03 Forming expressions

Forming expressions from statements

Why should we learn expressions at all? Well, as we have encountered so far, expressions help us simplify complex situations and make them shorter. Additionally, an expression always tells us something about a situation in a mathematical way. The operators used in expressions are a direct translation of the meaning that a statement provides. For example, if an expression says double the amount of an unknown number, we first need a placeholder variable for the number and then multiply that by 2. If we choose the placeholder for that number as “g”, then double the number becomes 2g. You might be thinking, “But the expressions we’ve learned so far are easy (like the one above). How can they get any simpler?” That’s true, the expressions we’ve learned are basic. But sometimes we need to use multiple operators like addition, subtraction, and multiplication and many more in the same situation. Getting them mixed up is very easy when we are trying to form expressions. The idea is to go word by word and understand which operation goes where.

Let’s say we want to figure out what “5 less than 7 times a number” means. When we see the word “less”, that means we need to subtract something. And when we see “times”, that means we need to multiply. We can rewrite it to say “subtract 5 from 7 multiplied to a number”. So, we need to multiply 7 by a number we don’t know yet. We can call that number 𝑥. Then, we need to subtract 5 from that result. Let’s write an expression for this. To write this as an expression, we start with 7𝑥 (that’s 7 times 𝑥). Then we subtract 5 from that to get 7𝑥 - 5. It’s important to do the multiplication first, and then the subtraction. But be careful! If the statement was “7 times 5 less than a number”, then we would write it as 7(𝑥 - 5). See the difference? Here, 7 is multiplied to a difference of 5 and a number.

Simplification of expressions

Earlier, we came across the idea of needing to simplify expression, either to find the value or to find the simplest form of expression. Back then, we worked with pretty simple expressions so the idea of thinking about which operator to work on first was not the issue. But there are times when we need to simplify expressions that are a lot more complicated. That’s where PEMDAS comes in. It helps us figure out how to write and solve expressions.

PEMDAS is an acronym for the following: P for Parenthesis
E for Exponents
M for Multiplication
D for Division
A for Addition
S for Subtraction

The operators have precedence based on how they appear on the name PEMDAS. But multiplication and division have the same priority, and so does addition and subtraction.
How do we simplify 7(4 + 5) ÷ 14 - 8^2?
We start with the parenthesis = 7 x 10 ÷ 14 -8^2
Then the exponent = 7 x 10 ÷ 14 - 64
Then the multiplication = 70 ÷ 14 - 64 (since that is the leftmost before division)
Then division = 5 - 64

We do not have addition, so we go with subtraction = -59

Once you memorize PEMDAS, it makes simplification easy, right?

You would do the same for algebraic expressions as well. Just make sure to keep the variable as it is. For example: 2(14a - 7a) x 3 - 9 P = 2 x 7a x 3 - 9 M = 42a - 9

We stop here since we cannot subtract 9 from an unknown value of 42a. Each step transforms the previous expression into a new one. This is called symbolic transformation, and it will be used more going further.