Pre-Algebra · Solving Equations

Solving Equations, Step by Step

Hey Caleb! This guide walks through solving equations one skill at a time. Read a section, look at the examples, then try the quiz at the end. Your progress saves on this phone, so you can stop and come back anytime.

1What Is an Equation?

An equation says that two things are equal. Think of the = sign as the middle of a balance scale: whatever is on the left side weighs exactly the same as what's on the right side.

x + 3 8 =

Solving an equation means finding the number that x stands for, the one number that makes both sides truly equal. On this scale, x = 5, because 5 + 3 = 8.

To find x, you get it alone on one side. You do that with inverse operations, which are operations that undo each other:

If x has…Undo it by…
adding (+ 3)subtracting (- 3)
subtracting (- 8)adding (+ 8)
multiplying (5x)dividing (÷ 5)
dividing (x/4)multiplying (· 4)
🔑 Key ideaWhatever you do to one side, you must do to the other side. That keeps the scale balanced. Use the opposite (inverse) operation to undo what's being done to x.

Quick refresh: negative numbers

Negatives cause more wrong answers than anything else in equations, so here are the rules in one place:

Example 1: Is x = 5 a solution of x + 3 = 8?

  1. x + 3 = 8Start with the equation.
  2. 5 + 3 = 8Replace x with 5.
  3. 8 = 8Both sides match, so yes: x = 5 is the solution.

Example 2: Adding and subtracting negatives

  1. -7 + 12 = 5Different signs: 12 − 7 = 5. The 12 is bigger and positive, so the answer is positive.
  2. 6 - (-5) = 6 + 5Subtracting a negative is the same as adding a positive.
  3. 6 + 5 = 11Now just add.

Example 3: Multiplying and dividing negatives

  1. (-3)(-4) = 12Same signs (both negative) → positive.
  2. -18 ÷ 3 = -6Different signs → negative.
  3. -20 ÷ (-5) = 4Same signs → positive.
⚠️ Common mistakes
  • Doing something to only one side. If you subtract 3 on the left, subtract 3 on the right too.
  • -4 - 9 is -13, not 5 or −5. Think money: you owe $4, then spend $9 more, so now you owe $13.
  • Two negatives make a positive only when you multiply or divide. -4 + (-9) is still negative.
↑ ProgressNext: One-step (+/−) →

2One-Step Equations: Adding & Subtracting

In these equations, a number is being added to or subtracted from x. To get x alone, undo that number with the opposite operation, on both sides.

🔑 Key ideaIf a number is added to x, subtract it from both sides.
If a number is subtracted from x, add it to both sides.

The blue highlights in the steps show what was done to both sides.

Example 1: x − 8 = −3

  1. x - 8 = -38 is being subtracted from x.
  2. x - 8 [[+ 8]] = -3 [[+ 8]]Add 8 to both sides to undo the − 8.
  3. x = 5On the left, −8 + 8 = 0, leaving just x. On the right, −3 + 8 = 5.
  4. 5 - 8 = -3Check: put 5 back in for x. It works! ✓

Example 2: x + 9 = 4

  1. x + 9 = 49 is being added to x.
  2. x + 9 [[- 9]] = 4 [[- 9]]Subtract 9 from both sides.
  3. x = -54 − 9 = −5. Answers can be negative!
  4. -5 + 9 = 4Check ✓

Example 3: 14 = x + 20 (x on the right)

  1. 14 = x + 20x can be on the right side. That's fine, since the = works both ways.
  2. 14 [[- 20]] = x + 20 [[- 20]]Subtract 20 from both sides.
  3. -6 = x14 − 20 = −6.
  4. x = -6Flip it around if you like. It means the same thing.
⚠️ Common mistakes
  • Using the same operation instead of the opposite. For x - 8 = -3, subtracting 8 gives x - 16 = -11, which is further from the answer. Undo minus with plus.
  • Sign slips: -3 + 8 is 5, not −11 or 11.
↑ ProgressNext: One-step (× / ÷) →

3One-Step Equations: Multiplying & Dividing

A number written right next to x, like 5x, means 5 times x. To undo multiplying, divide. When x is divided by a number, like x/4, undo it by multiplying.

🔑 Key idea5x means 5 · x → divide both sides by 5.
x/4 means x ÷ 4 → multiply both sides by 4.
Keep the sign: for -3x, divide by −3.

