Consistency Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Consistency.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A system is consistent when some set of values satisfies every equation at once.

Common stuck point: 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.

Sense of Study hint: Ask: Is there at least one set of values that makes every equation true at the same time?

Worked Examples

Example 1

easy

Is the system

\begin{cases} x + y = 5 \\ 2x + 2y = 8 \end{cases}

consistent?

Answer

Inconsistent (no solution)

First step

Step 1: Simplify equation 2:

x + y = 4

Full solution

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

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

\begin{cases} x + 2y = 3 \\ 2x + 4y = k \end{cases}

consistent?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

\begin{cases} 3x - y = 7 \\ x + 2y = 1 \end{cases}

consistent?

Example 2

medium

\begin{cases} x + y + z = 1 \\ x + y + z = 3 \end{cases}

consistent?

Example 3

easy

\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}

consistent?

Example 4

easy

A system simplifies to

0 = 7

. Is it consistent?

Example 5

easy

A system simplifies to

0 = 0

. Is it consistent?

Example 6

easy

Does a consistent system always have exactly one solution?

Example 7

easy

A single equation

x + y = 3

— is it consistent?

Example 8

easy

Two parallel lines with different intercepts — consistent or inconsistent?

Example 9

easy

Coincident lines (same line twice) — consistent or inconsistent?

Example 10

easy

If a system has a solution, what is it called?

Example 11

medium

Classify

\begin{cases} 2x - y = 3 \\ -4x + 2y = 1 \end{cases}

as consistent or not.

Example 12

medium

Classify

\begin{cases} x + 2y = 4 \\ 3x + 6y = 12 \end{cases}

Example 13

medium

\begin{cases} x + y + z = 1 \\ x + y + z = 2 \end{cases}

consistent?

Example 14

medium

After elimination a system gives

0 = 0

and

z

free. How many solutions, and is it consistent?

Example 15

medium

For what

b

\begin{cases} x + y = 2 \\ x + y = b \end{cases}

consistent?

Example 16

medium

A word problem yields

\begin{cases} a + c = 10 \\ a + c = 14 \end{cases}

. What went wrong / is it solvable?

Example 17

medium

Is a system with more equations than unknowns always inconsistent?

Example 18

challenge

Determine all

k

for which

\begin{cases} x + y = 1 \\ 2x + 2y = k \end{cases}

is consistent.

Example 19

challenge

3\times3

system row-reduces to a row

[0\ 0\ 0\ |\ 5]

. What does this row mean?

Example 20

challenge

Explain why checking consistency must come before counting degrees of freedom.

Example 21

medium

\begin{cases} 3x - y = 2 \\ -3x + y = -2 \end{cases}

consistent?

Example 22

medium

A model gives

\begin{cases} p + q = 8 \\ p - q = 2 \\ p = 5 \end{cases}

. Is it consistent?

Example 23

easy

Solve

\begin{cases} x=4 \\ x+y=10 \end{cases}

— is it consistent?

Example 24

medium

For what

k

\begin{cases} 3x-2y=5 \\ 6x-4y=k \end{cases}

consistent?

Example 25

medium

For what

k

\begin{cases} 3x-2y=5 \\ 6x-4y=k \end{cases}

inconsistent?

Example 26

medium

Classify

\begin{cases} x+y+z=1 \\ 2x+2y+2z=3 \end{cases}

Example 27

medium

Classify

\begin{cases} x+y=5 \\ x-y=1 \\ 2x=6 \end{cases}

Example 28

medium

Classify

\begin{cases} x+y=5 \\ x-y=1 \\ x+y=6 \end{cases}

Example 29

medium

For what

a

\begin{cases} ax + y = 1 \\ x + y = 2 \end{cases}

consistent with a unique solution?

Example 30

medium

In the system from X13, what value of

a

makes the system inconsistent?

Example 31

medium

\begin{cases} 2x+y=7 \\ 4x+2y=14 \\ x-y=1 \end{cases}

consistent?

Example 32

medium

A system

A\mathbf{x}=\mathbf{b}

is consistent iff

\text{rank}(A) =

Example 33

medium

Augmented matrix has a row

[0\ 0\ 0\ |\ 0]

. Does this make the system inconsistent?

Example 34

hard

Find all

k

for which

\begin{cases} x+ky=1 \\ kx+y=1 \end{cases}

is consistent.

Example 35

hard

Classify

\begin{cases} x+y+z=3 \\ x+2y+3z=6 \\ x+3y+5z=9 \end{cases}

Example 36

hard

For what

b

\begin{cases} x+y+z=1 \\ x+y+z=b \end{cases}

consistent?

Example 37

hard

3\times 4

augmented matrix row-reduces to

\begin{pmatrix}1 & 0 & 2 & | & 4\\ 0 & 1 & -1 & | & 3\\ 0 & 0 & 0 & | & 0\end{pmatrix}

. Is the system consistent? How many solutions?

Example 38

hard

3\times 4

augmented matrix reduces to

\begin{pmatrix}1 & 0 & 2 & | & 4\\ 0 & 1 & -1 & | & 3\\ 0 & 0 & 0 & | & 1\end{pmatrix}

. Consistency?

Example 39

hard

A word problem: total cost $50 for

x

adults at $5 and

y

kids at $3, with the count constraint

x+y=12

and

5x+3y=60

. Consistent?

Example 40

challenge

Find all

(a,b)

for which

\begin{cases} x+2y=3 \\ 2x+4y=a \\ x+y=b \end{cases}

is consistent.

Example 41

challenge

A homogeneous system

A\mathbf{x}=\mathbf{0}

— is it always consistent?

Example 42

challenge

For what

k

\begin{cases} x+y+z=1 \\ x+ky+z=2 \\ x+y+kz=3 \end{cases}

inconsistent?

← Back to Consistency Formula Explained Practice Mode

Related Concepts

Systems of Equations Contradiction Redundancy

Background Knowledge

These ideas may be useful before you work through the harder examples.

systems of equations