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Australia
Year 7

6.02 Building algebraic expressions

Lesson

Introduction

When writing a numeric expression we use numbers and basic operations to build up a number sentence that can be later calculated. Algebraic expressions are the same as numeric expressions except that they also use some new algebraic tools. These new algebraic tools are pronumerals and coefficients.

Pronumerals and coefficients

We use algebraic expressions when we want to write a number sentence but don't know all the numbers involved.

For example: What is the total weight of a cat and a dog?

A cat and dog are on a set of scales. At the bottom it says total weight equals cat plus dog which equals c plus d.

In this example we know that the total weight will be the weight of the cat added to the weight of the dog, but we don't know the number for either of these.

What we do instead is pretend that we know what these numbers are and replace them with pronumerals. In this case, let's use c for the weight of the cat and d for the weight of the dog.

Now we can write the number sentence as:\text{ Total weight}=c+d

This is an algebraic expression as it is a number sentence that uses pronumerals in the place of some numbers.

A pronumeral is a symbol, commonly a letter, that is used in the place of a numeric value.

Don't use mathematical operation symbols as pronumerals otherwise things won't make sense.

For example: '+-\lt = \%' could be an algebraic expression, but it is also very confusing.

Coefficients are used in algebraic expressions to represent how many sets of a pronumeral we have. They are written in front of a pronumeral without a multiplication symbol like so: 3u=3 \times u The pronumeral is u with a coefficient of 3.

Notice how we don't need the multiplication symbol to represent multiple sets of a pronumeral. This is because there is no danger of mixing up a coefficient next to a pronumeral with any other term, whereas if we did this for numbers they would get mixed up with two digit numbers (for example: 3\times 4=12 \neq 34)

A coefficient is a numeral that is placed before and multiplies a pronumeral in an algebraic term.

Coefficients are a bit different from multiplication though, since they also include the sign of the term. This means that a negative term, -6q for example, has a coefficient of -6.

We can see this more clearly in a longer expression.

Consider the expression: 4x-3y+7z

By breaking up the expression into its individual terms we can determine the coefficients of each pronumeral.

TermCoefficientPronumeral
4x4x
-3y-3y
+7z7z

From this we can see that the coefficient of y is -3, since it is a negative term, and the coefficient of z is 7, since it is a positive term. If there is no sign in front of a term we assume that the term is positive, so we know that the coefficient of x is 4.

We can also have algebraic terms where the coefficient is a fraction.

Consider: v\div 4 = \dfrac{v}{4}=\dfrac{1}{4}\times v

Since  dividing by a number is the same as multiplying by its reciprocal  , dividing by 4 gives us a coefficient of \dfrac{1}{4}.

What about variables that don't appear to have coefficients?

Consider the term x.

Since x is equal to 1 \times x which is also equal to 1x, it actually has a coefficient of 1. Whenever a pronumeral has no written coefficient, its coefficient can be assumed to be 1.

Similarly, the coefficient of -x is -1.

Examples

Example 1

Find the coefficient of y in:

a

4y

Worked Solution
Create a strategy

Recall that the definition of a coefficient is "a numeral or pronumeral that is placed before and multiplies a pronumeral in an algebraic expression".

Apply the idea

4y is the simplified result of 4\times y.

The coefficient is 4.

b

-4y

Worked Solution
Create a strategy

Recall that the definition of a coefficient is "a numeral or pronumeral that is placed before and multiplies a pronumeral in an algebraic expression".

Apply the idea

-4y is the simplified result of -4\times y.

The coefficient is -4.

Idea summary

A coefficient is a number that is placed before and multiplies a pronumeral in an algebraic term.

Basic operations in algebra

Aside from the use of coefficients in multiplication, the basic operations work almost the same for algebraic terms as they do for numbers.

Between pronumerals and numbers we have:

Word expressionAlgebraic expressionSimplified algebraic expression
\text{Three more than}\,xx+3x+3
\text{Three less than}\,xx-3x-3
\text{The quotient of }\ x\, \text{and three}x\div 3\dfrac{x}{3}
\text{The product of }\ x\, \text{and three}x\times 33x

Between pronumerals and other pronumerals we have:

Word expressionAlgebraic expressionSimplified algebraic expression
y\,\text{more than}\,xx+yx+y
y\,\text{less than}\,xx-yx-y
\text{The quotient of }\ x\,\text{and}\, yx\div y\dfrac{x}{y}
\text{The product of }\ x\,\text{and}\, yx\times yxy

As we can see from the tables, addition and subtraction in algebraic expressions does not usually simplify.

The only time they will simplify is when they are  like terms  .

It should also be noted that the division doesn't actually simplify but is instead written as a fraction, which is slightly more compact and removes the need to use brackets in more complicated expressions, for example: 4\div \left(x+3\right)=\frac{4}{x+3}

Technically, the multiplication between a pronumeral and a number also uses a more compact form by removing the multiplication symbol, but in this case using a coefficient is considered simplifying.

The same can be said for the multiplication between different pronumerals except, in this case, there is no coefficient and instead we have two pronumerals.

These operations will work the same way when applying more than one of them.

Examples

Example 2

What does the expression 8x mean?

Worked Solution
Create a strategy

A coefficient is multiplied by the variable next to it.

Apply the idea

The expression 8x means 8 is multiplied by x.

Example 3

Write an algebraic expression for the following phrase "eight more than the quotient of 9 and x".

Worked Solution
Create a strategy

Translate the terms into mathematical symbols and operations.

Apply the idea

The phrase "eight more than" indicates that we are to add 8.

The "quotient of 9 and x" indicates division, which we can write as a fraction with 9 as the numerator and x as the denominator.

We can combine the whole description into a single expression:\frac{9}{x}+8

Idea summary

Operations with pronumerals:

Word expressionAlgebraic expressionSimplified algebraic expression
\text{Three more than}\,xx+3x+3
\text{Three less than}\,xx-3x-3
\text{The quotient of }\ x\, \text{and three}x\div 3\dfrac{x}{3}
\text{The product of }\ x\, \text{and three}x\times 33x
y\,\text{more than}\,xx+yx+y
y\,\text{less than}\,xx-yx-y
\text{The quotient of }\ x\,\text{and}\, yx\div y\dfrac{x}{y}
\text{The product of }\ x\,\text{and}\, yx\times yxy

Outcomes

ACMNA175

Introduce the concept of variables as a way of representing numbers using letters

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