8.01 Introduction to variables

The basic concept of algebra is using symbols, called variables or pronumerals, in the place of numerical values that we don't know. We normally uses letters of the alphabet as our variables.

We can use these pictures in the same way that we would use numbers, except they have some special rules for how we write them.

In algebra, adding variables works the same as adding numbers.

Suppose we have one box containing p apples. If we add 3 more apples, how many apples do we have in total?

If we look at this problem in words we have:\text{Number of apples}= \text{one box of } p\, \text{apples plus}\, 3 \text{ more apples}

We can simplify this to:\text{Number of apples}= p\, \text{apples plus}\, 3 \text{ more apples}

Which we can write mathematically as:\text{Number of apples}= p + 3

Since we don't know what number the variable p stands for we can't simplify this expression any further.

Now what if we are adding different variables together?

Suppose we have a box containing p apples and a bag containing q apples. How many apples do we have altogether?

We can simplify this problem to: \text{Number of apples}= p\, \text{apples plus}\, q \text{ apples}

Which we can write mathematically as:\text{Number of apples}= p + q

Again, because we don't know the value of either variable, we can't simplify the expression any further.

But what if we add the same variables together?

Suppose that we have one box containing p apples and then we get another box containing p apples.

We can write p apples plus p more apples as:\text{Number of apples}= p + p

This is same as saying that we have 2 sets of p apples.

Remember that adding the same number multiple times is the same as multiplying it. That is what is happening here, two sets of p apples can be written as:\text{Number of apples}= p + p=2p

Examples

Example 1

The key shows how many oranges are in each sized box.

a

If Justin has a oranges, which box does he have?

A

B

Worked Solution

Create a strategy

We can use the key to determine which box contains a oranges.

Apply the idea

According to the key the small box contains a oranges and the large box contains b oranges.

So the correct answer is option A.

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b

If Laura has b oranges, which box does she have?

A

B

Worked Solution

Create a strategy

We can use the key to determine which box contains b oranges.

Apply the idea

According to the key the small box contains a oranges and the large box contains b oranges.

So the correct answer is option B.

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c

If Vincent has the same amount of oranges as Justin and Laura combined, how many oranges does he have altogether?

A

ab

B

a + b

Worked Solution

Create a strategy

Add together the amount in each box from parts (a) and (b).

Apply the idea

When adding the number of oranges together we get the sum of a and b as an expression of a+b.

The correct answer is option B.

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Multiplying variables also works the same as multiplying numbers except for one key difference.

Notice how we wrote two sets of p as 2p instead of 2 \times p. When multiplying numbers and variables together we can simplify the result by writing the number in front of the variable to represent the number of sets we have.

This way of writing multiplication only works for algebraic terms. If we try to do this for numbers we will get the wrong answer.

For example: 4 \times 7 = 28, not 47

We saw in the addition section that adding the same variable multiple times is the same as multiplication.

For example: if we have 5 boxes containing p apples each, then the total number of apples will be equal to 5 sets of p.

We can write this mathematically as:\text{Number of apples}= 5 \times p=5p

Examples

Example 2

Xanthe is selling some oranges in boxes and some oranges individually.

a

How many oranges does each symbol represent?

Worked Solution

Create a strategy

Use the key to check how many oranges each symbol is equal to.

Apply the idea

Using the key we can see that the small box is equal to x oranges, the large box is equal to y oranges and the individual orange is equal to 1 orange.

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b

Using the values from part (a), how many oranges does the image represent?

Worked Solution

Create a strategy

We can use the key to write the total as an expression using multiplication.

Apply the idea

We can see that we have three sets of x, four sets of y and three sets of 1 orange.

We can write the whole expression as: 3x+4y+3

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