Introduction to equations

​This is a balance scale. It has two bowls that hang from an arm. When the bowls are evenly balanced like now, we know that both sides weigh the same. In the right bowl, there are ten one-kilogram weights. In the left, there are two one-kilogram weights and a secret box.

Now we want to know how much the secret box weighs. We can see that the left side weighs the same as the right one. How much does the box weigh? If the box together with two kilograms weighs ten kilograms, the box must weigh eight kilograms since two plus eight is equal to ten. If we remove two kilograms from each side, the scales remain even.

And if we open the box and look inside, we find one, two, three, four, five, six, seven, eight kilograms. ​ ​What we just did with the scale we can also do in regular mathematics with numbers by writing an equation. At first, the scale looked like this. The secret box plus two one-kilogram weights weigh the same as ten. The equal sign means the same thing as the scale that is imbalanced, that the value on both sides is the same. An equation is a mathematical tool that works the same way as a balance scale.

What is on the left has the same value as what is on the right. With this simple property, equations help us find the values of unknown quantities. The unknown quantity, one in a secret box, is usually represented by a letter, often X, like this. Now we'll solve the equation. A simple way of solving an equation is the index finger method. ​Start by covering X with your index finger.

You can now ask yourself what is under my finger if that plus two equals ten. Eight. What you have under your index finger has to be eight. Under your index finger is X. Therefore X  equals eight.

We solved the equation. Was that too easy? Let's take another one. In the left bowl, there is a ten kilogram weight and a five kilogram weight. On the right, there is a secret box and three one-kilogram weights.

How much does the secret box weigh? We write an equation from left to right. Ten plus five equals X plus one, plus one, plus one. First we can simplify this equation a bit by summing up the ten and five on the left and the ones on the right. Now you can put your index finger over X.

Fifteen equals something under the finger plus three. So, what number do you add to three to get 15? What you have under your index finger has to be 12, and under your index finger you have X. So, X equals 12. In the answer you general write X to the left of the equals sign and the value to the right.

But if you write 12 equals X  instead of X equals 12 it still means the exact same thing. That is the meaning of the equals sign. ​ ​An equation is just an equality. It consists of two sides whose numerical values are the same. Equations almost always have an unknown value, a number whose value we don't know, but can calculate from the equation. The unknown value is usually represented by a letter.

For example, X. There are equations with several unknown values, and those where you can't calculate their numerical value, but we are getting ahead of ourselves. Start by practicing with simple equations with one variable, and continue after you master them.