Example 1: 5x = 35

  1. 5x = 35x is multiplied by 5.
  2. 5x [[÷ 5]] = 35 [[÷ 5]]Divide both sides by 5.
  3. x = 75x ÷ 5 is just x, and 35 ÷ 5 = 7.
  4. 5(7) = 35Check ✓

Example 2: x/4 = 7

  1. x/4 = 7x is divided by 4.
  2. x/4 [[· 4]] = 7 [[· 4]]Multiply both sides by 4.
  3. x = 28Dividing by 4 and multiplying by 4 cancel out. 7 · 4 = 28.
  4. 28/4 = 7Check ✓

Example 3: −3x = 18

  1. -3x = 18x is multiplied by −3. The negative sign goes with the 3.
  2. -3x [[÷ (-3)]] = 18 [[÷ (-3)]]Divide both sides by −3.
  3. x = -6Positive ÷ negative = negative.
  4. -3(-6) = 18Check: negative times negative is positive ✓

Answers don't have to be whole numbers. For 2x = 7, x = 7/2, which is 3.5. You can type either one.

⚠️ Common mistakes
  • Subtracting instead of dividing: for 5x = 35, the 5 is multiplying, so x is not 30. Divide to get 7.
  • Dropping the negative: -3x = 18 gives x = -6, not 6.
  • For x/4 = 7, dividing by 4 gives the wrong answer. x is already divided by 4, so multiply.
↑ ProgressNext: Two-step →

4Two-Step Equations

Now x has two things happening to it. In 2x + 3, x is multiplied by 2, then 3 is added. To undo it, work backwards: undo the adding or subtracting first, then the multiplying or dividing.

Think of socks and shoes: socks go on first, then shoes. To take them off, the shoes come off first. Undo the last thing first.

🔑 Key idea: undo in reverse orderStep 1: Undo the + or − (get the x-term by itself).
Step 2: Undo the × or ÷ (get x alone).

Example 1: 2x + 3 = 13

  1. 2x + 3 = 13x is times 2, then plus 3.
  2. 2x + 3 [[- 3]] = 13 [[- 3]]Step 1: undo + 3 by subtracting 3 from both sides.
  3. 2x = 10Now it's a one-step equation.
  4. 2x [[÷ 2]] = 10 [[÷ 2]]Step 2: undo × 2 by dividing both sides by 2.
  5. x = 5Solved.
  6. 2(5) + 3 = 13Check: 10 + 3 = 13 ✓

Example 2: 4x − 7 = −19

  1. 4x - 7 = -19x is times 4, then minus 7.
  2. 4x - 7 [[+ 7]] = -19 [[+ 7]]Undo − 7 by adding 7 to both sides.
  3. 4x = -12−19 + 7 = −12 (different signs: 19 − 7 = 12, and the bigger number was negative).
  4. 4x [[÷ 4]] = -12 [[÷ 4]]Divide both sides by 4.
  5. x = -3Negative ÷ positive = negative.
  6. 4(-3) - 7 = -19Check: −12 − 7 = −19 ✓

Example 3: x/3 + 2 = 5

  1. x/3 + 2 = 5x is divided by 3, then plus 2.
  2. x/3 + 2 [[- 2]] = 5 [[- 2]]Undo + 2 by subtracting 2 from both sides.
  3. x/3 = 3One step left.
  4. x/3 [[· 3]] = 3 [[· 3]]Undo ÷ 3 by multiplying both sides by 3.
  5. x = 9Solved.
  6. 9/3 + 2 = 5Check: 3 + 2 = 5 ✓
⚠️ Common mistakes
  • Dividing just part of a side. For 2x + 3 = 13, dividing only the 2x by 2 (and not the 3) breaks the balance. Get rid of the + 3 first.
  • Sign slips with negatives: -19 + 7 = -12, not −26.
↑ ProgressNext: Distribute & combine →

5Distributive Property & Combining Like Terms

Sometimes an equation needs cleaning up before you can solve it. Two cleanup tools:

🔑 Key ideaClean up each side first (distribute, then combine like terms). Then solve it like a two-step equation.

Example 1: 3(x + 2) = 21

  1. 3(x + 2) = 21The 3 multiplies everything in the parentheses.
  2. 3x + 6 = 21Distribute: 3 · x = 3x and 3 · 2 = 6.
  3. 3x + 6 [[- 6]] = 21 [[- 6]]Subtract 6 from both sides.
  4. 3x = 1521 − 6 = 15.
  5. x = 5Divide both sides by 3.
  6. 3(5 + 2) = 21Check: 3 · 7 = 21 ✓

Shortcut: you could also divide both sides by 3 first to get x + 2 = 7, so x = 5.

