Algebra is a mathematical language used to generalise patterns and relationships that we observe in the world around us.
Exploration
As an example of how algebra is used, we will consider the perimeter of a square:
We know that the perimeter of a shape is the total distance around its edge. We also know that a square has four equal sides.
To create a model that will represent any square (i.e. a generalised square) we will use the variable $m$m to represent the length of each side.
We can then form a general expression for the perimeter of any square:
| Perimeter | $=$= | $m+m+m+m$m+m+m+m |
(Add together the length of each side) |
| $=$= | $4m$4m |
(Simplify) |
If we want to evaluate the perimeter of a particular square, we simply substitute a value for $m$m into the expression $4m$4m:
- For a square with side length $6$6 cm, substitute $m=6$m=6 to find the perimeter. This means we replace the letter $m$m, each time it occurs, with the number $6$6.
Perimeter $=$= $4m$4m $=$= $4\times6$4×6 (Substitute $m=6$m=6)
$=$= $24$24 cm - For a square with side length $10$10 cm, substitute $m=10$m=10 to find the perimeter. In this case, we replace the letter $m$m, each time it occurs, with the number $10$10.
Perimeter $=$= $4m$4m $=$= $4\times10$4×10 (Substitute $m=10$m=10)
$=$= $40$40 cm
So $4m$4m is a general expression for the perimeter of any square, and substituting a value for $m$m will give us the perimeter of a particular square.
Understanding the language of algebra
In algebra, we use letters in combination with numbers and other mathematical symbols to create expressions, equations or formulas that model a particular problem or situation. This makes it much easier to understand the problem and to find a solution.
The letters represent variables (values that change) or constants (values that stay the same). These letters are sometimes referred to as pronumerals or unknowns.
In the exploration above, the side length of our square was initially unknown, because our model was used to represent all possible squares. We could have chosen any letter to represent the side length. The important thing is that we defined what the letter represented before we created an expression.
The language of algebra has been designed for clear and efficient communication of mathematical ideas. It is a language understood anywhere in the world and is based on the following conventions:
- When multiplying two numbers together, we use a multiplication sign, such as $2\times3$2×3. When we multiply a number by a variable, or when we multiply variables together, we leave out the multiplication sign. As an example, $4\times m$4×m is written as $4m$4m, and $a^2\times b$a2×b is written as $a^2b$a2b.
- When we multiply a number by one or more variables, we write the number first and then the variables. For example, $p\times3\times q$p×3×q is written as $3pq$3pq.
- If we multiply a variable by itself, we simplify the expression using an exponent. So the expression $m\times m\times m$m×m×m, is written as $m^3$m3 instead of $mmm$mmm.
- If we multiply one or more variables by $1$1, we don't need to write the $1$1. For example, instead of writing $1\times x$1×x or $1x$1x, we can just write $x$x. In a similar way, $-1\times x$−1×x is written as $-x$−x.
- We usually avoid the division symbol $\div$÷, and instead indicate division using fractions. So instead of writing $t\div12$t÷12, we would write $\frac{t}{12}$t12.
- Improper fractions are preferred over mixed numbers because they are easier to work with in algebraic expressions. For this reason, we would write $\frac{21}{5}$215 instead of $4\frac{1}{5}$415.
Evaluating by substitution
To evaluate an expression, equation or formula, we substitute values for the variables. Substitution means we replace each occurrence of a letter with a particular value.
Use the following applet to practise evaluating an expression given the value of $x$x.
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Worked examples
example 1
If $x=3$x=3, evaluate $6x-4$6x−4.
Think: This means we replace every $x$x in the expression with the number $3$3.
Do:
| When $x=3$x=3, | |||
| $6x-4$6x−4 | $=$= | $6\times3-4$6×3−4 |
(Substitute $x=3$x=3) |
| $=$= | $18-4$18−4 |
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| $=$= | $14$14 |
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Example 2
If $x=6$x=6 and $y=0.5$y=0.5, evaluate $6x-2y-12$6x−2y−12.
Think: Just like before, we replace the letter $x$x with the number $6$6, and the letter $y$y with the number $0.5$0.5. Make sure to use the correct order of operations when evaluating the expression.
