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Educational adventures for curious kids

Math 1 · Lesson 4 · part 3 of 4

To solve: get x alone

To solve an equation means to get the letter by itself, so the last line reads x = the answer .

To solve: get x alone

To solve an equation means to get the letter by itself, so the last line reads x = the answer. An equation is a balance, so whatever you do to one side you must do to the other. After Step 1, there are only two things that can be on the letter: a number added to it, and its coefficient. Undo them in that order.

  1. 1RewriteStep 1. No subtraction, no division, every 1 written in.
  2. 2Add the oppositeof any number that is added to x, on both sides. That number and its opposite make 0, the do-nothing number, so it disappears.
  3. 3Multiply by the reciprocalof the coefficient on x, on both sides. The coefficient and its reciprocal make 1, the do-nothing number, so x is alone.
  4. 4CheckPut your answer back into the original equation. Both sides should match.

Why that order? Think of getting dressed: socks first, then shoes, so to undo it the shoes come off first. The number added on is the outside layer. The number you add or multiply by is shown in this color.

Example 1: 2x − 6 = 8

Step 1 rewrite2x + (−6) = 8Subtracting 6 is adding −6. The coefficient of x is 2, and −6 is added.
Step 2 add the opposite2x + (−6) + 6 = 8 + 6The opposite of −6 is 6. On the left, (−6) + 6 = 0, so 2x + 0 = 2x. Now 2x = 14.
Step 3 multiply by the reciprocal12 × 2x = 12 × 14The reciprocal of 2 is 12. On the left, 12 × 2 = 1, so 1x = x.
Answerx = 7
Check2 × 7 + (−6) = 14 + (−6) = 8 Both sides match, so x = 7 is right.

Example 2: x ÷ 5 + 1 = 3

Step 1 rewrite15x + 1 = 3Dividing by 5 is multiplying by 15. The coefficient of x is 15, and 1 is added.
Step 2 add the opposite15x + 1 + (−1) = 3 + (−1)The opposite of 1 is −1. On the left, 1 + (−1) = 0. Now 15x = 2.
Step 3 multiply by the reciprocal5 × 15x = 5 × 2The reciprocal of 15 is 5. On the left, 5 × 15 = 1, so 1x = x.
Answerx = 10
Check15 × 10 + 1 = 2 + 1 = 3 Both sides match, so x = 10 is right.

A fraction bar is not a written division sign. x3 already is the coefficient 13 on x, so Step 1 has nothing to rewrite there. Only a written ÷ sign is a division to rewrite. If there are parentheses, such as 3(x + 2) = 21, distribute first (hand the 3 to everyone inside, Lesson 2) and then it is an ordinary equation.

Where to go next

Next: Practice ProblemsTry them, get help one step at a time, then print how it went. All the partsSolving Equations