3. Algebra and Exponents

Lesson

Algebra is a key tool in mathematics that allows us to write ideas and general statements concisely. In algebra we use letters or symbols called variables to represent unknown values.

From information given we can use algebra to form:

- Expressions - a group of mathematical symbols representing a number or quantity. Some examples of expressions include $2x$2
`x`, $5y+7$5`y`+7, and $\frac{a}{4}-2b$`a`4−2`b`. Expressions do**not**contain equality symbols. - Equations - a mathematical statement that shows that two expressions are equal. Some examples of equations are $2x=8$2
`x`=8, $3x+5=2x-7$3`x`+5=2`x`−7, and $5y^2-4=2x$5`y`2−4=2`x`. Equations**always**include an equality symbol. - Formulas - a rule written using symbols that describe a relationship between different quantities. Some examples of formulas are $A=L\times W$
`A`=`L`×`W`, $a^2+b^2=c^2$`a`2+`b`2=`c`2, and $y=2x+5$`y`=2`x`+5.

Remember!

- Addition, which has the symbol "$+$+", can be expressed by words such as "more than", "sum", "plus", "add" and "increased by".
- Subtraction, which has the symbol "$-$−", can be expressed by words such as "less than", "difference", "minus", "subtract" and "decreased by".
- Multiplication, which has the symbol "$\times$×", can be expressed by words such as "groups of", "times", "product" and "multiply".
- Division, which has the symbol "$\div$÷", can be expressed by words such as "quotient" and "divided by". We usually represent division using fractions instead of using the "$\div$÷" operator.
- Equality, which has the symbol "$=$=", can be expressed by words such as "is", "equal to" and "the same as". A number sentence needs one of these to be an equation!

Write down an expression to represent the number of matchsticks there are if we have four full boxes and twelve loose matchsticks.

**Think:** Let $m$`m` be the number of matchsticks in a full box. Then we have $4$4 lots of $m$`m` plus $12$12 additional matches.

**Do:** "$4$4 lots of $m$`m`" means $4\times m$4×`m`, and "$12$12 additional matches" means we are going to add $12$12 to this amount.

So we have $4\times m+12$4×`m`+12, which we can write more simply as $4m+12$4`m`+12.

Write down an equation in simplest form to represent "$v$`v` is $5$5 less than $3$3 lots of $u$`u`".

**Think:** What symbol, number or variable can we use to represent each part of the sentence?

**Do:** "$v$`v` is" means that $v$`v` will be on one side of the "$=$=" sign and everything else will be on the other side. "$3$3 lots of $u$`u`" means $3\times u$3×`u`, and "$5$5 less than" means we are going to subtract $5$5 from this amount, using the "$-$−" operator.

So we have $v=3\times u-5$`v`=3×`u`−5, which we can write more simply as $v=3u-5$`v`=3`u`−5.

Careful!

The order of the numbers in the sentence is not necessarily the same as the order in the equation!

In the example above, "$5$5 less than" meant that $5$5 was to be **subtracted from** the following term "$3$3 lots of $u$`u`". So the equation was written as $v=3u-5$`v`=3`u`−5.

Write an equation in simplest form for:

$y$`y` is $x$`x` divided by $3$3 plus $12$12.

Write an equation for:

$y$`y` equals five times the sum of $x$`x` and ten.

Write an algebraic expression for '$15$15 less than the quotient of $p$`p` and $6$6'.

The other day Tina bought $v$`v` chart-topping albums from Entertainment R Us for $\$52$$52. Write an expression for the average cost of each album.

Create algebraic expressions to generalize relationships expressed in words, numbers, and visual representations, in various contexts.