Are you ready to master dependent independent and inconsistent equations? Take our fun Free Quiz: Dependent, Independent & Inconsistent Equations to test your grasp of consistent and inconsistent equations, challenge yourself with an inconsistent vs dependent equations study, and sharpen your skills with real-life systems of equations quiz problems. Discover how well you can spot when systems have one solution, infinite solutions, or none at all, and get instant, real-time feedback to propel your progress. Pair this with our independent and dependent variables quiz or jump into a linear equations quiz now and boost your algebra confidence!
Which term describes a system of equations that has exactly one solution?
Dependent
Independent
Inconsistent
Redundant
An independent system has exactly one solution because the lines intersect at a single point. This is contrasted with dependent (infinitely many solutions) and inconsistent (no solutions). For more, see Khan Academy.
What is the classification of the system 2x + 4y = 8 and x + 2y = 4?
Independent
Dependent
Inconsistent
Overdetermined
These two equations are multiples of each other, so they represent the same line. Such a system has infinitely many solutions and is called dependent. See Khan Academy.
How many solutions does an inconsistent system of linear equations have?
Infinitely many
Exactly one
None
At least two
An inconsistent system has no solution because the lines are parallel and never meet. It is characterized by contradictions when attempting to solve. More details at Khan Academy.
Which system is independent?
x + y = 3 and 2x + 2y = 6
x + y = 3 and x + y = 4
x + y = 3 and x - y = 1
2x + 4y = 8 and x + 2y = 4
x + y = 3 and x - y = 1 intersect at exactly one point, giving one unique solution. Therefore the system is independent. See Khan Academy.
Which describes a dependent system?
Parallel lines
Coincident lines
Intersecting lines
Perpendicular lines
Dependent systems correspond to coincident lines, meaning all equations describe the same line. They have infinitely many solutions. Learn more at Khan Academy.
The system 3x + 2y = 6 and 6x + 4y = 12 is:
Independent
Dependent
Inconsistent
Underdetermined
The second equation is twice the first, so they represent the same line. That makes the system dependent with infinitely many solutions. See Khan Academy.
If two equations have no points in common, the system is called:
Dependent
Independent
Inconsistent
Coincident
No common points means the lines are parallel, so there is no solution. Such systems are inconsistent. For more details visit Khan Academy.
Which pair of equations is inconsistent?
x + 2y = 4 and 2x + 4y = 8
y = 2x + 1 and y = 2x - 3
x - y = 2 and y = x - 2
2x + y = 5 and 4x + 2y = 10
These two lines have the same slope but different intercepts, so they are parallel and never meet. The system has no solution and is inconsistent. More at Khan Academy.
What type of system is represented by x - y = 1 and 2x - 2y = 2?
Independent
Dependent
Inconsistent
Overdetermined
The second equation is just twice the first, so they represent the same line. Hence, the system is dependent with infinite solutions. See Khan Academy.
Which scenario yields an independent system?
Same slope, same intercept
Different slopes
Same slope, different intercept
Vertical and horizontal lines
Lines with different slopes intersect at exactly one point, giving a unique solution. Therefore the system is independent. Refer to Khan Academy.
Solve and classify: x + y = 5 and x - y = 1.
(3,2); independent
(2,3); dependent
(5,0); inconsistent
(0,5); independent
Adding gives 2x = 6, so x = 3 and y = 2. The unique solution makes the system independent. Explanation and steps at Khan Academy.
Which classification fits the system 2x - y = 4, 4x - 2y = 9?
Dependent
Independent
Inconsistent
Overdetermined
Doubling the first equation gives 4x - 2y = 8, which conflicts with 4x - 2y = 9. The contradiction yields no solution, making it inconsistent. More at Khan Academy.
Determine the nature of the system: 3x + y = 7, 6x + 2y = 14.
Independent
Dependent
Inconsistent
Underdetermined
The second is 2× the first, so both equations coincide. There are infinitely many solutions, indicating the system is dependent. See Khan Academy.
Find and classify the solution of x - 2y = 3 and 2x - 4y = 6.
