Graphical Method of Linear Equations

Discover the three types of solutions and master the graphical approach to solving linear equation systems!

Understanding Solution Types

Linear equation pairs can have one solution,
infinitely many solutions, or no solution at all!

✖️
Intersecting Lines
Lines meet at ONE point
Unique Solution
(Consistent)
a₁/a₂ ≠ b₁/b₂
∞
Coincident Lines
Lines overlap completely
Infinite Solutions
(Dependent)
a₁/a₂ = b₁/b₂ = c₁/c₂
∥
Parallel Lines
Lines never meet
No Solution
(Inconsistent)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Intersecting Lines - Unique Solution

━ x + 3y = 6 ━ 2x - 3y = 12 ● Solution (6, 0)

Coincident Lines - Infinite Solutions

━ 5x - 8y + 1 = 0 ━ 3x - (24/5)y + 3/5 = 0

Parallel Lines - No Solution

━ x + 2y - 4 = 0 ━ 2x + 4y - 12 = 0

🛍️ Champa's Shopping Problem

Champa went to a sale to purchase pants and skirts.

When asked how many of each she bought, she said:
"The number of skirts is two less than twice the number of pants."

"Also, the number of skirts is four less than four times the number of pants."

Building the Equations

Let x = number of pants
Let y = number of skirts

y = 2x - 2
y = 4x - 4

Solving Algebraically

Since both equal y:

2x - 2 = 4x - 4

-2 + 4 = 4x - 2x

2 = 2x

x = 1

Substitute back:

y = 2(1) - 2 = 0

y = 0

Graphical Solution

━ y = 2x - 2 ━ y = 4x - 4 ● Solution (1, 0)
Final Answer
Pants (x) = 1
Skirts (y) = 0

Champa bought 1 pair of pants
and did not buy any skirts

Click any button below to explore different solution types!
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📐 Key Concepts
A pair of linear equations can be consistent (having solutions) or inconsistent (having no solution). Use the ratio test to determine the type of solution without graphing!