Types of Solutions

Algebra 1 · Axiom Academy

Two lines can cross once, never meet, or be the very same line — and each outcome shows up as a clean signal in the algebra. 1. One Solution — the lines cross When two lines have different slopes , they are tilted differently, so they are guaranteed to meet at exactly one point . That single (x,y) pair is the one and only place that satisfies both equations — the solution to the system. Different slopes ( -1 and +1 ) → they must cross once Add the two equations to eliminate y : (x+y)+(x-y)=8+2 gives 2x=10 , so x=5 and then y=3 . Solving lands on a single point — that is the algebraic fingerprint of one solution. 2. No Solution — parallel lines When two lines share the same slope but different y -intercepts , they march in the same direction forever and stay a fixed gap apart. They never cross , so no point lies on both lines — the system has no solution . Same slope ( 2 ), different intercepts → never meet Set them equal: 2x+1 = 2x-3 . Subtract 2x from both sides and the variable vanishes, leaving 1 = -3 — a statement that is simply false . A false numeric equation is the algebra's way of saying "these lines never agree." 3. Infinitely Many — the same line twice Sometimes two equations that look different describe the exact same line . One equation is just a multiple of the other, so every point on the line satisfies both — there are infinitely many solutions . Second equation is the first → one line

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