Practice Consistency in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
Solve {x=4x+y=10\begin{cases} x=4 \\ x+y=10 \end{cases} — is it consistent?

Example 2

easy
After elimination, a system reduces to 0=40=4. Consistent or inconsistent?

Example 3

easy
A system simplifies to 0=00 = 0. Is it consistent?

Example 4

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

Example 5

hard
A word problem: total cost $50 for xx adults at $5 and yy kids at $3, with the count constraint x+y=12x+y=12 and 5x+3y=605x+3y=60. Consistent?

Example 6

hard
A 3×43\times 4 augmented matrix reduces to (102∣401−1∣3000∣1)\begin{pmatrix}1 & 0 & 2 & | & 4\\ 0 & 1 & -1 & | & 3\\ 0 & 0 & 0 & | & 1\end{pmatrix}. Consistency?

Example 7

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

Example 8

easy
Is {x+y=5x−y=1\begin{cases} x + y = 5 \\ x - y = 1 \end{cases} consistent?

Example 9

medium
For what bb is {x+y=2x+y=b\begin{cases} x + y = 2 \\ x + y = b \end{cases} consistent?

Example 10

challenge
Determine all kk for which {x+y=12x+2y=k\begin{cases} x + y = 1 \\ 2x + 2y = k \end{cases} is consistent.

Example 11

medium
Is {3x−y=2−3x+y=−2\begin{cases} 3x - y = 2 \\ -3x + y = -2 \end{cases} consistent?

Example 12

hard
A 3×43\times 4 augmented matrix row-reduces to (102∣401−1∣3000∣0)\begin{pmatrix}1 & 0 & 2 & | & 4\\ 0 & 1 & -1 & | & 3\\ 0 & 0 & 0 & | & 0\end{pmatrix}. Is the system consistent? How many solutions?

Example 13

hard
For what bb is {x+y+z=1x+y+z=b\begin{cases} x+y+z=1 \\ x+y+z=b \end{cases} consistent?

Example 14

medium
Classify {x+2y=43x+6y=12\begin{cases} x + 2y = 4 \\ 3x + 6y = 12 \end{cases}.

Example 15

easy
Is {3x−y=7x+2y=1\begin{cases} 3x - y = 7 \\ x + 2y = 1 \end{cases} consistent?

Example 16

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

Example 17

medium
After elimination a system gives 0=00 = 0 and zz free. How many solutions, and is it consistent?

Example 18

easy
A system simplifies to 0=70 = 7. Is it consistent?

Example 19

medium
Is {x+y+z=1x+y+z=3\begin{cases} x + y + z = 1 \\ x + y + z = 3 \end{cases} consistent?

Example 20

challenge
A 3×33\times3 system row-reduces to a row [0 0 0 ∣ 5][0\ 0\ 0\ |\ 5]. What does this row mean?
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