Identify the number of solutions to a linear system

A linear system in two variables can have exactly one solution, no solution, or infinitely many solutions, determined by whether its equations represent intersecting, parallel distinct, or coincident lines. Equivalent reasoning appears through graphs, substitution, or elimination: a contradiction indicates no solution, while an identity indicates infinitely many solutions; these cases distinguish independent, inconsistent, and dependent systems and support solving systems algebraically. More advanced systems with additional variables or parameters are not included.

Detailed Explanation: Identify the number of solutions to a linear system

To identify the number of solutions, use elimination to remove one variable. Then examine the result:

A true equation with variables, such as $x=3$ , gives exactly one solution.
A false statement, such as $0=4$ , gives no solution.
A true identity, such as $0=0$ , gives infinitely many solutions.

Worked example

Determine the number of solutions:

\begin{aligned} 3x+2y&=8\\ 6x+4y&=20 \end{aligned}

Multiply the first equation by $2$ :

6x+4y=16

Now compare this with the second equation:

6x+4y=20

Subtract the first equation from the second:

(6x+4y)-(6x+4y)=20-16

0=4

This is a contradiction because $0$ cannot equal $4$ . Therefore, the system has no solution.

Graphically, the equations represent two distinct parallel lines: they never intersect.

Remember:

If elimination produces a variable equation, the lines intersect once, so there is one solution.
If elimination produces a contradiction like $0=4$ , there are no solutions.
If elimination produces an identity like $0=0$ , the equations describe the same line, so there are infinitely many solutions.

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Types of Solutions - Equation Pair to Zero, One, or Infinite

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