2. Expressions & 3D Geometry

Lesson

We have already learned about the concept of terms and the different components that make up a mathematical expression. Now we are going to learn how to manipulate expressions involving algebraic terms so that we can generate new expressions that are equivalent, or equal in value, to the original.

Before we begin

Remember to look at the operator directly to the left of a term to see whether it is positive or negative. If there is no operator at the beginning of an expression, it means that the first term is positive.

For example, in the expression $4x+2-x$4`x`+2−`x`, there are two positive terms ($4x$4`x` and $2$2) and one negative term ($-x$−`x`).

In algebra, variables are used to represent unknown numbers. This means that all of the rules that we have already learned about adding, subtracting, multiplying and dividing numbers apply to algebraic terms as well! Let's take a look at what this means for each operation.

The order in which we add numbers doesn't matter. For example, $5+6$5+6 is the same as $6+5$6+5, both of which add up to $11$11. This is called the Commutative Property of Addition. This same property applies to algebraic terms. So $a+7$`a`+7 is the same as $7+a$7+`a`, for example, and $x+3y$`x`+3`y` is the same as $3y+x$3`y`+`x`.

This also means that when adding multiple terms we can group them in any order that we like. This is called the Associative Property of Addition. For example, $2m+1+8$2`m`+1+8 is the same as $\left(2m+1\right)+8$(2`m`+1)+8 but it would be better to group the terms as $2m+\left(1+8\right)$2`m`+(1+8), because we can add the $1$1 and the $8$8 to get a simpler equivalent expression of $2m+9$2`m`+9.

So we can see how using these properties to our advantage can help us rewrite expressions in a much simpler way.

The order in which we subtract numbers **does** matter. For example, $10-2$10−2 is not the same as $2-10$2−10, since $10-2=8$10−2=8 and $2-10=-8$2−10=−8. Similarly, $x-y$`x`−`y` is, in most cases, not the same as $y-x$`y`−`x`.

Remember that the operator directly to the left of a term is the sign of that term. If we think about subtraction as adding a negative term, we can see what is happening more clearly. $x-y$`x`−`y` can be rewritten as $x+\left(-y\right)$`x`+(−`y`), which is the same as $-y+x$−`y`+`x`, but **not** the same as $y-x$`y`−`x`. So if we are careful we can use the properties of addition to rewrite expressions that involve subtraction or negatives.

Careful!

Remember we can only add and subtract like terms. Like terms are terms that have the same variables with the same exponents, however the coefficients do not have to be the same.

For example, $2x$2`x` and $-4x$−4`x` are like terms but $2x$2`x` and $2x^2$2`x`2 are not like terms because the variables do not have the same exponents. Also, $2x$2`x` and $2y$2`y` are not like terms because the terms have different variables.

Just like when we add numbers, the order in which we multiply numbers doesn't matter. For example, $6\times4$6×4 is the same as $4\times6$4×6, and they are both equal to $24$24. This is called the Commutative Property of Multiplication. This property is true when variables are included as well. For example, $r\times5$`r`×5 is the same as $5\times r$5×`r`, and $ab$`a``b` is the same as $ba$`b``a`.

Just like in addition we can also use the Associative Property of Multiplication to group numbers or variables that are being multiplied in different ways. For example, $h\times4\times9$`h`×4×9 can be grouped as $\left(h\times4\right)\times9$(`h`×4)×9 but it is better to group it as $h\times\left(4\times9\right)$`h`×(4×9), which is equal to $h\times36$`h`×36 or $36h$36`h`.

Remember!

- Anything multiplied by $1$1 is equal to itself e.g., $1\times5=5$1×5=5, $r\times1=r$
`r`×1=`r`, $1\times5t=5t$1×5`t`=5`t`and so on. - Anything multiplied by $0$0 is equal to $0$0 e.g. $12\times0=0$12×0=0, $b\times0=0$
`b`×0=0, $9x\times0=0$9`x`×0=0 and so on.

The order in which we divide numbers **does** matter. Dividing $2$2 pizzas into $8$8 slices is not the same as dividing $8$8 pizzas into $2$2 gigantic slices! If we write this mathematically, $2\div8=\frac{1}{4}$2÷8=14 which is not the same as $8\div2=4$8÷2=4. This is also the case when there are variables involved. In general, $m\div n$`m`÷`n` is **not** equal to $n\div m$`n`÷`m`.

Summary

- We can reverse the order of addition and multiplication, even involving algebraic terms.
- We cannot reverse the order of subtraction and division.

Is the expression on the left side of the number sentence equivalent to the expression on the right?

$v+8=8+v$`v`+8=8+`v`

**Think:** Are the terms on the left-hand side of the number sentence the same as those on the right?

**Do:** The Commutative Property of Addition states that we can reverse the order of terms that are being added. Since the same terms $v$`v` and $8$8 are being added on both sides of this number sentence, the expressions are equivalent.

Is this equation balanced?

$5-n=n-5$5−`n`=`n`−5

**Think:** Are the terms on the left-hand side of the equation the same as those on the right?

**Do:** Remember we cannot reverse the order of subtraction. On the left-hand side of the equation, the $5$5 is positive, and the $n$`n` is negative. On the right-hand side of the equation, the $n$`n` is positive, and the $5$5 is negative. So this equation is not balanced.

Is the left side of this number sentence equivalent to the right side?

$v\div9=9\div v$`v`÷9=9÷`v`

**Think:** Could we write these terms as equivalent fractions?

**Do**: $\frac{v}{9}$`v`9 is not the same as $\frac{9}{v}$9`v`, so the left and right sides of the number sentence are not equivalent.

Is $\left(m\times7\right)\times3$(`m`×7)×3 equivalent to $m\times\left(7\times3\right)$`m`×(7×3)?

**Think:** Are we multiplying the same numbers and variables in both expressions?

**Do**: We have the variable $m$`m` and the numbers $7$7 and $3$3 in both expressions. Since the order of multiplication doesn't matter, expressions are equivalent. (Both are equal to $21m$21`m`.)

Find the missing term that would make the equation true.

$dn+\frac{1}{7}=\editable{}+dn$`d``n`+17=+`d``n`

**Think: **What term is on the left-hand side of the equation that is missing from the right?

**Do:** Considering that the Commutative Property of Addition is being used here missing term is $\frac{1}{7}$17.

Answer each of the following questions.

Is $\frac{1}{6}b\times6$16

`b`×6 equivalent to $b$`b`for every value of $b$`b`?Yes

ANo

BIs $bg$

`b``g`the same as $gb$`g``b`for every value of $b$`b`and $g$`g`?Yes

ANo

BIs $2b$2

`b`equivalent to $b-b+b$`b`−`b`+`b`for every value of $b$`b`?Yes

ANo

B

Using the commutative property of addition, write an expression that is different but equivalent to $8\left(4+m\right)$8(4+`m`).

$8\left(4+m\right)$8(4+

`m`)$=$=$\editable{}$($\editable{}$+$\editable{}$)

Apply the properties of operations to generate equivalent expressions.