Consistency Formula

The Formula

A system is consistent if its solution set S \neq \emptyset

When to use: The constraints don't contradict each other—there's some answer that works.

Quick Example

x + y = 5 \quad \text{and} \quad x - y = 1 is consistent (solution: x = 3, y = 2).

Notation

Consistent: S \neq \emptyset (at least one solution exists). Inconsistent: S = \emptyset (no solution). Indicated by reaching 0 = c (c \neq 0) during simplification.

What This Formula Means

A system of equations is consistent when there exists at least one set of variable values that satisfies every equation simultaneously.

The constraints don't contradict each other—there's some answer that works.

Formal View

A system A\mathbf{x} = \mathbf{b} is consistent iff \mathbf{b} \in \mathrm{Col}(A), equivalently \mathrm{rank}(A) = \mathrm{rank}([A \mid \mathbf{b}]). Otherwise, the system is inconsistent and S = \emptyset.

Worked Examples

Example 1

easy
Is the system \begin{cases} x + y = 5 \\ 2x + 2y = 8 \end{cases} consistent?

Solution

  1. 1
    Step 1: Simplify equation 2: x + y = 4.
  2. 2
    Step 2: Compare: x + y = 5 and x + y = 4. Contradiction!
  3. 3
    Step 3: No values of x, y satisfy both. The system is inconsistent.

Answer

Inconsistent (no solution)
A system is consistent if at least one solution exists. Here the two equations demand x + y equal both 5 and 4 simultaneously, which is impossible.

Example 2

medium
For what value of k is \begin{cases} x + 2y = 3 \\ 2x + 4y = k \end{cases} consistent?

Common Mistakes

  • Ignoring a contradiction like 0 = 5 and continuing to solve — this means no solution exists
  • Confusing 'consistent with infinitely many solutions' and 'consistent with exactly one solution'
  • Assuming a system is inconsistent just because it is difficult to solve

Why This Formula Matters

Checking consistency is the first step when solving any system of equations — it tells you whether a solution even exists before you invest effort solving. In engineering, an inconsistent system means conflicting requirements that must be revised. In linear algebra, consistency links to whether \mathbf{b} lies in the column space of A.

Frequently Asked Questions

What is the Consistency formula?

A system of equations is consistent when there exists at least one set of variable values that satisfies every equation simultaneously.

How do you use the Consistency formula?

The constraints don't contradict each other—there's some answer that works.

What do the symbols mean in the Consistency formula?

Consistent: S \neq \emptyset (at least one solution exists). Inconsistent: S = \emptyset (no solution). Indicated by reaching 0 = c (c \neq 0) during simplification.

Why is the Consistency formula important in Math?

Checking consistency is the first step when solving any system of equations — it tells you whether a solution even exists before you invest effort solving. In engineering, an inconsistent system means conflicting requirements that must be revised. In linear algebra, consistency links to whether \mathbf{b} lies in the column space of A.

What do students get wrong about Consistency?

Inconsistency often shows as 0 = 5 or similar contradiction.

What should I learn before the Consistency formula?

Before studying the Consistency formula, you should understand: systems of equations.

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