Laws of arithmetic with algebraic terms
We have already learnt about the concept of terms and the different components that make up a mathematical expression. Now we are going to learn how to read number sentences involving algebraic terms so that we can make sure they are balanced.
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 a number sentence, it means that the first term is positive.
For example, in the number sentence $4x+2-x$4x+2−x, there are two positive terms ($4x$4x and $2$2) and one negative term ($-x$−x).
Note that the value of an algebraic term could change if we know what numbers the variables represent. For example, the term $-3y$−3y is a negative term, since the operator in front is a minus sign. If we know that the variable $y$y represents the number $-4$−4, however, then the value of the term will be positive, since $-3\times\left(-4\right)=+12$−3×(−4)=+12.
The General Idea
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.
Adding terms
The order in which we add number 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. The 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+3y is the same as $3y+x$3y+x.
This also means that we can add multiple terms in any order that we like. For example, $\left(2m+1\right)+8$(2m+1)+8 is the same as $2m+\left(1+8\right)$2m+(1+8), which is equal to $2m+9$2m+9.
Subtracting terms
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 not the same as $y-x$y−x in general.
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, which is not the same as $y-x$y−x.
Multiplying terms
Just like when we add terms, the order in which we multiply two 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 true for algebraic terms as well. For example, $r\times5$r×5 is the same as $5\times r$5×r, and $ab$ab is the same as $ba$ba.
This is true for more than one multiplication at a time as well. For example, $\left(h\times4\right)\times9$(h×4)×9 is the same as $h\times\left(4\times9\right)$h×(4×9), which is equal to $36h$36h.
- 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$1×5t=5t 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$9x×0=0 and so on.
Dividing terms
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$2÷8 is not the same as $8\div2$8÷2. This is also the case when there are algebraic terms involved. In general, $m\div n$m÷n is not equal to $m\div n$m÷n.
- We can reverse the order of addition and multiplication, even involving algebraic terms.
- We cannot reverse the order of subtraction and division.
Examples
Question 1
Is this number sentence true or false?
$v+8=8+v$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: 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 equation, this number sentence is TRUE.
Question 2
Is this number sentence true or false?
$5-n=n-5$5−n=n−5
Think: Are the terms on the left-hand side of the number sentence the same as those on the right?
Do: 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 number sentence is FALSE.
Question 3
Is this number sentence true or false?
$v\div9=9\div v$v÷9=9÷v
Think: Could we write these terms as equivalent fractions?
Do: $\frac{v}{9}$v9 is not the same as $\frac{9}{v}$9v, so this number sentence is FALSE.
Question 4
Is this number sentence true or false?
$\left(m\times7\right)\times3=m\times\left(7\times3\right)$(m×7)×3=m×(7×3)
Think: Are we multiplying the same numbers and variables on both sides of the equation?
Do: We have the variable $m$m and the numbers $7$7 and $3$3 on both sides of the equation. Since the order of multiplication doesn't matter, this number sentence is TRUE. (Both sides are equal to $21m$21m.)
Question 5
Find the missing term that would make the number sentence true.
$dn+\frac{1}{7}=\editable{}+dn$dn+17=+dn
Think: What term is on the left-hand side of the number sentence that is missing from the right?
Do: This missing term is $\frac{1}{7}$17.
Let's have a look at these worked example videos.
Question 6
Is this number sentence true or false: $1\times3m=3m$1×3m=3m
Question 7
Is this number sentence true or false: $\left(3+7\right)+a=3+\left(7+a\right)$(3+7)+a=3+(7+a)
Question 8
Find the missing term that would make the number sentence true:
$0+\editable{}=9g$0+=9g