[3/4 4/4]

Answer

[2/3 -1/3]
[-3/3 4/3]

Answer

[1/2 -1/2]
[3/2 4/2]

Question 10

Determine whether the following vectors are linearly independent or linearly dependent:

v1 = [1, 2]
v2 = [3, 6]

Accepted Answer

Linearly dependent

Answer

Cannot be determined

Answer

None of the above

Answer

Linearly independent

Question 11

Use the Gram-Schmidt process to orthogonalize the following set of vectors:

v1 = [1, 2, 3]
v2 = [2, 4, 6]

Accepted Answer

[1/√14, 2/√14, 3/√14], [0, 4/√28, 6/√28]

Answer

[1/3, 2/3, 3/3], [2/3, 4/3, 6/3]

Answer

[1/2, 2/2, 3/2], [2, 4, 6]

Answer

[1/4, 2/4, 3/4], [2/4, 4/4, 6/4]

Question 12

Which of the following systems of linear equations is represented by the augmented matrix

[1 2 -1 | 4]
[0 1  3 | 2]
[0 0  1 | -1] ?

Accepted Answer

x + 2y - z = 4, y + 3z = 2, z = -1

Answer

x + 2y - z = 4, y + 3z = 2, -z = 1

Answer

x + 2y - z = 4, 2y + 3z = 2, z = -1

Answer

x + y - z = 4, y + 3z = 2, z = -1

Question 13

If a system of linear equations has infinitely many solutions, what can be said about the graphs of the equations?

Accepted Answer

The graphs of the equations coincide.

Answer

The graphs of the equations are parallel.

Answer

The graphs of the equations intersect at a single point.

Answer

The graphs of the equations are perpendicular.

Question 14

Which of the following matrices is in row echelon form?

Accepted Answer

[1 2 3]
[0 1 4]
[0 0 1]

Answer

[1 0 0]
[0 2 1]
[0 0 1]

Answer

[1 2 3]
[0 0 4]
[0 1 1]

Answer

[0 1 0]
[1 0 1]
[0 0 1]

Question 15

What is the inverse of the matrix A =

[1 2]
[0 1] ?

Accepted Answer

[1 -2]
[0  1]

Answer

The inverse does not exist.

Answer

[-1 -2]
[ 0 -1]

Answer

[1  2]
[0 -1]

Question 16

A system of linear equations is said to be inconsistent if:

Accepted Answer

It has no solution.

Answer

It has infinitely many solutions.

Answer

It has exactly one solution.

Answer

The determinant of the coefficient matrix is zero.

Question 17

Which of the following row operations is NOT allowed when performing Gaussian elimination?

Accepted Answer

Dividing a row by zero.

Answer

Swapping two rows.

Answer

Adding a multiple of one row to another row.

Answer

Multiplying a row by a nonzero constant.

Question 18

If the determinant of the coefficient matrix of a system of linear equations is non-zero, what can be concluded about the system's solution?

Accepted Answer

The system has a unique solution.

Answer

The system has infinitely many solutions.

Answer

The system has no solution.

Answer

The system is homogeneous.

Question 19

Which of the following is NOT a type of matrix used in solving systems of linear equations?

Accepted Answer

Vandermonde matrix

Answer

Adjacency matrix

Answer

Identity matrix

Answer

Augmented matrix

Question 20

What is the determinant of the matrix [2 1; 3 4]?

Accepted Answer

5

Answer

11

Answer

2

Answer

-1

Question 21

Which of the following is a valid method for solving a system of linear equations?

Accepted Answer

Cramer's rule

Answer

Euler's method

Answer

Simpson's method

Answer

Lagrange multipliers

Question 22

What is the solution to the system of equations: 
 2x + 3y = 11
 x - y = 3

Accepted Answer

x = 4, y = 1

Answer

x = 3, y = 2

Answer

x = 1, y = 4

Answer

x = 2, y = 3

Question 23

What is the reduced row echelon form of the matrix [1 2 3; 4 5 6; 7 8 9]?

Accepted Answer

[1 0 0; 0 1 0; 0 0 1]

Answer

[1 2 3; 0 1 2; 0 0 1]

Answer

[0 0 1; 1 0 0; 0 1 0]

Answer

[0 1 0; 0 0 1; 1 2 3]

Question 24

Which of the following is a necessary condition for a system of linear equations to have a unique solution?

Accepted Answer

The coefficient matrix has a non-zero determinant.

Answer

The number of equations is equal to the number of variables.

Answer

The system is consistent.

Question 25

What is the geometric interpretation of a system of linear equations in two variables?

Accepted Answer

A set of lines in the plane.

Answer

A set of points in the plane.

Answer

A circle in the plane.

Answer

A parabola in the plane.

Question 26

Which of the following transformations preserves the solution set of a system of linear equations?

Accepted Answer

Adding a multiple of one equation to another.

Answer

Multiplying an equation by a non-zero constant.

Answer

Replacing a variable with a linear expression in terms of another variable.

Answer

Swapping two equations.

Question 27

What is the number of solutions to the system of equations: 
 2x + 3y = 6
 4x + 6y = 12

Accepted Answer

Infinitely many

Answer

No solution

Answer

Two solutions

Answer

Unique solution

Question 28

Which of the following systems of equations represents a line parallel to the y-axis?

Accepted Answer

x = 2, y is free

Answer

2x + y = 4

Answer

y = 2, x is free

Answer

x + 2y = 6