Test Your Skills With A Linear Equation In Three Variables Quiz

2.

How many solutions does a consistent linear system with three variables have?

Infinite
One
No solution
Exactly two

Correct Answer

A. One

Explanation

A consistent linear system in three variables has equations that intersect at a single point. For example:
Equation 1: x + y + z = 6
Equation 2: 2x - y + z = 7
Equation 3: x - 2y - z = -3
Solving this system shows all three planes meet at a single point, confirming one unique solution. Inconsistent systems have no common points, while dependent systems involve overlapping planes, leading to infinitely many solutions.

Rate this question:

3.

Which is true for a system of linear equations in three variables?

Planes intersect at a line
Planes intersect at a point
Planes are parallel
Planes are coincident

Correct Answer

A. Planes intersect at a point

Explanation

For a system of three equations, the intersection of the corresponding planes determines the solution. Consider:
Plane 1: x + y + z = 6
Plane 2: x - y + z = 4
Plane 3: x + y - z = 2
These planes intersect at a single point if consistent, form a line if dependent, or never intersect if inconsistent. When solved, the system results in a unique intersection point representing the solution.

Rate this question:

1

4.

What does the solution of a linear system in three variables geometrically represent?

A point
A plane
A line
No solution

Correct Answer

A. A point

Explanation

The solution to three-variable equations geometrically represents the point of intersection of three planes. For example:
Plane 1: x + y + z = 6
Plane 2: 2x - y + z = 7
Plane 3: x - 2y - z = -3
Solving, the values for x, y, and z simultaneously satisfy all equations, locating the intersection point. If planes are parallel or do not meet, no solution exists.

Rate this question:

5.

If x + y + z = 6, x - y + z = 4, and x + y - z = 2, what is z?

2
3
4
5

Correct Answer

A. 2

Explanation

Solve x + y + z = 6, x - y + z = 4, and x + y - z = 2 for z:
Subtract the second equation from the first: (x + y + z) - (x - y + z) = 6 - 4, simplifying to 2y = 2 → y = 1.
Substitute y = 1 into x + y + z = 6: x + 1 + z = 6 → x + z = 5.
Subtract the third equation from the first: (x + y + z) - (x + y - z) = 6 - 2 → 2z = 4 → z = 2.
Thus, z = 2.

Rate this question:

6.

What is the determinant used for in solving systems of three linear equations?

To check consistency
To find solutions
To eliminate variables
To verify answers

Correct Answer

A. To find solutions

Explanation

Determinants are essential in solving linear systems using matrices. For example:
Matrix A =
1 1 1
2 -1 1
1 -2 -1
The determinant of A is calculated as follows:
det(A) = 1(-1)(-1) + 1(1)(1) + 1(2)(-2) − 1(1)(-1) − 1(2)(1) − 1(-1)(-2)
det(A) = 1 + 1 − 4 − (-1) − 2 − 2
det(A) = 3
Since det(A) is not equal to zero, the system is consistent and has a unique solution. A determinant of zero would indicate dependency among rows, resulting in no unique solution.

Rate this question:

7.

If a system has no solution, what is the system called?

Dependent
Inconsistent
Consistent
Unique

Correct Answer

A. Inconsistent

Explanation

Inconsistent systems arise when equations have no common solution. For example:
Plane 1: x + y + z = 6
Plane 2: x + y + z = 8
These planes are parallel, as they have identical coefficients for variables but different constants. Parallel planes never intersect, meaning no point satisfies both equations. This inconsistency ensures the system has no solution.

Rate this question:

8.

Which method is not used for solving three-variable linear systems?

Substitution
Elimination
Matrix method
Factoring

Correct Answer

A. Factoring

Explanation

Factoring is used for quadratic equations like x² - 5x + 6 = 0. Linear systems like x + y + z = 6 require substitution, elimination, or matrices. For example:
Substitute values into x + y + z = 6.
Eliminate y or z using another equation.
Factoring doesn’t apply here because there are no powers or products of variables to decompose.

Rate this question:

9.

What is the rank of a coefficient matrix with three equations and three unknowns if it has a unique solution?

0
1
2
3

Correct Answer

A. 3

Explanation

The rank of a coefficient matrix indicates the number of independent equations. For a system:
Matrix A =
1 1 1
2 -1 1
1 -2 -1
The rank of A is determined by row-reducing it to echelon form. If the rank of A equals the number of variables (3), all rows are linearly independent, and the system has a unique solution. If the rank is less than the number of variables, it indicates dependency among the rows, leading to infinitely many solutions. If the rank is inconsistent with the augmented matrix, the system has no solution. Thus, rank analysis provides insight into the solvability of the system.

Rate this question:

10.

If x + y + z = 9, 2x - y + z = 7, and x - 2y - z = -3, what is x?

2
3
4
5

Correct Answer

A. 3

Explanation

Solve x + y + z = 9, 2x - y + z = 7, and x - 2y - z = -3 for x:
Add equations 1 and 2: (x + y + z) + (2x - y + z) = 9 + 7 → 3x + 2z = 16.
Subtract equation 3 from equation 1: (x + y + z) - (x - 2y - z) = 9 - (-3) → 3y + 2z = 12 → y = 4 - z.
Substitute y = 4 - z into 3x + 2z = 16, solve for x:
3x + 2(2) = 16 → 3x = 12 → x = 4.

Rate this question:

Test Your Skills With a Linear Equation in Three Variables Quiz

Which of these represents a linear equation in three variables?

Quiz Preview

How many solutions does a consistent linear system with three variables have?

Which is true for a system of linear equations in three variables?

What does the solution of a linear system in three variables geometrically represent?

If x + y + z = 6, x - y + z = 4, and x + y - z = 2, what is z?

What is the determinant used for in solving systems of three linear equations?

If a system has no solution, what is the system called?

Which method is not used for solving three-variable linear systems?

What is the rank of a coefficient matrix with three equations and three unknowns if it has a unique solution?

If x + y + z = 9, 2x - y + z = 7, and x - 2y - z = -3, what is x?