Expressions & Variables

Algebra is the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations. We can combine these variations of numbers and symbols (that we do or do not know) to give us an expression. An expression can be any combination of known numbers and/or unknown numbers.

Question 2

To make sure you understand what an expression is saying when they include variables , such as x, simply explain what is happening in these expressions.

2x - 7

Try describing the expression in words.

Multiply the x by 2, then, subtract 7

{3x - 8} / 3

Remembering BEDMAS

The x is being multiplied by 3, then that is being subtracted by 8, all of this is then divided by 3.

5x2 -4

BEDMAS, so exponent first.

x is to the power of two, or x is being multiplied by itself twice, then it is multiplied by 5, then 4 is subtracted.

{2x3 + 1} / 4

Write the brackets around the numerator.

x is multiplied by itself three times, then it is multiplied by 2, the 1 is added, this is then divided by 4.

Now that we can understand an expression, we can start to evaluate them. Below, first explain the steps involved, then do the calculations involved for the given value of x.

[Extra Tip: subbing in a value of, say x = 6 or x = 4, into an expression, like 2x + 1 can be thought of as, subbing in positions of football players into a team. If there are two Quarterbacks John and Mike (think of it like q = john or q = mike), and the Team is the Tigers, we can sub in q = john or q = mike into the Quarterback position. The Team is still the Tigers, but the makeup of the Tigers becomes different when we change the Quarterback (or change the value of q).]

Question 3.1

2x - 9, when x = 4

Explain: (remembering the order of operations)

Multiply x by 2, then,

Subtract that by 9

Calculate:

When we are putting in a value of x, like x = 4, look to where the x is on the expression, and replace it with a 4

2x - 9

2(4) - 9 =

Multiply the 4 by 2

8 - 9 =

Then subtract the 9

= -1

Question 3.2

5x2 -12, when x = -4

Explain: Remembering BEDMAS, so exponents first

First, x2 or x is multiplied by itself.

Then multiply that by 5,

Then subtract 12.

Calculate:

5x2 -12

x = -4

Subbing in gives us, (using extra brackets can help us see the order of operations)

5((-4)2) - 12 =

First we do exponents,

5 (16) - 12 =

Then multiply by 5,

80 - 12 =

And lastly subtract,

= 68

Question 3.3

{3(x - 3)}/3 + 4, when x = -6

Explain:

Subtract x by 3,

Multiply that by 3,

Divide that 3,

Calculate:

When we see something in an expression being divided like 3(x - 3) divided by 3, we are allowed to write brackets around the numerator 3(x-3), this can help us see what the order of operations is. Division always implies that there are ”invisible” brackets around the numerator.

(3( -6 - 3))/3 +4 =

Doing the subtraction in the brackets first,

(3(-9))/3 + 4 =

Then Multiply by the 3,

(-27)/3 + 4 =

Divide by 3,

-9 + 4 =

= -5

Calculate the value of the expressions below for the given value of x.

Question 4.1

3x2 - 4x - 5 when x = 3

Now we have two places of x in the expression, however, the method is still the same, sub the x into both places in the expression.

3(32) - 4(3) - 5 =

BEDMAS as always!

Exponent first,

3(9) - 4(3) - 5 =

Multiply,

27 - 12 - 5 =

Lastly subtract,

= 10

Question 4.2

{3(x - 6)}/(x - 4) when x = 0

First, try writing brackets around the numerator to help identify what our order of operations is.

(3(x - 6))/(x - 4),

Now, sub 0 into the value of x,

(3(0 - 6))/(0 - 4),

[It is an easy mistake to think that subbing in a value of x = 0 into an expression would result in that expression equalling 0, as multiplying something by zero equals zero, however, if we correctly follow our order of operations we will see that this is not the case!]

Order of operations tell us we must sort the brackets first,

First, let's start with the numerator, (3(0 - 6)), a reminder, when there are brackets inside of brackets, you start working from the inside out. In this case, we start with the (0 - 6)

So doing our numerator first,

(3(0 - 6))/(0 - 4) =

(3(-6)) / (0 - 4) =

Then the denominator

-18 / (0 - 4) =

-18 / -4 =

Now that we have sorted the brackets, we divide, giving us,

9 / 2

Question 4.3

{2x2 + 2}/5 - 3 when x = 2

Writing brackets around the numerator

(2x2 + 2)/5 - 3

Then sub in our x = 2 into the x slots,

(2(2)2 + 2)/5 - 3

Then, we calculate the numerator first, (2(2)2 + 2), remembering the order of operations.