Do:
| When $x=6$x=6 and $y=0.5$y=0.5, | |||
| $6x-2y-12$6x−2y−12 | $=$= | $6\times6-2\times0.5-12$6×6−2×0.5−12 |
(Substitute $x=6$x=6 and $y=0.5$y=0.5) |
| $=$= | $36-1-12$36−1−12 |
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| $=$= | $23$23 |
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Example 3
If $p=-4$p=−4 and $q=-5$q=−5, evaluate $p-q^2$p−q2:
Think: Once again, we will replace $p$p with $-4$−4 and $q$q with $-5$−5 in the expression.
Do:
| When $p=-4$p=−4 and $q=-5$q=−5, | |||
| $p-q^2$p−q2 | $=$= | $-4-\left(-5\right)^2$−4−(−5)2 |
(Substitute $p=-4$p=−4 and $q=-5$q=−5) |
| $=$= | $-4-25$−4−25 |
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| $=$= | $-29$−29 |
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Reflect: Notice that in this case the variable $q$q is squared. When we replace $q$q with a negative number, we place it inside a pair of brackets, to make it clear that the value being squared is negative. It also helps us avoid any confusion with the subtraction sign immediately before it in the expression.
When we substitute negative values into expressions, it can be helpful to enclose the negative value in a pair of brackets. This is particularly important when the negative value is being raised to a power or when an operator like $+$+, $-$− or $\times$×, comes immediately before it in the expression.
Evaluating more complex expressions
Some patterns and relationships are described by more complex expressions. Here are a few examples:
- A car salesperson's earnings from selling a car valued at $\$y$$y, is given by $0.02\left(y-20000\right)$0.02(y−20000).
- The amount of oxygen in the air $x$x km above sea level can be approximated by $160-15x$160−15x.
- For a person $h$h cm tall and weighing $w$w kg, their body surface area, in m2, can be estimated using $\sqrt{\frac{hw}{3600}}$√hw3600.
Notice that these expressions involve more than one operation, and sometimes more than one variable, so we need to make sure we follow the correct order of operations when substituting values.
Worked examples
example 4
Find the value of the expression $0.02\left(y-20000\right)$0.02(y−20000) when $y=30000$y=30000:
Solution
| When $y=30000$y=30000, | |||
| $0.02\left(y-20000\right)$0.02(y−20000) | $=$= | $0.02\left(30000-20000\right)$0.02(30000−20000) |
(Substitute $y=30000$y=30000) |
| $=$= | $0.02\times10000$0.02×10000 |
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| $=$= | $200$200 |
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example 5
Find the value of the expression $160-15x$160−15x when $x=2$x=2:
Solution
| When $x=2$x=2, | |||
| $160-15x$160−15x | $=$= | $160-15\times2$160−15×2 |
(Substitute $x=2$x=2) |
| $=$= | $160-30$160−30 |
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| $=$= | $130$130 |
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example 6
Find the value of the expression $\sqrt{\frac{hw}{3600}}$√hw3600 when $h=160$h=160 and $w=60$w=60. Round your answer to two decimal places if necessary.
Solution
| When $h=160$h=160 and $w=60$w=60, | |||
| $\sqrt{\frac{hw}{3600}}$√hw3600 | $=$= | $\sqrt{\frac{160\times60}{3600}}$√160×603600 |
(Substitute $h=160$h=160 and $w=60$w=60) |
| $=$= | $\sqrt{\frac{9600}{3600}}$√96003600 |
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| $=$= | $1.632$1.632... |
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| $=$= | $1.63$1.63 (2 d.p.) |
(Round to two decimal places) |
Practice questions
Question 1
If $m=-3$m=−3 and $n=4$n=4, evaluate the following:
$mn-\left(m-n\right)$mn−(m−n)
$m^2+9n$m2+9n
Question 2
Evaluate $\frac{2a\times9}{5b}$2a×95b when $a=25$a=25 and $b=-2$b=−2.
Find the exact value in simplest form.
Question 3
Consider the expression $x\left(3y+4\right)+42$x(3y+4)+42.
Evaluate $x\left(3y+4\right)+42$x(3y+4)+42 at $x=4$x=4 and $y=-2$y=−2.
Evaluate $x\left(3y+4\right)+42$x(3y+4)+42 at $x=-43$x=−43 and $y=-1$y=−1.