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$2x, $5y+7$5y+7, and $\frac{a}{4}-2b$a4−2b. Expressions do not contain equality symbols.
- Equations - a mathematical statement that shows that two expressions are equal. Some examples of equations are $2x=8$2x=8, $3x+5=2x-7$3x+5=2x−7, and $5y^2-4=2x$5y2−4=2x. 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, $a^2+b^2=c^2$a2+b2=c2, and $y=2x+5$y=2x+5.
- 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!
Worked examples
Example 1
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$4m+12.
Example 2
Write down an equation in simplest form to represent "$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=3u−5.
Practice questions
Question 1
Write an equation in simplest form for:
$y$y is $x$x divided by $3$3 plus $12$12.
Question 2
Write an equation for:
$y$y equals five times the sum of $x$x and ten.
Question 3
Write an algebraic expression for '$15$15 less than the quotient of $p$p and $6$6'.
Question 4
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.
Finding unknown values
Once we can write worded problems as algebraic equations, we can often solve them by finding the unknown values. We have seen this when finding unknown sides in the measurement chapter. Let's look at some further examples of this:
Worked examples
Example 3
In the equation $2n-4=10$2n−4=10, what is the value of $n$n?
Think: One way of writing this equation in words is "Double a number and subtract four then the result is ten". What number would we need to start with for this to be true?
Do: Working backwards we can see that four more than ten must be twice our starting number. So $2n=14$2n=14, halving both sides we get $n=7$n=7. Writing this out in logical steps helps reduce errors, allows others to see our thinking and provides a framework for more complicated equations when reversing the steps might not be as straightforward. So formally we can set our solving process out as follows:
| $2n-4$2n−4 | $=$= | $10$10 |
Write out the original equation. |
| $2n$2n | $=$= | $10+4$10+4 |
Add $4$4 to both sides |
| $2n$2n | $=$= | $14$14 |
Simplify the right-hand side. |
| $n$n | $=$= | $7$7 |
Divide both sides by $2$2. |
Remember that to keep an equation balanced, we must perform the same operation to the whole expression on each side of the equation.
Practice questions
Question 5
A number (call it $n$n) plus ten equals eighteen.
Write the sentence using mathematical symbols.
What number is $n$n?
Question 6
A number (call it $n$n) is divided by two to give five.
Write the sentence using mathematical symbols.
What number is $n$n?