Before you read

Answer first — see what you already know.

How many solutions does this system have?
${y = 3 x + 5 y = 3 x - 2$

A
One solution
B
No solution
C
Infinitely many solutions
D
Two solutions

Check your answer

Answer:

Before you read

Answer first — see what you already know.

How many solutions?
${4 x + 2 y = 10 2 x + y = 5$

A
One solution
B
No solution
C
Infinitely many solutions
D
Cannot be determined

Check your answer

Answer:

Theory

Three Cases for Systems

When you solve a system of two linear equations, there are three possible outcomes:

Case	Graph	What happens when solving	Example
One solution	Lines intersect at one point	You find $x = a$ , $y = b$	Most systems
No solution	Lines are parallel (never meet)	You get a false statement like $0 = 5$	Same slope, different y-intercept
Infinitely many	Lines are the same line	You get a true statement like $0 = 0$	Same slope AND same y-intercept

Key insight: it all comes down to the slopes.
- Different slopes $\to$ one solution (lines must cross)
- Same slope, different intercept $\to$ no solution (parallel)
- Same slope, same intercept $\to$ infinitely many (same line)

No solution: parallel lines never intersect

Theory

Recognizing the Cases Without Solving

For a system in standard form:
$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$

Compare the ratios of the coefficients:

Condition	Result
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$	One solution
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$	No solution
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Infinitely many

Example 1

How many solutions?
$\begin{cases} 2x + 4y = 10 \\ x + 2y = 3 \end{cases}$

Ratios: $\frac{2}{1} = 2$ , $\frac{4}{2} = 2$ , $\frac{10}{3} \approx 3.3$

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

No solution (parallel lines).

No solution

Example 2

How many solutions?
$\begin{cases} 3x - 6y = 12 \\ x - 2y = 4 \end{cases}$

Ratios: $\frac{3}{1} = 3$ , $\frac{-6}{-2} = 3$ , $\frac{12}{4} = 3$

All ratios equal $\to$ infinitely many solutions (same line).

Infinitely many solutions

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