Interpreting Intersections and Solution Types

A system of two linear equations in two variables can have exactly one solution, no solution, or infinitely many solutions.

One solution: lines intersect at exactly one point. Slopes are different.
No solution: lines are parallel and distinct. Slopes are equal, intercepts differ.
Infinitely many solutions: lines are the same line. Both slope and intercept match after simplification.

The panels below show what each case actually looks like on a graph.

Algebraic tests:

\begin{aligned} \text{If } &a_1x + b_1y = c_1,\quad a_2x + b_2y = c_2,\\ \text{compare ratios } &\frac{a_1}{a_2},\ \frac{b_1}{b_2},\ \frac{c_1}{c_2}.\\ &\text{One solution if ratios of }a\text{ and }b\text{ are not equal.}\\ &\text{No solution if } \frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2}.\\ &\text{Infinitely many if } \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}. \end{aligned}

Graph meaning:

Intersection point $(x,y)$ satisfies both equations.
In contexts, $x$ and $y$ carry units. The coordinates answer the question in those units.

One Solution

No Solution

Infinitely Many Solutions

Core Skills

Convert to slope intercept form to compare slopes and intercepts.
Use elimination or substitution to detect contradictions like $0=5$ or identities like $0=0$ .
Interpret the ordered pair in context with correct units.

\begin{cases} y = 2x + 1\\ y = -x + 10 \end{cases}

Set equal:

2x + 1 = -x + 10 \Rightarrow 3x = 9 \Rightarrow x = 3

. Then

y = 2(3) + 1 = 7

. The lines have different slopes, so they meet once at

\boxed{(3,7)}

\begin{cases} 2x - 3y = 6\\ 4x - 6y = 10 \end{cases}

Multiply the first equation by 2 to compare:

4x - 6y = 12

. The second is

4x - 6y = 10

. Same left side, different constants. This gives

12 = 10

, a contradiction. Slopes match and intercepts differ, so no solution. Lines are parallel.

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

Multiply the second by 3:

3x + 6y = 12

. The equations are identical. Every point on the line is a solution, so infinitely many solutions.

Two plans for streaming:

\begin{cases} \text{Plan A cost } C = 8 + 2m\\ \text{Plan B cost } C = 2 + 3m \end{cases}

where

m

is the number of movies and

C

is dollars. Set equal:

8 + 2m = 2 + 3m \Rightarrow m = 6

. Then

C = 8 + 2(6) = 20

. At

m=6

movies, both plans cost

20

dollars. The intersection

(6, 20)

means 6 movies and 20 dollars.

Different slopes gives one intersection and one solution.
Equal slopes with different intercepts gives no solution.
Proportional equations describe the same line and give infinitely many solutions.
In word problems, attach units to the coordinates and state what the point means.