Jugami Pre-Algebra

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Simple Equations

Find the mystery number by keeping both sides in balance.

An equation is a statement that two things are equal, like x + 3 = 7. Solving it means finding the number x stands for. Here, x = 4, because 4 + 3 = 7.

This page shows how to think of an equation as a balance scale, how to undo operations to get x by itself, how to check your answer, and why the method works. Then try the practice problems and the Balance Builder game.

Jump to practicePlay Balance Builder
A balance scale with an x box and 3 unit cubes on the left and 7 unit cubes on the right, level, showing x + 3 = 7 xx + 3 = 7

Variables, expressions and equations

A variable is a letter, like x or n, that stands for a number we don’t know yet. An expression is a calculation written with numbers, variables and operations, like 3x + 2. It has no equals sign, so it isn’t true or false; it just has a value once you know x.

An equation says two expressions are equal, like 3x + 2 = 14. The value of the variable that makes it true is the solution. In 3x, the number 3 is the coefficient; it means 3 times x.

Example 1: Expression or equation?

Which is an equation: 3x + 2 or 3x + 2 = 14? What is the value of 3x + 2 when x = 4?

  1. 3x + 2 has no equals sign: it is an expression.
  2. 3x + 2 = 14 states that two things are equal: it is an equation.
  3. When x = 4: 3 × 4 + 2 = 12 + 2 = 14. So x = 4 is the solution of 3x + 2 = 14.

The balance idea

Think of an equation as a balance scale that is perfectly level. The left side of the equals sign sits on one pan and the right side on the other. Whatever you do, the scale must stay level, so whatever you do to one side, you do to the other.

A level balance scale: the left pan holds an x box and 3 unit cubes, the right pan holds 7 unit cubes, so x + 3 = 7 xx + 3 = 7left sideright side
x + 3 = 7 as a level balance: an unknown box and 3 cubes weigh the same as 7 cubes.
Two balance scales, one above the other. Top: x plus 3 cubes balances 7 cubes. Bottom: after taking 3 cubes off each side, x balances 4 cubes, so x = 4 xxx + 3 = 7x = 4take 3 cubes off each side
Taking the same 3 cubes off both pans keeps the scale level and leaves x = 4.

Example 2: Solving with the balance

Solve x + 3 = 7.

  1. The x has 3 added to it. To get x alone, take 3 off both sides.
  2. x + 3 − 3 = 7 − 3, so x = 4.
  3. Check: 4 + 3 = 7. ✓

Inverse operations and one-step equations

Inverse operations undo each other. Adding and subtracting undo each other, and multiplying and dividing undo each other. To solve, ask “what was done to x?” and do the inverse to both sides.

  • x + 5 = 12 → subtract 5 from both sides → x = 7
  • x − 5 = 12 → add 5 to both sides → x = 17
  • 5x = 20 → divide both sides by 5 → x = 4
  • x/5 = 4 → multiply both sides by 5 → x = 20

Example 3: Subtraction, with negatives

Solve x − 8 = −3.

  1. 8 was subtracted from x, so add 8 to both sides.
  2. x = −3 + 8. Different signs: 8 − 3 = 5, positive (see adding integers).
  3. x = 5. Check: 5 − 8 = −3. ✓

Example 4: Multiplication

Solve −4x = 28.

  1. x was multiplied by −4, so divide both sides by −4.
  2. x = 28 ÷ (−4). Different signs give a negative answer.
  3. x = −7. Check: −4 × (−7) = 28. ✓

Example 5: Division

Solve x/5 = 6.

  1. x was divided by 5, so multiply both sides by 5.
  2. x = 6 × 5 = 30.
  3. Check: 30 ÷ 5 = 6. ✓

Two-step equations

In 2x + 3 = 11, two things were done to x: first it was multiplied by 2, then 3 was added. To undo them, work backwards, like taking off your shoes before your socks: undo the adding first, then the multiplying.

Algebra tiles: 2 long x-tiles and 3 small unit tiles on the left of an equals sign, 11 unit tiles on the right, showing 2x + 3 = 11 xx=2 x-tiles + 3 unit tiles11 unit tiles2x + 3 = 11
Algebra tiles for 2x + 3 = 11. Remove 3 unit tiles from each side, then split the 8 that are left equally between the 2 x-tiles.

Example 6: A two-step equation

Solve 2x + 3 = 11.

  1. Undo + 3: subtract 3 from both sides. 2x = 8.
  2. Undo × 2: divide both sides by 2. x = 4.
  3. With tiles: take 3 unit tiles off each side, leaving 2 x-tiles = 8 units, so each x-tile is 4.

Checking a solution

You can always check an answer by substituting it: put the number in place of the variable in the original equation and see if both sides come out equal.

Is x = 4 a solution of 2x + 3 = 11? Left side: 2 × 4 + 3 = 11. Right side: 11. They match, so yes. Is x = 5 a solution? 2 × 5 + 3 = 13, and 13 ≠ 11, so no.

Writing an equation from a word problem

Choose a letter for the unknown, say what it means, and write the sentence as an equation. A bar model helps you see it.