Unique; independent
None; inconsistent
Infinitely many; dependent
Two; special case
The second is twice the first, so they represent the same line. This yields infinite solutions, making the system dependent. More detail at Khan Academy.
Which system is inconsistent?
x + y = 4 and x + y = 4
x - y = 2 and -x + y = -2
3x + y = 5 and 6x + 2y = 11
2x - y = 1 and 4x - 2y = 2
Doubling 3x + y = 5 gives 6x + 2y = 10, which contradicts the second equation. Hence no solution and the system is inconsistent. See Khan Academy.
Solve: x/2 + y/3 = 1, x/2 – y/3 = 2. What is the classification?
(2, ?1); independent
(?2, 1); independent
(3, 0); inconsistent
Infinite; dependent
Adding gives x = 2; substituting yields y = –1. Unique solution means the system is independent. Steps at Khan Academy.
Which classification applies to 5x + 5y = 10 and x + y = 2?
Independent
Dependent
Inconsistent
Overdetermined
Dividing the first equation by 5 gives x + y = 2, the same as the second. This infinite solution scenario is dependent. See Khan Academy.
Decide the system type: x – 3y = 2 and 2x – 6y = 5.
Dependent
Independent
Inconsistent
Coincident
Doubling the first gives 2x – 6y = 4 which conflicts with the second (5), so no solution exists. The system is inconsistent. More at Khan Academy.
Substitute y = 2x + 1 into 4x – 2y = 3 gives 4x – 2(2x+1)=3 ? 4x–4x–2=3 ? –2=3 contradiction. Actually no real solution ? inconsistent. (Note: correct answer should be inconsistent.) For correct walkthrough see Khan Academy.
Which system has infinitely many solutions?
x + y = 3 and x – y = 1
2x + y = 4 and 4x + 2y = 8
x – y = 2 and x – y = 3
3x + 3y = 6 and x + y = 0
The second equation is exactly twice the first, representing the same line and thus infinite solutions. This is a dependent system. More info at Khan Academy.
Classify the 3×3 system if its determinant is zero but equations are multiples.
Independent
Dependent
Inconsistent
Singular unique
Zero determinant with proportional equations yields infinite solutions, so the system is dependent. A zero determinant alone only indicates non-invertibility. See Khan Academy Linear Algebra.
If two equations in a 3×3 system conflict, what is the classification?
Independent
Dependent
Inconsistent
Overdetermined
A conflict in any subset means no solution for the full system, making it inconsistent. Even if others align, one contradiction is enough. More at Khan Academy.
Which indicates a dependent system in matrix form?
Row echelon with pivot in each row
Zero row after elimination
All nonzero rows
Unique pivots and RHS match
A zero row with zero on the right after row reduction indicates a free variable and infinitely many solutions. That means dependence. See Khan Academy Linear Algebra.
Identify the system type: x + 2y + 3z = 1, 2x + 4y + 6z = 2, x – y + z = 0.
Independent
Dependent
Inconsistent
Overdetermined
The second equation is twice the first, introducing a redundant equation. The third adds no conflict, so infinite solutions occur and the system is dependent. More at Khan Academy.
When is a homogeneous system always dependent?
Non-zero determinant
Zero determinant
Unique trivial solution
No solution
A homogeneous system (Ax=0) with zero determinant has infinitely many nontrivial solutions, making it dependent. A nonzero determinant gives only the trivial solution. See Khan Academy Linear Algebra.
Which indicates inconsistency after Gaussian elimination?
A row of zeros equals zero
No pivot in last column
A row of zeros equals nonzero
Pivots in every column
A row like [0 0 … 0 | c] with c?0 indicates a contradiction and no solutions. This is the hallmark of inconsistency. More at Khan Academy.
Classify: x + y + z = 3, 2x + 2y + 2z = 6, 3x + y + z = 4.
Independent
Dependent
Inconsistent
Underdetermined
The second is twice the first, but the third doesn’t align, creating a contradiction. Thus no solution exists and the system is inconsistent. See Khan Academy.
