Algebra

Lesson

We've previously learnt about the concept of negative numbers and how to simplify and evaluate equations with directed numbers. Now we are going to combine this with our knowledge of adding and subtracting algebraic terms.

There are a few important things to remember when working with directed numbers:

- Subtracting a negative term is the same as adding a positive term. For instance, the expression $3-\left(-2a\right)$3−(−2
`a`) can be written more simply as $3+2a$3+2`a`. - Adding a negative term is the same as subtracting a positive term. So $h^2+\left(-3\right)$
`h`2+(−3) is the same as just $h^2-3$`h`2−3.

Handy Hint

It may help to visualise a number line when you are working with directed numbers. For example, the number line below shows $-3-5$−3−5 (subtracting a number from a negative number):

We can see that both jumps are moving in the same direction. So the result of $-8$−8 is even further from zero, in the negative direction.

As we have seen before, only like terms can be added or subtracted. This is still true even when directed numbers are involved.

- For example, the expression $-2t-3t$−2
`t`−3`t`could be simplified, since the two terms are like terms (they have the same variable part). - On the other hand, $-x-7y-4xy$−
`x`−7`y`−4`x``y`cannot be simplified, since each term has a different variable part.

Combining like terms with directed numbers works exactly like we have already been doing, but with the rules of directed numbers included. Consider the earlier expression $-2t-3t$−2`t`−3`t`:

Combining algebraic terms |
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$2t$2t |
$+$+ | $3t$3t |
$=$= | $5t$5t |

Combining directed numbers |
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$-2$−2 | $-$− | $3$3 | $=$= | $-5$−5 |

Both at the same time |
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$-2t$−2t |
$-$− | $3t$3t |
$=$= | $-5t$−5t |

So we can combine like terms by thinking about what happens to the coefficients.

Simplify the expression $5y-7y$5`y`−7`y`.

Think: There are only two terms in this expression and they both have the same variable, $y$`y`, so they are like terms.

Do: We can combine the like terms by looking at their coefficients. Since $5-7=-2$5−7=−2, we have that $5y-7y=-2y$5`y`−7`y`=−2`y`.

Simplify the expression $-3ab+4a-b-6ab$−3`a``b`+4`a`−`b`−6`a``b`.

Think: The variables used in this expression are a and b. Which terms have the same combination of these variables?

Do: The two terms $-3ab$−3`a``b` and $-6ab$−6`a``b` are like terms, since they have the same variable part. They can be combined to give $-3ab-6ab=-9ab$−3`a``b`−6`a``b`=−9`a``b`.

For the other two terms, one has only $a$`a` and the other has only $b$`b`, so they are not like terms. Therefore we have $-3ab+4a-b-6ab=-9ab+4a-b$−3`a``b`+4`a`−`b`−6`a``b`=−9`a``b`+4`a`−`b`.

Simplify: $-2m-9m$−2`m`−9`m`

Simplify the expression: $3x-\left(-4x\right)-2x$3`x`−(−4`x`)−2`x`

Generalise the properties of operations with fractional numbers and integers.