Exponents, 22 = 4

(2(4) + 2)/5 + 3 =

Multiply 2(4)

(8 + 2)/5 + 3 =

Addition, as we are still working inside the brackets, (8 + 2)

10/5 + 3 =

Division,

2 + 3 = 5

Question 4.4

{{-3x/4} + 4(x-2)} / {2x2 + 1} when x = 4

First, lets sub our x value in, and add brackets around the numerator.

({-3(4)/4} + 4( (4) - 2 ) ) / {2(4)2 + 1}

This question has a lot going on in it, it can help if we write the numerator on its completely on its on, calculate its value, then sub it back into the full expression.

Writing the numerator on its own.

{-3(4)/4} + 4((4) - 2),

Now we can see there is brackets, multiplication, division, addition and subtraction.

So, brackets first,

{-3(4)/4} + 4(2)

Multiplication next (however, it would also be correct to do division next as well).

-3(4) = - 12 & 4(2) = 8

-12/4 + 8 =

Division,

-3 + 8 =

5

Question 5.1

Sarah has decided recently learned how to make chairs after taking a carpentry class and wants to start a business on the side. She figures she can make 3 chairs per week and already has 12 chairs that she made while she was in the carpentry class. Sarah worked out an expression to calculate how many chairs she will have after a certain number of weeks, 3w + 12, where w is the number of weeks she has been working. (assume that she is not yet selling any of the chairs she makes)

How many chairs will Sarah have after 5 weeks?

[First, let's just understand fully what the expression 3w + 12 means. Breaking it down, 3w means 3 times the number of weeks she has been working, as every week, she makes 3 new chairs, so times the number of weeks by 3. The + 12 is because she has started with 12 chairs already, so no matter what week she is in, she will always have those 12 chairs to add to her total]

So, what we know from the question,

The number of chairs she has after a certain number of weeks (represented by w) is equal to 3w + 12

The question wants to know how many chairs there are after 5 weeks. 5 weeks is symbolized as w =5,

So, we substitute in the value of w = 5 into the expression,

3(5) + 12 =

Remembering order of operations,

15 + 12 =

27

So, when w = 5 (or after 5 weeks) Sarah will have 27 chairs.

Question 5.2

After her first month, Sarah realises that after every week, she gets faster and faster at making chairs as her skill improves. Sarah has now come up with a more accurate expression to calculate the number of chairs she has, {w2}/80 + 3w +12, still with w = number of months worked. How many chairs will she have after 1 year?

So what do we know now from the question,

The new expression is {w2}/80 + 3w + 12, with still w = weeks worked,

How many chairs will she have after one year

When units like, years and months or hours and minutes or yards and miles are used in questions, you can often be asked to calculate something in one part of the question using one unit and then be asked to calculate something else in another part of the question in a different unit. This often is not made obvious and is used to try and make sure you have a good understanding of your units and are paying attention to the question.

All we have to do here is remember what units we are working in are (weeks) and change one year into that. So one year = 52 weeks, or w = 52.

As before, sub that into the expression {w2}/80 + 3w + 12,

{522} / 80 + 3(52) + 12

Exponents first,

2704 / 80 + 3(52) + 12

Multiplication,

2704 / 80 + 156 + 12

Division,

33.8 + 156 + 12

201.8

So, after 52 weeks (w = 52) or one year, sarah will have 201.8 chairs.

A skill you will develop with more “wordy” questions like this is to identify what the math is asking us to do and what is largely unimportant. For this one, it might cause a distraction thinking about her speeding up and her skill changing after one month, however all we need to be able to do is identify what the question is asking us and what the numbers and expressions are that we need to work with are.

Question 6

The following work is an evaluated expression done incorrectly:

Evaluate x2 - 3(x - 2) when x = 2

Incorrect work:

x2 - 3(x - 2) =

22 - 3(2 - 2) =

22 - 3(0) =

4 - 3(0) =

1(0) =

0

Identify the mistake made and why it was wrong

This question is about identifying the order of operations and finding the mistake

x2 - 3(x - 2) =

22 - 3(2 - 2) =

The values for x are subbed in which is correct

22 - 3(0) =

^ The brackets are done first, which is correct

4 - 3(0) =

The exponents are done next, which is correct

1(0) =

The subtraction was done next, which is incorrect, as we can see there was a multiplication to do first, 3(0).

0

Evaluate the expression correctly

Follow the same steps until we reach the incorrect one:

x2 - 3(x - 2) =

22 - 3(2 - 2) =

22 - 3(0) =

4 - 3(0) =

Here we remember not to make the same mistake, by multiplying first,

4 - 0 = 4