When a 3×3 system has rank 2 and augmented rank 2, it is:
Independent
Dependent
Inconsistent
Singular unique
Equal coefficient and augmented ranks less than unknown count imply infinite solutions. The system is dependent. More at Khan Academy Linear Algebra.
Which condition ensures an independent system in higher dimensions?
Coefficient matrix is singular
Full rank equal to variables
Zero determinant
Augmented rank > coefficient rank
Full rank equal to number of variables (non-zero determinant) ensures a unique solution, i.e., an independent system. See Khan Academy Linear Algebra.
In ?? a system has 3 independent equations and rank 3. How many degrees of freedom remain?
0
1
3
4
Degrees of freedom = variables (4) – rank (3) = 1, so one free parameter. This yields infinitely many solutions. More at Khan Academy Linear Algebra.
Which statement is true for a nonhomogeneous system with a singular coefficient matrix?
Always independent
Always inconsistent
Dependent or inconsistent
Unique solution exists
A singular coefficient matrix (zero determinant) means either no solutions or infinite solutions. Thus it can be dependent or inconsistent. See Khan Academy Linear Algebra.
If a system’s augmented matrix rank exceeds its coefficient matrix rank, then the system is:
Dependent
Independent
Inconsistent
Underdetermined
Augmented rank > coefficient rank indicates a contradiction upon elimination, hence no solution. The system is inconsistent. For more see Khan Academy Linear Algebra.
0
{"name":"Which term describes a system of equations that has exactly one solution?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"Which term describes a system of equations that has exactly one solution?, What is the classification of the system 2x + 4y = 8 and x + 2y = 4?, How many solutions does an inconsistent system of linear equations have?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}
Study Outcomes
Differentiate Dependent, Independent & Inconsistent Systems -
Compare graphs and solution sets to distinguish dependent, independent, and inconsistent systems of equations within the dependent independent and inconsistent framework.
Identify Inconsistent Equations -
Recognize an inconsistent equation by spotting contradictions and understanding why certain systems have no valid solutions.
Classify System Consistency -
Evaluate systems of equations to classify them as consistent or inconsistent, strengthening your grasp of consistent and inconsistent equations.
Solve Systems to Determine Solutions -
Use algebraic methods to solve linear systems and interpret whether they yield a unique solution, infinitely many solutions, or none.
Apply Substitution and Elimination Methods -
Implement substitution and elimination techniques to efficiently solve systems and deepen your understanding of inconsistent vs dependent equations.
Enhance Algebraic Reasoning Skills -
Build critical thinking and problem-solving abilities through targeted practice in this systems of equations quiz.
Cheat Sheet
Classification of Systems -
In any system of linear equations you'll encounter three outcomes: independent (a single unique solution), dependent (infinitely many solutions), and inconsistent (no solution). Remember "one, many, none" to map to independent, dependent, and inconsistent. This framework is outlined in MIT OpenCourseWare's linear algebra notes.
Spotting an Inconsistent Equation -
An inconsistent equation arises when two lines share the same slope but different y-intercepts (e.g., y=2x+1 and y=2x−3), so they never meet. A quick mnemonic is "parallel but apart" to recall inconsistent vs dependent equations. Khan Academy's systems of equations quiz resources emphasize checking slope - intercept form to confirm no intersection.
Identifying Dependent Systems -
Dependent systems occur when equations represent the same line (e.g., y=3x+2 and 2y=6x+4), yielding infinitely many solutions. Think "coincident twins" to remember dependent independent and inconsistent topics. Purdue University's online algebra notes show how scaling one equation verifies dependence.
Matrix Rank and Consistency -
Use the rank criterion: if rank(A)=rank([A|b])=n, the system is consistent; if rank(A)=rank([A|b])
Solving Strategies and Error Checks -
Employ substitution or elimination, then verify solutions back in original equations to catch an inconsistent equation or confirm dependence. A handy tip: if elimination yields 0=5, flag inconsistency; if you get 0=0, expect infinite solutions (dependence). Checking your work this way is stressed in the University of California's math tutoring guides.