Example 2: 2x + 5x − 4 = 17

  1. 2x + 5x - 4 = 17Two x-terms on the left.
  2. 7x - 4 = 17Combine like terms: 2x + 5x = 7x.
  3. 7x - 4 [[+ 4]] = 17 [[+ 4]]Add 4 to both sides.
  4. 7x = 2117 + 4 = 21.
  5. x = 3Divide both sides by 7.
  6. 2(3) + 5(3) - 4 = 17Check: 6 + 15 − 4 = 17 ✓

Example 3: −2(x − 4) = 14

  1. -2(x - 4) = 14Distribute the −2, sign included.
  2. -2x + 8 = 14−2 · x = −2x, and −2 · (−4) = +8 (negative times negative).
  3. -2x + 8 [[- 8]] = 14 [[- 8]]Subtract 8 from both sides.
  4. -2x = 614 − 8 = 6.
  5. x = -3Divide both sides by −2.
  6. -2(-3 - 4) = 14Check: −2 · (−7) = 14 ✓
⚠️ Common mistakes
  • Only multiplying the first term: 3(x + 2) is 3x + 6, not 3x + 2.
  • Losing the sign: -2(x - 4) is -2x + 8, because negative × negative = positive.
  • Combining unlike terms: 3x + 4 is not 7x.
↑ ProgressNext: x on both sides →

6Variables on Both Sides (and Checking Your Answer)

When x shows up on both sides, your first job is to get all the x-terms on one side. Subtract one of the x-terms from both sides, and then it's a two-step equation you already know how to solve.

Tip: move the smaller x-term. That way, the x-term you keep stays positive.

🔑 Key idea1. Get the x-terms on one side. 2. Get the numbers on the other side. 3. Divide. 4. Check: plug your answer into the original equation. If both sides come out equal, you're right.

Example 1: 5x + 4 = 2x + 16

  1. 5x + 4 = 2x + 16x on both sides. 2x is the smaller x-term.
  2. 5x + 4 [[- 2x]] = 2x + 16 [[- 2x]]Subtract 2x from both sides.
  3. 3x + 4 = 165x − 2x = 3x, and the x on the right is gone.
  4. 3x + 4 [[- 4]] = 16 [[- 4]]Subtract 4 from both sides.
  5. 3x = 1216 − 4 = 12.
  6. x = 4Divide both sides by 3.
  7. 5(4) + 4 = 24Check the left side with x = 4.
  8. 2(4) + 16 = 24Check the right side with x = 4. Both are 24 ✓

Example 2: 3x − 5 = 7x + 11

  1. 3x - 5 = 7x + 113x is the smaller x-term.
  2. 3x - 5 [[- 3x]] = 7x + 11 [[- 3x]]Subtract 3x from both sides.
  3. -5 = 4x + 11Now x is only on the right.
  4. -5 [[- 11]] = 4x + 11 [[- 11]]Subtract 11 from both sides.
  5. -16 = 4x−5 − 11 = −16.
  6. x = -4Divide both sides by 4.
  7. 3(-4) - 5 = -17Check left: −12 − 5 = −17.
  8. 7(-4) + 11 = -17Check right: −28 + 11 = −17. They match ✓

Example 3: Checking catches a mistake

A friend says x = 3 solves 2(x + 1) = x + 7. Let's check.

  1. 2(3 + 1) = 8Left side with x = 3.
  2. 3 + 7 = 10Right side with x = 3. 8 and 10 don't match, so x = 3 is wrong.
  3. 2x + 2 = x + 7Solve it properly. First distribute the 2.
  4. 2x + 2 [[- x]] = x + 7 [[- x]]Subtract x from both sides.
  5. x + 2 = 72x − x = x.
  6. x = 5Subtract 2 from both sides.
  7. 2(5 + 1) = 5 + 7Check: 12 = 12 ✓
⚠️ Common mistakes
  • Subtracting an x-term from only one side.
  • Writing 5x - 2x as 3 instead of 3x. Don't drop the x!
  • Checking with a later step instead of the original equation. Always check in the original.
↑ ProgressNext: Final review →

★Final Mixed Review

This quiz mixes every type of equation from the guide. For each one, first figure out what kind it is (one-step? two-step? parentheses? x on both sides?), then use the matching steps. If you get stuck, scroll back up to that section.

↑ Back to progress