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8.1 | Writing Formulas and Substituting in a Formula |

In these revision notes for Writing Formulas and Substituting in a Formula, we cover the following key points:

- What is a mathematical sentence?
- What are mathematical operators?
- Where does a mathematical sentence differ from a wordy sentence?
- What are formulas? Why do we use them?
- How to write a formula? What do we call its components?
- What types of variables are there in a formula?
- How to substitute the variables in a formula?
- What is the procedure used when solving an exercise containing formulas?
- How to convert wordy problems into formulas and vice-versa?
- How to find the inverse formula when the original formula is known?

**A mathematical sentence is a fact (it may be either true or false) that combines two expressions (written in mathematical symbols) connected through a comparison operator between them.** This comparison operator may be one of the following:

Equal to (=);

Greater than (>);

Smaller than (<);

Greater than or equal to (≥); and

Smaller than or equal to (≤)

Mathematical sentences are used to express word sentences in a much shorter way, where words are replaced with math symbols.

**A formula is a fact or a rule written in mathematical symbols**. It contains a set of instructions on how to calculate an unknown quantity in terms of one or more known ones. The unknown quantity (otherwise known as the **independent variable**) is related to the known ones (**dependent variables**) through the equal sign ' = '.

Variables in formulas are combined with each other through mathematical operators such as addition, subtraction, multiplication, division, raise in power, logarithm, exponentiation, etc. Only the multiplication symbol ('·' or '×') is not written in a formula but is implied. Likewise, the division symbol (÷) is replaced by the fraction bar ('-' or '/').

We can either create a formula by reading and interpreting the description or interpret a formula in order to express in words the phenomenon described by it.

We use formulas to express scientific phenomena and also to describe daily activities in a shorter way, especially when these activities are recurring and have the same routine. The next step after correctly writing a formula is to substitute the known variables in it. In this way, we open the path to the calculation of the unknown quantity.

The steps one has to follow when dealing with formulas are:

**Step 1:** Identify the variables that are present in the situation under consideration.

**Step 2:** Assign a different letter to every variable including the one you are going to calculate.

**Step 3:** Write down the correct formula by including all variables identified in the first step.

**Step 4:** Substitute the other variables with numbers and calculate the value of the variable you are interested in by using the BEDMAS (PEMDAS) rule.

Wordy problems contain the necessary information to produce one or more formula; you just have to identify the variables participating in the event.

Sometimes, the independent variable in a formula is known and the value of one of dependent variables is required. In this case, we need to find a new formula based on the original one but which expresses the new unknown quantity in terms of the known ones. **This is known as the inverse formula**.

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