Skill: 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.