How do you solve a system of two linear equations?

Set the two equations equal: m1*x + b1 = m2*x + b2, then solve for x = (b2 - b1) / (m1 - m2). Substitute x back into either equation to find y. This works when the slopes are different (one unique solution).

When does a system of equations have no solution?

A system has no solution when the two lines are parallel — they have the same slope but different y-intercepts. Graphically, the lines never cross. Algebraically, you get a contradiction like 0 = 5.

When does a system have infinite solutions?

A system has infinitely many solutions when both equations describe the same line — same slope AND same y-intercept. Every point on the line satisfies both equations.

Systems of Equations Solver

Adjust slopes and intercepts for two lines. Watch the intersection move, disappear (parallel), or merge (same line).

Two Lines

Presets

Line 1: m₁ = 1.0

1.0

b₁ = 2.0

2.0

Line 2: m₂ = -0.5

-0.5

b₂ = -1.0

-1.0

Solution

Equation 1	--
Equation 2	--
Case	--
Solution	--
Relationship	--

The Math Behind It

Solving Systems

A system of two linear equations: y = m₁x + b₁ and y = m₂x + b₂
To find intersection: set equal, solve m₁x + b₁ = m₂x + b₂
Solution: x = (b₂ - b₁) / (m₁ - m₂)
Then find y by plugging x back into either equation

Three Cases

One solution: Lines intersect at exactly one point (different slopes)
No solution: Lines are parallel — same slope, different intercepts
Infinite solutions: Lines are the same — same slope AND same intercept

Try This

Make slopes equal — watch intersection vanish (parallel)
Then match intercepts — lines merge (infinite solutions)
Set m₁ × m₂ = -1 for perpendicular lines

Systems of Equations Solver

Two Lines

Solution

The Math Behind It

Solving Systems

Three Cases

Try This

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