Chapter 8: Operations with binary numbers

Chapter 10: Simplification of algebraic expressions

Chapter 9: Using symbols

9.1 Open and closed number sentences

Closed number sentences

A number sentence is a group of numbers and mathematical symbols (such as $+$ , $-$ or $=$ ) that gives information about the numbers. Here is an example of a number sentence:

$3 + 4 = 7$

This is called a closed number sentence, because all the information is given in the sentence. This closed number sentence is always true.

Here is an example of a closed number sentence that is false:

$5 + 8 = 2$

closed number sentence A closed number sentence gives all the information about the numbers, and it is always true or always false.

Here are some more examples of closed number sentences:

$9 \div 3 = 3$ (true)

$7\times 12 = 84$ (true)

$4 \times 8 = 8 + 8 + 8 + 8 + 8$ (false)

Exercise 9.1: Decide whether closed sentences are true or false

State whether the closed sentence is always true or false.

$4\times 10 = 40$
True
$102\div 2 = 12$
False
$30 + 30 + 30 = 3\times 30$
True
$9\div 18 = 2$
False
$9\div 18 = \frac{1}{2}$
True

Open number sentences

Here is another example of a number sentence:

$\square + 3 = 7$

This is called an open number sentence, because not all the information is given in the sentence. We do not know if this sentence is true or false because we do not know what is in the box.

open number sentence An open number sentence does not give all the information about the numbers, but has information missing that we must find.

Worked example 9.1: Finding missing information in an open sentence

Find the value of the square in this open number sentence:

$\square + 3 = 7$

Step 1: Ask yourself the question, "What do I need to add to 3 to get 7?"

Using your knowledge and mental maths, you will find that you need to add 4 to 3 to get 7.
Step 2: Write down the value of the square.
$\square = 4$

Exercise 9.2: Find missing information in open sentences

For each open number sentence, find the number that makes the sentence true.

$17 - 9 = \square$
8
$3 + 9 = \square$
12
$2\times 12 = \square$
24
$9-\square= 2$
7
$\bigcirc=\frac{1}{2}+ \frac{1}{2}$
1
$10\times\square = 100$
10
$63\div9=\square$
7
$30 + 30 + \bigcirc = 3\times 30$
30
$\square\times6=30$
5
$5 + \bigcirc + 18 = 30$
7

Variables in open sentences

So far we have worked with open sentences that contain a symbol such as a $\square$ or a $\bigcirc$ . Instead of pictures like these, in mathematics it works better to use letters to represent missing information. These letters are called variables, because their value can vary.

variable A variable is a symbol for a number we do not know yet. It is usually a letter such as $x$ or $y$ .

If we have an expression, such as $x+5$ , we cannot know the value of the $x$ unless it is given. If we give the $x$ different values, then we can make number sentences that are true.

If $x$ = 1, then $x+5 = 1 + 5 = 6$

If $x$ = 3, then $x+5 = 3 + 5 = 8$

If $x$ = 0, then $x+5 = 0 + 5 = 0$

The value of the $x$ can vary, and that is why it is called a variable.

Exercise 9.3: Find the value of the expression

Find the value of the following expressions if $x=3$ .

$17 - x$
14
$3 + x$
6
$2\times x\times 0$
0
$9-x+2$
8
$\frac{1}{x}+ \frac{1}{x}$
$\frac{2}{3}$
$10\times x$
30
$60\div x$
20
$30 + 30 + x$
63
$x\times6$
18
$5 + x + 18$
26

9.2 Solving open sentences that have variables

Variables and open sentences form part of a section of mathematics we call algebra. To show that there is information that we must find, an open sentence can use a picture symbol (such as a box or a circle) or a variable (a letter representing any number).

Here are some examples of open sentences using variables:

$3 + x = 5$ $p - 4 = -3$ $n - 17 = 20$ $3 \times y = 15$ $b \div 20 = 5$

In these open sentences, we can find the correct value for the unknown variable that will make the sentence true. This is called solving the open sentence.

If the sentence that you need to be solve is straightforward, one option is to use mental maths to solve it.

Worked example 9.2: Finding the value of a variable using mental maths

Find the value of $x$ in this open number sentence:

$x + 7 = 9$

Step 1: You need to find the value of $x$ that will make the statement true. To do this, you can ask yourself the question, "What number added to 7 will give me 9?"

Using your knowledge and mental maths, you will find that you need to add 2 to 7 to get 9.
Step 2: This means that the value of $x$ is 2, so write your answer:
$x=2$

Using inverse operations to solve open sentences

Another way to find the value of $x$ in $x + 7 = 9$ , is to use the inverse operation of addition. The inverse operation of addition is subtraction. In Chapter 5 you learnt that subtraction may be seen as another form of addition. For example, $6-4$ is the same as $6+(-4)$ . The numbers $4$ and $-4$ are called additive inverses. When we add them, the answer is zero: $4+(-4)=0$ .

The inverse operation of multiplication is division. In Chapter 6 you learnt that dividing by a fraction is the same as multiplying by the reciprocal of the fraction. For example, $4\div\frac{2}{3}$ is the same as $4\times\frac{3}{2}$ . A fraction and its reciprocal are called multiplicative inverses. When we multiply them, the answer is 1: $\frac{2}{3}\times\frac{3}{2}=1$ .

inverse operation An inverse operation does the opposite of another operation. Addition and subtraction are inverse operations, and multiplication and division are inverse operations.