A bar model. The top bar is the $33 goal. The bottom bar, the same length, is 4 equal parts labeled w plus a $5 part, so 4w + 5 = 33 4w + 5 = 33goal: $33wwww$54 weeks of savings = 33 − 5 = 28
The goal bar and the savings bar are the same length: 4 weeks of w dollars plus $5 equals $33.

Example 7: Saving up

Maya has $5. She saves the same amount each week for 4 weeks and then has $33. How much does she save each week?

  1. Let w = dollars saved each week. Equation: 4w + 5 = 33.
  2. Undo + 5: 4w = 28. Undo × 4: w = 7.
  3. She saves $7 a week. Check: 4 × 7 + 5 = 33. ✓

Equations with negatives and fractions

The steps don’t change when the numbers are negative or fractions; you just use the rules from earlier units. Review Integers for the sign rules and Fractions for multiplying and dividing fractions. To undo multiplying by a fraction, multiply by its reciprocal (flip it), since a fraction times its reciprocal is 1.

Example 8: A fraction coefficient

Solve (2/3)x = 10.

  1. x was multiplied by 2/3. Multiply both sides by the reciprocal, 3/2.
  2. (3/2) × (2/3)x = 1x = x, and 10 × 3/2 = 30/2 = 15.
  3. x = 15. Check: (2/3) × 15 = 30/3 = 10. ✓

How do we know doing the same thing to both sides works?

An equation says the left side and the right side are the same number, just written two ways. If two things are the same number, then adding 3 to each gives the same result, subtracting 3 from each gives the same result, and so on (dividing by anything except 0). So the new equation is still true for the same x.

And because each step can be undone (subtract 3 can be undone by add 3), no solutions are lost or added along the way. That’s why the x you end with really solves the equation you started with, and why checking always works.

Two scales, one above the other. Top: x plus 3 cubes balances 7 cubes and stays level. Bottom: 3 cubes were taken off the right side only, so x plus 3 cubes outweighs 4 cubes and the scale tips down on the left xxSame change on both sides: still levelTake 3 from the right side only: it tips
Change only one side and the scale tips: the two sides are no longer equal.

History: where “algebra” and “=” come from

In Baghdad in the early 800s, the scholar Muhammad ibn Musa al-Khwarizmi, working at the House of Wisdom, wrote a book called Hisab al-jabr w’al-muqabala, often translated as “the book of calculation by completion and balancing.” Al-jabr (“completion,” or restoring) was his step for clearing away subtracted terms by moving them to the other side. The word in the title became our word algebra, and his own name gave us the word algorithm.

The equals sign came much later. In 1557 the Welsh doctor and mathematician Robert Recorde introduced “=” in his book The Whetstone of Witte, choosing two parallel lines “bicause noe 2 thynges can be moare equalle.” Before that, people wrote out words like “is equal to.”

Did you know?

Recorde’s equals sign was much longer than ours, and it didn’t catch on right away. For well over a century, into the 1700s, many writers used other symbols, such as “ae” for the Latin aequalis.

Practice problems

Solve each one, check it by substituting, then open it to compare.

1. x + 9 = 4

Subtract 9 from both sides: x = 4 − 9 = −5. Check: −5 + 9 = 4. ✓

2. y − 6 = −10

Add 6 to both sides: y = −10 + 6 = −4. Check: −4 − 6 = −10. ✓

3. 7n = −56

Divide both sides by 7: n = −56 ÷ 7 = −8. Check: 7 × (−8) = −56. ✓

4. m/3 = −4

Multiply both sides by 3: m = −4 × 3 = −12. Check: −12 ÷ 3 = −4. ✓

5. 3x − 5 = 16

Add 5: 3x = 21. Divide by 3: x = 7. Check: 3 × 7 − 5 = 16. ✓

6. x/4 + 2 = −1

Subtract 2: x/4 = −3. Multiply by 4: x = −12. Check: −12 ÷ 4 + 2 = −3 + 2 = −1. ✓

7. Concert tickets cost $3 each plus a $6 fee for the whole order. Jo paid $27. How many tickets did Jo buy? Write an equation and solve it.

Let t = number of tickets: 3t + 6 = 27. Subtract 6: 3t = 21. Divide by 3: t = 7 tickets. Check: 3 × 7 + 6 = 27. ✓

8. x − 3/5 = 1/5

Add 3/5 to both sides: x = 1/5 + 3/5 = 4/5. Check: 4/5 − 3/5 = 1/5. ✓

Game: Balance Builder

Ten equations to solve for x, one-step and two-step. Every answer is a whole number. Type it, like 4 or -4, and press Enter. One point per correct answer. Your best score is saved only on this device. About 4 minutes.

Talk About It

  • Why do we undo the adding before the multiplying in 2x + 3 = 11? What happens if you divide by 2 first?
  • Is “3x + 2” true or false? Why can’t you answer?
  • Make up a word problem that 5n − 4 = 31 could describe.
  • Why can’t you divide both sides of an equation by 0?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami. Names and prices in the examples are made up for practice.