Solving Linear Equations

Calculus Readiness · Axiom Academy

LESSON Solving Linear Equations An equation is a balance scale — do the same thing to both sides, and the scale stays level until the variable stands alone. Picture 2x + 3 = 11 as a balanced scale: two x -blocks and three single blocks on the left weigh the same as eleven single blocks on the right. To find x , peel away everything around it — but to keep the scale level, remove the same weight from both pans . Subtract 3 from both sides — undo the “+3” Divide both sides by 2 — undo the “×2” When x sits on both pans, gather it onto one side first. In 2x + 5 = x + 8 , take one x -block off each pan at once — the scale stays level, and now the unknown lives only on the left. Then clear the spare units the same way. Both pans hold x -blocks. Remove one x from each side, leaving x + 5 = 8 . Now take 5 single blocks off both pans, isolating x = 3 . Removing the same thing from equal sides leaves them equal — the balance is never broken. Substitute x = 3 : left 2(3)+5 = 11 , right 3 + 8 = 11 . Level. Taking one x -block off both pans is exactly “subtract x from both sides.” The physical act and the algebra are the same thing — that’s why the scale never tips. Parentheses are a bundle . The left pan of 3(x + 2) = 18 holds three identical (x + 2) bundles. Unpack them — that’s distributing — to get 3x + 6 , and the rest is the same balance you already know. Distribute: three copies of (x + 2) spread out into 3x + 6 — every bundle contributes one x -block and two units.

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