This closed number sentence is true:

$2 + 7 = 9$

We know that this closed number sentence is also true:

$2 = 9 - 7$

Notice that we get from the first number sentence to the second number sentence using the inverse operation of addition, which is subtraction:

\begin{align} 2 + 7 &= 9 \newline 2 + 7\, \color{red}{-\;7} &= 9\space \color{red}{-\;7} \newline 2 + 0 &= 9 - 7 \newline 2 &= 9 - 7 \end{align}

Here is an example of a closed number sentence using multiplication:

$2 \times 3 = 6$

We know that this closed number sentence is also true:

$2 = 6 \div 3$

Notice that we get from the first number sentence to the second number sentence using the inverse operation of multiplication, which is division:

\begin{align} 2 \times 3 &= 6 \newline 2 \times 3\; \color{red}{\div\;3} &= 6\space \color{red}{\div\;3} \newline 2 \times 1 &= 6 \div 3 \newline 2 &= 6 \div 3 \end{align}

We may use inverse operations to find the value of a variable in an open number sentence. We aim to write the number sentence in this format:

$variable=\text{value}$

Worked example 9.3: Solving an open sentence using the inverse of addition

Find the value of $x$ in this open sentence:

$x + 12 = 29$

Step 1: Use an inverse operation for any number that is added to or subtracted from the variable.
$x + 12 - 12 = 29 - 12$
Step 2: Simplify the sentence from Step 1 until you get to the format $\boldsymbol{variable}=\textbf{ value}$ .
\begin{align} x + 12 - 12 &= 29 - 12 \newline x + 0 &= 17 \newline x &= 17 \end{align}

Worked example 9.4: Solving an open sentence using the inverse of multiplication

Find the value of $y$ in this open sentence:

$y \times 3 = 27$

Step 1: Use an inverse operation for any number that the variable is multiplied or divided by.
$y \times 3 \div 3 = 27 \div 3$
Step 2: Simplify the number sentence from Step 1 until you get to the format $\boldsymbol{variable}=\textbf{ value}$ .
\begin{align} y \times 3 \div 3 &= 27 \div 3 \newline y \times 1 &= 9 \newline y &= 9 \end{align}

Note that an open sentence is not regarded as solved when the variable is negative, for example $-x=5$ . The variable must be positive in your final answer, for example $x=-5$ .

The format $variable=\text{value}$ may be swapped around. Therefore, $x=7$ and $7=x$ has the same meaning.

Worked example 9.5: Solving an open sentence using the inverse operation on the variable

Find the value of $x$ :

$18 - x = 14$

Step 1: Use an inverse operation to get a positive variable.
$18 -x + x= 14 + x$
Step 2: Simplify the number sentence from Step 1 until it has only one positive variable.
\begin{align} 18 -x + x &= 14 + x \newline 18 + 0 &= 14 + x \newline 18 &= 14 + x \end{align}
Step 3: Use an inverse operation for the number that is added to or subtracted from the variable.
$18 - 14 = 14 - 14 + x$
Step 4: Simplify the number sentence from Step 1 until you get to the format $\boldsymbol{variable}=\textbf{ value}$ .
\begin{align} 18 - 14 &= 14 -14 + x \newline 4 &= 0 + x \newline 4 &= x \newline x &= 4 \end{align}

Exercise 9.4: Solve open sentences

Find the value of the variable in the following open sentences:

$y + 7 = 30$
\begin{align} y + 7 - 7 &= 30 - 7 \newline y &= 23 \end{align}
$3 + x = 15$
\begin{align} 3 - 3 + x &= 15 - 3 \newline x &= 12 \end{align}
$z - 8 = 20$
\begin{align} z - 8 + 8 &= 20 + 8 \newline z &= 28 \end{align}
$q - 4 = -5$
\begin{align} q - 4 + 4 &= -5 + 4 \newline q &= -1 \end{align}
$17 - n = 20$
\begin{align} 17 - n + n &= 20 + n \newline 17 &= 20 + n \newline 17 - 20 &= 20 - 20 + n \newline 3 &= n \newline n &= 3 \end{align}
$15 - p = -5$
\begin{align} 15 - p + p &= -5 + p \newline 15 &= -5 + p \newline 15 + 5 &= -5 + 5 + p \newline 20 &= p \newline p &= 20 \end{align}
$x \times 8 = 48$
\begin{align} x \times 8 \div 8 &= 48 \div 8 \newline x &= 6 \newline \end{align}
$3 \times y = 15$
\begin{align} 3 \div 3 \times y &= 15 \div 3 \newline y &= 5 \newline \end{align}
$c\div 20 = 5$
\begin{align} c \div 20 \times 20 &= 5 \times 20 \newline c &= 100 \newline \end{align}
$40 \div x = 8$
\begin{align} 40 \div x \times x &= 8 \times x \newline 40 &= 8 \times x \newline 40 \div 8 &= 8 \div 8 \times x \newline 5 &= x \newline x &= 5 \end{align}

Solve open sentences with two operations

Open number sentences may have more than one operation, for example $4-2+x=10$ . To solve number sentences like these, you must remember the correct order of operations that you learnt last year:

Brackets
Of
Divide
Multiply
Add
Subtract

Worked example 9.6: Solving open sentences with two operations

Find the value of $x$ :

$x \div 2 + 3 = 8$

Step 1: Decide which inverse operation to do first.

According to the order BODMAS, you must calculate $x \div 2$ before $2+3$ .
This means $x \div 2$ must "stay together". You may insert brackets to help you.
We must use the inverse of addition first.
$(x \div 2) +3 - 3 = 8 - 3$
Step 2: Simplify the number sentence from Step 1 until you have only one operation.
\begin{align} (x \div 2) +3 - 3 &= 8 - 3 \newline x \div 2 &= 5 \end{align}
Step 3: Use an inverse operation and simplify to get to the format $\boldsymbol{variable}=\textbf{ value}$ .
\begin{align} x \div 2 &= 5 \newline x \div 2 \times 2 &= 5 \times 2 \newline x &= 10 \end{align}

Exercise 9.5: Solve open sentences with two operations