Mathwords logoReference LibraryMathwords

Inconsistent System of Equations

Inconsistent System of Equations

A system of equations which has no solutions.

Note: Attempts to solve inconsistent systems typically result in impossible statements such as 0 = 3.

 

Example: System {2x−3y=1, 4x−6y=1} is inconsistent. There are no solutions to the system.

 

See also

Consistent system of equations, overdetermined system of equations, underdetermined system of equations, linear system of equations

Key Formula

a1x+b1y=c1a2x+b2y=c2Inconsistent when a1a2=b1b2≠c1c2a_1x + b_1y = c_1 \\[4pt] a_2x + b_2y = c_2 \\[6pt] \text{Inconsistent when } \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}
Where:
  • a1,a2a_1, a_2 = Coefficients of x in the first and second equations
  • b1,b2b_1, b_2 = Coefficients of y in the first and second equations
  • c1,c2c_1, c_2 = Constants on the right side of the first and second equations

Worked Example

Problem: Determine whether the following system is inconsistent: 2x + 3y = 6 4x + 6y = 15
Step 1: Check the ratio condition. Compare the coefficients and constants of both equations.
a1a2=24=12,b1b2=36=12,c1c2=615=25\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{6}{15} = \frac{2}{5}
Step 2: Since the coefficient ratios are equal but differ from the constant ratio, the system is inconsistent. To confirm, try solving by elimination. Multiply the first equation by 2.
2(2x+3y)=2(6)  ⟹  4x+6y=122(2x + 3y) = 2(6) \implies 4x + 6y = 12
Step 3: Subtract the second equation from this result.
(4x+6y)−(4x+6y)=12−15  ⟹  0=−3(4x + 6y) - (4x + 6y) = 12 - 15 \implies 0 = -3
Step 4: The statement 0=−30 = -3 is false. This confirms no solution exists.
0≠−30 \neq -3
Answer: The system is inconsistent — it has no solution. Graphically, the two lines are parallel and never intersect.

Another Example

This example uses a system of three equations (instead of two) to show that inconsistency can arise even when some equations in the system are compatible with each other. Only one conflicting equation is enough to make the entire system inconsistent.

Problem: Determine whether the following system of three equations is inconsistent: x + y + z = 2 2x + 2y + 2z = 4 3x + 3y + 3z = 9
Step 1: Observe that the second equation is exactly 2 times the first equation, so it provides no new information.
2(x+y+z)=2(2)  ⟹  2x+2y+2z=4  ✓2(x + y + z) = 2(2) \implies 2x + 2y + 2z = 4 \; \checkmark
Step 2: Now compare the third equation to 3 times the first equation.
3(x+y+z)=3(2)  ⟹  3x+3y+3z=63(x + y + z) = 3(2) \implies 3x + 3y + 3z = 6
Step 3: The third equation says 3x+3y+3z=93x + 3y + 3z = 9, but our derivation gives 3x+3y+3z=63x + 3y + 3z = 6. Subtract to isolate the contradiction.
6=9  ⟹  0=36 = 9 \implies 0 = 3
Step 4: This is a false statement, so the system has no solution and is inconsistent.
0≠30 \neq 3
Answer: The system is inconsistent. Even though two of the three equations agree, the third contradicts them, so no solution exists.

Frequently Asked Questions

What is the difference between an inconsistent and a consistent system of equations?
A consistent system has at least one solution — either exactly one (independent) or infinitely many (dependent). An inconsistent system has no solution at all. When you solve an inconsistent system, you reach a contradiction like 0=50 = 5, whereas a consistent system yields actual values for the variables.
How can you tell if a system is inconsistent by looking at a graph?
For a system of two linear equations in two variables, the lines are parallel but not identical. They have the same slope but different y-intercepts, so they never cross. Since there is no point of intersection, there is no solution.
Can a system of nonlinear equations be inconsistent?
Yes. Inconsistency is not limited to linear systems. For example, the system x2+y2=1x^2 + y^2 = 1 and x2+y2=4x^2 + y^2 = 4 is inconsistent because no point can lie on two circles of different radii centered at the same point. Any system — linear or nonlinear — with no common solution is inconsistent.

Inconsistent System vs. Consistent System

Inconsistent SystemConsistent System
Number of solutionsZero — no solution existsOne or infinitely many solutions
Graph (2 linear equations)Parallel lines that never intersectLines that intersect at one point, or overlap entirely
Algebraic resultA false statement like 0 = 3Actual variable values, or a dependent identity like 0 = 0
Coefficient ratio test (2×2 linear)a₁/a₂ = b₁/b₂ ≠ c₁/c₂a₁/a₂ ≠ b₁/b₂ (independent) or a₁/a₂ = b₁/b₂ = c₁/c₂ (dependent)

Why It Matters

Recognizing inconsistent systems matters in algebra courses because it prevents you from wasting time searching for a solution that doesn't exist. In real-world modeling — such as balancing chemical equations, circuit analysis, or fitting data — an inconsistent system signals that your constraints are contradictory and the model needs revision. On standardized tests, you are often asked to classify a system as consistent or inconsistent before solving, which requires understanding this concept.

Common Mistakes

Mistake: Confusing an inconsistent system (no solution) with a dependent system (infinitely many solutions).
Correction: Both situations involve proportional coefficients, but the key difference is in the constants. If the constant ratio matches the coefficient ratios, the system is dependent (infinitely many solutions). If it does not match, the system is inconsistent (no solutions). Always check all three ratios.
Mistake: Concluding a system is inconsistent after getting 0 = 0 during elimination.
Correction: The statement 0=00 = 0 is true, not false. It means the equations are dependent and have infinitely many solutions. An inconsistent system produces a false statement like 0=50 = 5. Do not confuse a true identity with a contradiction.

Related Terms