Elimination with Three Variables

Algebra 2 · Axiom Academy

LESSON Elimination with Three Variables One variable at a time: collapse a system down to a , solve it, then back-substitute for the last unknown. 1. Cancel a variable to shrink the system The whole method rests on one move: add a multiple of one equation to another so a chosen variable cancels . We will knock out z . Take two pairs of equations, scale each so their z -terms are equal and opposite, and combine — every z disappears, leaving equations in x and y only. the same coefficients, written as an augmented matrix Elimination is the same move at a smaller size . The two leftover equations have y -terms +2y and -2y — equal and opposite already, so just add them and y cancels, leaving a single equation in x . Dividing gives x = 0.2 . Substituting that back into -x + 2y = -8 gives 2y = -7.8 , so y = -3.9 . The eliminated variable is not lost — it is waiting in the original equations . Drop the known x and y into equation (1) and every term but one becomes a number, leaving a single equation in z . You reduced a 3×3 system to a 2×2, solved it, and climbed back up — the same eliminate-then-back-substitute pattern works for any number of variables. Scroll up to revisit any step.

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