Consistent System of Equations — Definition & Examples

Consistent System of Equations

A system of equations that has at least one solution.

Key Formula

\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}

Where:

$a_1, b_1, c_1$ = Coefficients and constant of the first equation
$a_2, b_2, c_2$ = Coefficients and constant of the second equation
$x, y$ = Unknown variables to solve for
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ = Condition for exactly one solution (independent system)
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ = Condition for infinitely many solutions (dependent system)

Worked Example

Problem: Determine whether the following system is consistent, and if so, find the solution: x + y = 5 and 2x − y = 1.

Step 1: Check the consistency condition. Compare the ratios of the coefficients.

\frac{a_1}{a_2} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{1}{-1} = -1

Step 2: Since the ratios are not equal (1/2 ≠ −1), the lines are not parallel. The system has exactly one solution, so it is consistent and independent.

\frac{1}{2} \neq -1

Step 3: Solve by adding the two equations together to eliminate y.

(x + y) + (2x - y) = 5 + 1 \implies 3x = 6 \implies x = 2

Step 4: Substitute x = 2 back into the first equation to find y.

2 + y = 5 \implies y = 3

Step 5: Verify in the second equation: 2(2) − 3 = 4 − 3 = 1. ✓ Both equations are satisfied.

2(2) - 3 = 1 \checkmark

Answer: The system is consistent with exactly one solution: (x, y) = (2, 3).

Another Example

This example shows a consistent system that is dependent (infinitely many solutions), unlike the first example which was independent (exactly one solution). Both types are consistent because at least one solution exists.

Problem: Determine whether the following system is consistent: 2x + 4y = 10 and x + 2y = 5.

Step 1: Compare the ratios of all coefficients and the constants.

\frac{a_1}{a_2} = \frac{2}{1} = 2, \quad \frac{b_1}{b_2} = \frac{4}{2} = 2, \quad \frac{c_1}{c_2} = \frac{10}{5} = 2

Step 2: All three ratios are equal. This means the first equation is just 2 times the second equation. The two equations represent the same line.

2(x + 2y) = 2(5) \implies 2x + 4y = 10

Step 3: Since the lines overlap completely, every point on the line x + 2y = 5 is a solution. For instance, (1, 2) and (5, 0) both work.

x + 2y = 5 \implies y = \frac{5 - x}{2}

Answer: The system is consistent with infinitely many solutions. Any point (x, y) satisfying x + 2y = 5 is a solution. This is called a dependent system.

Frequently Asked Questions

What is the difference between a consistent and inconsistent system of equations?

A consistent system has at least one solution — the graphs of the equations intersect at one or more points. An inconsistent system has no solution at all — the graphs never intersect. For two linear equations in two variables, an inconsistent system produces parallel lines that never meet.

How do you know if a system of equations is consistent without solving it?

For a system of two linear equations, compare the ratios of the coefficients. If a₁/a₂ ≠ b₁/b₂, the system is consistent with one solution. If a₁/a₂ = b₁/b₂ = c₁/c₂, the system is consistent with infinitely many solutions. Only when a₁/a₂ = b₁/b₂ but ≠ c₁/c₂ is the system inconsistent.

Can a consistent system have infinitely many solutions?

Yes. A consistent system has at least one solution, but it can also have infinitely many. When the equations are dependent (one is a multiple of another), every point on the shared line or plane is a solution. Such a system is called consistent and dependent.

Consistent System vs. Inconsistent System

	Consistent System	Inconsistent System
Definition	Has at least one solution	Has no solution
Graphical meaning (2 linear eqns)	Lines intersect or overlap	Lines are parallel and never meet
Coefficient condition	a₁/a₂ ≠ b₁/b₂, or all three ratios are equal	a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Number of solutions	Exactly one (independent) or infinitely many (dependent)	Zero
Example	x + y = 5 and x − y = 1 → solution (3, 2)	x + y = 5 and x + y = 3 → no solution

Why It Matters

Classifying a system as consistent or inconsistent is one of the first things you learn in algebra when working with simultaneous equations. It tells you whether a solution exists before you spend time trying to find one. This concept extends into linear algebra, where determining consistency of larger systems using matrices and row reduction is a central technique in many applied fields.

Common Mistakes

Mistake: Confusing 'consistent' with having exactly one solution.

Correction: A consistent system has at least one solution. It can have exactly one solution (independent) or infinitely many solutions (dependent). Both cases count as consistent.

Mistake: Thinking that if two equations look different, the system must be consistent.

Correction: Two equations can look different yet be parallel lines (e.g., x + y = 3 and 2x + 2y = 10). Always check the coefficient ratios rather than relying on appearance.

Related Terms

Inconsistent System of Equations — The opposite case: a system with no solutions
Simultaneous Equations — Equations solved together; may be consistent or not
Solution — The values that satisfy all equations in the system
Underdetermined System of Equations — Fewer equations than unknowns; often infinitely many solutions
Overdetermined System of Equations — More equations than unknowns; may still be consistent
Linear System of Equations — Most common type of system tested for consistency