Consistency Formula

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

The Formula

A system is consistent if its solution set Sβ‰ βˆ…S \neq \emptyset

When to use: The constraints don't contradict each otherβ€”there's some answer that works.

Quick Example

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

Notation

Consistent: Sβ‰ βˆ…S \neq \emptyset (at least one solution exists). Inconsistent: S=βˆ…S = \emptyset (no solution). Indicated by reaching 0=c0 = c (cβ‰ 0c \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 otherβ€”there's some answer that works.

Formal View

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

Worked Examples

Example 1

easy
Is the system {x+y=52x+2y=8\begin{cases} x + y = 5 \\ 2x + 2y = 8 \end{cases} consistent?

Answer

Inconsistent (no solution)

First step

1
Step 1: Simplify equation 2: x+y=4x + y = 4.

Full solution

  1. 2
    Step 2: Compare: x+y=5x + y = 5 and x+y=4x + y = 4. Contradiction!
  2. 3
    Step 3: No values of x,yx, y satisfy both. The system is inconsistent.
A system is consistent if at least one solution exists. Here the two equations demand x+yx + y equal both 5 and 4 simultaneously, which is impossible.

Example 2

medium
For what value of kk is {x+2y=32x+4y=k\begin{cases} x + 2y = 3 \\ 2x + 4y = k \end{cases} consistent?

Common Mistakes

  • Equating consistency with a unique solution - infinitely many solutions still counts as consistent.
  • Calling a system inconsistent because it has free variables - free variables mean many solutions, not zero.
  • Missing the 0=c0=c signal - a false numeric statement during elimination means inconsistent, stop solving.

Why This Formula Matters

It's the first question worth asking about any system: an inconsistent system wastes effort because no answer exists. The tell is reaching a false statement like 0=30=3 during elimination, which signals S=βˆ…S=\emptyset and stops you cold. Recognizing it by "Is there at least one set of values that makes every equation true at the same time?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from contradiction and linear system behavior and degrees of freedom in a mixed problem set.

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 otherβ€”there's some answer that works.

What do the symbols mean in the Consistency formula?

Consistent: Sβ‰ βˆ…S \neq \emptyset (at least one solution exists). Inconsistent: S=βˆ…S = \emptyset (no solution). Indicated by reaching 0=c0 = c (cβ‰ 0c \neq 0) during simplification.

Why is the Consistency formula important in Math?

It's the first question worth asking about any system: an inconsistent system wastes effort because no answer exists. The tell is reaching a false statement like 0=30=3 during elimination, which signals S=βˆ…S=\emptyset and stops you cold. Recognizing it by "Is there at least one set of values that makes every equation true at the same time?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from contradiction and linear system behavior and degrees of freedom in a mixed problem set.

What do students get wrong about Consistency?

The procedure for consistency is the easy part; the trap is equating consistency with a unique solution. Asking "Is there at least one set of values that makes every equation true at the same time?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Consistency formula?

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

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