Algebra for Beginners: Equations as a Balance Scale

The equation is a balance

Look at an old scale with two pans. If both sides hold the same weight, the beam stays level. An equation is that scale written in symbols, and the equals sign is its middle. Put a box with a hidden weight x on the left pan, add 3 units, and put 7 units on the right pan: x + 3 = 7. The scale is level, so we know the hidden weight can be found. The sign = was introduced in 1557 by the Welsh mathematician Robert Recorde, who chose two parallel lines because, he explained, no two things could be more alike. Keep the picture in mind: every equation is a level beam.

Do the same on both sides

How do you find x without breaking the balance? Whatever you do to one pan, do exactly the same to the other. Take 3 away from the left pan and the beam tips, unless you also take 3 away from the right. After both removals the box x alone balances 7 − 3 = 4 units. So x = 4. Check by putting 4 back into the equation: 4 + 3 = 7, and the scale is level again. The same rule holds for other moves: you can add the same number to both sides, multiply both by the same number, or divide both by the same non-zero number.

Try your own scale

Open the sandbox under the picture. It builds an equation of the form a·x + b = c with three sliders: a from 1 to 5, b from 0 to 10, and c. The scale tilts until you solve it, then it settles level. The answer is x = (c − b) / a, so with a = 3, b = 4 and c = 19 you get x = (19 − 4) / 3 = 5, and the check 3 × 5 + 4 = 19 works. If c − b does not divide evenly, the answer is a fraction. With a = 2, b = 1 and c = 6, x = 5/2 = 2.5. A hidden weight can be a part of a unit.

Two steps on the scale

Now the left pan holds two boxes and one extra unit: 2x + 1 = 9. Work in order, undoing the last thing first. Take 1 from both pans: 2x = 8. Two boxes weigh 8, so halve both pans: x = 4. Check: 2 × 4 + 1 = 9. Here is another one: 3x + 5 = 20. Subtract 5 from both sides to get 3x = 15, then divide by 3 to get x = 5, and the check 3 × 5 + 5 = 20 works out. The plan is the same every time: first remove what was added, then undo the multiplier.

Al-Khwarizmi and al-jabr

Around AD 820, in the House of Wisdom in Baghdad, the scholar Muhammad ibn Musa al-Khwarizmi wrote a book on solving equations. Its title contained the word al-jabr, meaning restoring or completing: moving a subtracted amount to the other side, where it becomes added. For x − 3 = 5, restore by adding 3 to both sides, and x = 8. Latin turned the word into 'algebra'. His own name, Latinised as Algoritmi, gave us 'algorithm', because a Latin book on his Hindu numerals taught step-by-step rules of calculation. He wrote his rules in words; letters like x came much later.

Two unknowns, two scales

What if there are two hidden weights? Then you need two scales. Suppose x + y = 10 and x − y = 2. Put the scales side by side and add them: the y on one cancels the y on the other. We are left with 2x = 12, so x = 6. Now use the first scale: 6 + y = 10, so y = 4. Check the second one: 6 − 4 = 2. Both balance. This trick of adding or subtracting whole equations to cancel one unknown is called elimination. In words: two numbers add up to 10 and differ by 2. They are 6 and 4.

Another way: substitution

A second method is to swap one unknown for something you know. A cinema sold 100 tickets for 1000 in all. Adult tickets cost 12 and child tickets cost 8. Let a be the adults and c the children. Then a + c = 100 and 12a + 8c = 1000. From the first, c = 100 − a. Put that into the second: 12a + 8(100 − a) = 1000, so 4a + 800 = 1000, and a = 50. Then c = 50 as well. Check: 50 × 12 + 50 × 8 = 600 + 400 = 1000. One equation with one unknown is a scale you already know how to balance.

Algebra: equations as a balance