Test Your Skills With a Linear Equation in Three Variables Quiz

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| By Hansika
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| Attempts: 96
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  • 1/10 Questions

    Which of these represents a linear equation in three variables?

    • X + y + z = 10
    • X^2 + y + z = 5
    • Xy + z = 3
    • X/y + z = 2
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About This Quiz

Challenge your understanding of linear equations in three variables with this engaging quiz. Solve problems, explore consistent systems, and discover geometric representations.

Test Your Skills With A Linear Equation In Three Variables Quiz - Quiz

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  • 2. 

    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.

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  • 3. 

    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.

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  • 4. 

    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.

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  • 5. 

    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.

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  • 6. 

    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.

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  • 7. 

    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.

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  • 8. 

    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.

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  • 9. 

    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.

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  • 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.

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  • Current Version
  • Feb 14, 2025
    Quiz Edited by
    ProProfs Editorial Team
  • Jan 28, 2025
    Quiz Created by
    Hansika
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