No Solution & Infinite Solutions

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No Solution

A system of equations has no solution when the two lines are parallel. This means they:

Two lines that never intersect — they just go side-by-side forever.

Put both equations into slope-intercept form:
y = mx + b

If m (slope) is the same but b (y-intercept) is different → no solution

Equation 1: $y=2x+1$

Equation 2: $y=2x-3$

Both linear functions have a slope of 2. Same slope and different y-intercepts, so there’s no solution.

A system of equations has infinite solutions when the two equations represent the same exact line. That means:

It looks like just one line, even though it came from two equations.

Again, write both equations in slope-intercept form:
y = mx + b

If both equations are exactly the same after simplifying → infinite solutions

Equation 1: $y = 3x + 2$
Equation 2: $6y = 18x + 12$ → divide everything by 6
$y = 3x + 2$ → Same as Equation 1 → Infinite Solutions

Get both equations into slope-intercept form ( $y = mx + b$ )
Compare slopes:
- Same slope, different y-intercepts → ❌ No Solution
- Same slope, same y-intercepts → ♾ Infinite Solutions
- Different slopes → ✅ One Unique Solution (the lines intersect once)

You can also graph both equations in Desmos: