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Consistent vs.
Inconsistent Systems

Learn to classify systems of linear equations by their number of solutions: one solution, infinitely many, or none at all.

Linear Algebra

What Is a System of Equations?

A system of linear equations is a set of two or more equations with the same variables. We want to find values that satisfy all equations simultaneously.

Graphically, each linear equation is a line in the coordinate plane. The solution to the system is the set of points where the lines meet.

Three Possible Outcomes

One Solution
Lines intersect at exactly one point. The system is consistent and independent.
Infinitely Many
Lines are identical (overlap completely). The system is consistent and dependent.
No Solution
Lines are parallel and never meet. The system is inconsistent.

Visualizing the Three Cases

One Solution Infinitely Many No Solution

Formal Definitions

1
Consistent System: A system that has at least one solution. This includes systems with exactly one solution or infinitely many solutions.
2
Inconsistent System: A system that has no solution. No ordered pair satisfies all equations simultaneously.
3
Independent: A consistent system where equations represent different lines (one intersection point).
4
Dependent: A consistent system where equations represent the same line (infinitely many solutions).

The Slope-Intercept Test

When both equations are written in slope-intercept form \( y = mx + b \), you can classify the system by comparing slopes and y-intercepts:

Slopes Intercepts Result
Different slopes \( m_1 \neq m_2 \) Any One solution
Same slope, same intercept \( m_1 = m_2 \) \( b_1 = b_2 \) Infinitely many
Same slope, different intercept \( m_1 = m_2 \) \( b_1 \neq b_2 \) No solution

Dependent vs. Inconsistent

Students often confuse dependent (same line, infinitely many solutions) with inconsistent (parallel, no solution). Both involve equal slopes, but the distinction lies in the intercepts.

Dependent
Same slope and same intercept. The equations are multiples of each other. Every point on one line is also on the other.
Inconsistent
Same slope but different intercepts. The lines are parallel: they run side by side forever and never cross.

The Coefficient Ratio Method

For a system in standard form \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \), compare the ratios of corresponding coefficients:

1
If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the system is consistent and independent (one solution).
2
If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the system is consistent and dependent (infinitely many).
3
If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the system is inconsistent (no solution).

How to Classify a System

1
Rewrite both equations in slope-intercept form \( y = mx + b \), or identify the coefficients in standard form.
2
Compare slopes (or coefficient ratios). If slopes differ → consistent, one solution. Done.
3
If slopes are equal, compare intercepts (or the constant ratio). Same → dependent. Different → inconsistent.
4
Verify by solving: substitution or elimination will either give a unique answer, an identity like \( 0 = 0 \), or a contradiction like \( 0 = 5 \).

Example 1: One Solution

Classify the system: \( x + y = 5 \) and \( 2x - y = 1 \).

x y 0 1 2 3 4 5 6 7 8 1 2 3 4 5 (2, 3) \( x + y = 5 \) \( 2x - y = 1 \)

Example 2: Infinitely Many

Classify the system: \( 2x + 4y = 8 \) and \( x + 2y = 4 \).

x y 0 1 2 3 4 5 1 2 3 Same line! \( 2x + 4y = 8 \) \( x + 2y = 4 \)

Example 3: No Solution

Classify the system: \( 3x + 6y = 12 \) and \( x + 2y = 7 \).

x y 0 1 2 3 4 5 6 7 1 2 3 4 ↕ Parallel! \( 3x + 6y = 12 \) \( x + 2y = 7 \)

Algebraic Verification

When you solve a system algebraically (elimination or substitution), the result tells you the classification:

Unique values
You get \( x = a, \; y = b \). One solution → consistent, independent.
Identity
You get \( 0 = 0 \). Always true → infinitely many → consistent, dependent.
Contradiction
You get \( 0 = 5 \). Never true → no solution → inconsistent.

The Determinant Connection

For the system \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \), the determinant of the coefficient matrix is:

\( D \neq 0 \)
Unique solution exists. The system is consistent and independent.
\( D = 0 \)
Lines are either the same or parallel. Check constants to distinguish dependent from inconsistent.

Common Mistakes

✕ Confusing Parallel with Same Line
Both have equal slopes, but parallel lines have different intercepts (no solution), while the same line has equal intercepts (infinitely many).
✕ Forgetting to Simplify
Equations like \( 4x + 8y = 16 \) and \( x + 2y = 4 \) look different but are the same line. Always simplify before comparing.
✕ Misreading Contradictions
When elimination gives \( 0 = 5 \), some students think they made an error. This is actually the correct signal for no solution.

Summary Table

Property Consistent Independent Consistent Dependent Inconsistent
Solutions Exactly one Infinitely many None
Graph Lines intersect Lines overlap Lines parallel
Slopes \( m_1 \neq m_2 \) \( m_1 = m_2 \) \( m_1 = m_2 \)
Intercepts Any \( b_1 = b_2 \) \( b_1 \neq b_2 \)
Determinant \( D \neq 0 \) \( D = 0 \) \( D = 0 \)
Algebra result \( x = a, y = b \) \( 0 = 0 \) \( 0 = k \; (k \neq 0) \)

Question 1 of 5

Classify the system: \( y = 3x + 2 \) and \( y = 3x - 4 \).

Question 2 of 5

When solving by elimination, you arrive at the statement \( 0 = 0 \). What does this mean?

Question 3 of 5

Given the system \( 2x - y = 5 \) and \( 4x - 2y = 10 \), what is the determinant of the coefficient matrix?

Question 4 of 5

The system \( x + 3y = 9 \) and \( 2x - y = 4 \) has slopes \( m_1 = -\frac{1}{3} \) and \( m_2 = 2 \). This system is:

Question 5 of 5

Which of the following is true about inconsistent systems?