# Writing expressions

Learn to write algebraic expressions in and out of word problems.

## Why do we have math if we can describe things in words?

Sometimes in math we describe an expression with a phrase. For example, the phrase
"$2$ more than $5$"
can be written as the expression
$2 + 5$.
Similarly, when we describe an expression in words that includes a variable, we're describing an algebraic expression (an expression with a variable).
For example,
"$3$ more than $x$"
can be written as the algebraic expression
$x+ 3$.
But why? Why use math if we can describe things in words? One of the many reasons is that math is more precise and easier to work with than words are. This is a question you should keep thinking about as we dig deeper into algebra.

## Different words for addition, subtraction, multiplication, and division

Here is a table that summarizes common words for each operation:
OperationWordsExample algebraic expression
Additionplus, sum, more than, increased by$x + 3$
Subtractionsubtracted, minus, difference, less than, decreased by$p - 6$
Multiplicationtimes, product$8k$
Divisiondivided, quotient$a \div 9$
For example, the word "product" tells us to use multiplication. So, the phrase
"the product of $8$ and $k$"
can be written as
$8k$.

## Let's take a look at a trickier example

Write an expression for "$m$ decreased by $7$".
Notice that the phrase "decreased by" tells us to use subtraction.
So the expression is $m - 7$.
The word "decreased" is used, which means we should subtract, but we still need to be careful about which order to put the numbers in.
For example, "$m$ decreased by $7$" tells us to start at $m$ and decrease by $7$ which can be written as the expression $m - 7$.
On the other hand, the expression $7 - m$ is "$7$ decreased by $m$."

## Let's try some practice problems!

### Basic expressions

Write an expression for "$y$ more than $7$".
Write an expression for "$6$ minus $b$".
Write an expression for "$x$ divided by $9$".
Write an expression for "$2$ multiplied by $t$".
Write an expression for "the sum of $a$ and $4$".

### More complicated expressions

Write an expression for "$8$ less than the product of $7$ and $x$".
Write an expression for "the sum of $6$ and the product of $3$ and $d$".

### Word problems

Daniel makes $100$ dollars each week. He worked for $x$ weeks this summer.
How much money did Daniel make this summer?
Let's see what happens as the number of weeks increases:
$\blueD{1}$$100 \cdot \blueD{1} = 100$
$\blueD{2}$$100 \cdot \blueD{2} = 200$
$\blueD{3}$$100 \cdot \blueD{3} = 300$
$\blueD{x}$$100 \cdot \blueD{x} = 100x$
The answer: $100x$
Sebastian has $12$ more trophies than Megan. Megan has $t$ trophies.
How many trophies does Sebastian have?
$\blueD{10}$$\blueD{10}+12 = 22$
$\blueD{20}$$\blueD{20}+12 = 32$
$\blueD{30}$$\blueD{30}+12 = 42$
$\blueD{t}$$\blueD{t}+12$
The answer: $t+12$