Algebra

Lesson

Many real life situations involve problems with unknown amounts or values. By using algebra, we can express these kinds of problems as mathematical statements. Doing so allows us to more easily understand the problem, and we will later see a number of tools and tricks that can be used to solve such problems.

Say there is a certain amount of fuel left in the tank and at the petrol station we pump 30 litres of fuel to fill the tank. Now because we don't know exactly how much fuel there was in the tank to begin with, we can use a symbol such as a letter to represent the amount. These letters are called variables or pronumerals, and the most common ones are $x$`x` and $y$`y`. So we could represent the total amount of fuel in the tank by the expression $x+30$`x`+30. Mathematical expressions that use variables are called algebraic expressions.

An important convention to note is that when we multiply a number by a variable, we leave out the multiplication sign. For example, $2\times y$2×`y` can be written more simply as $2y$2`y`. When we do so, the number is often called the coefficient of the variable. So in the above example of $2y$2`y`, we would call the number $2$2 the **coefficient of** $y$`y`.

Similarly, we usually express division using fractions instead of the symbol $\div$÷. So we would write $\frac{x}{2}$`x`2 in place of $x\div2$`x`÷2.

We can use algebra to represent many different situations. For example, think of a store that sells laptops for $\$1000$$1000 each and smartphones for $\$500$$500 each. The amount they make in a day will depend on how many of each they sell.

If we let:

- $a$
`a`represent the number of laptops sold in a day, and - $b$
`b`represent the number of smartphones sold in a day,

then the total amount they make in a day is given by the expression $1000a+500b$1000`a`+500`b`.

If they sell $5$5 laptops and $6$6 smartphones in one day, then the total sales for the day will be:

$1000a+500b$1000a+500b |
$=$= | $1000\times5+500\times6$1000×5+500×6 |

$=$= | $5000+3000$5000+3000 | |

$=$= | $8000$8000 |

So the store will have made a total of $\$8000$$8000 in sales on this particular day.

If $x$`x` represents the number of peaches then write an expression for the number of peaches minus $17$17.

If $x$`x` represents the number of elephants than write an expression for $4$4 times the number of elephants plus $10$10.

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

Form and solve simple linear equations