Now you’ll learn how to multiply numbers by expressions that have variables in them. When you do that, you multiply the numbers with each other, and then the variables come along too. We’ll start by looking at the most fundamental way that this works.
The simplest expression we have that includes a variable is $x$. In front of this $x$, there’s an invisible 1. This 1 will play an important role in what is to come, so pay close attention!
The multiplication in the rule below might look a little strange. When I write $x$, it’s already equal to just one $x$. To also write a 1, to indicate that there’s just one x, is unnecessary.
Rule
$$1\cdot x=1x=x$$ |
When you know that $x=1x$, you can move onto the next step.
What does it mean to multiply a number by an expression with variables? Take, for example, the multiplication expression $3\cdot 2x$. This expression describes how many times you have the factor $2x$. It can be analyzed like this:
$$3\cdot 2x=\underset{\text{}2x\text{addedthreetimes}}{\underbrace{2x+2x+2x}}=6x$$ |
This method is kind of bothersome, though. Because of that, to multiply a number by an expression with variables, you just multiply the number with the number in front of the variable. Remember that when there isn’t any number in front of $x$, there’s still an invisible 1 there, and you can multiply your number by 1.
Example 1
$$\begin{array}{llll}\hfill 1\cdot x& =x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 2\cdot x& =2\cdot 1x=2x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 5\cdot x& =5\cdot 1x=5x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
When you multiply a number by $x$, you multiply that number by the invisible 1 in front of $x$. You can see what that looks like above.
Let’s take another step and see what really happens with expressions like the one above. This time you’re multiplying a number by an expression that consists of more than one $x$.
Rule
When you multiply a number by a lonely variable, you can just put the number in front of the variable.
When you multiply a number by a variable that already has a coefficient in front of it, you can just multiply the number and the coefficient by each other.
Example 2
Multiplying numbers by variables:
$$2\cdot 2x=4x$$ |
because $$\begin{array}{llll}\hfill 2\cdot 2x& =2x+2x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =4x.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
Example 3
$$4\cdot 2x=8x$$ |
Example 4
Multiplying numbers by variables:
$$4\cdot 5x=20x$$ |
because $$\begin{array}{llll}\hfill 4\cdot 5x& =5x+5x+5x+5x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =20x.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
Example 5
Multiplying numbers by variables:
$$10\cdot 9x=90x$$ |