Algebra

NZ Level 5

Simplify algebraic expressions involving distributive law

Lesson

We have learnt how to:

- expand brackets using the distributive law,
- collect like terms, and
- perform operations with positive and negative algebraic terms.

Now we can use all these skills to simplify expressions which involve the four operations and the distributive law.

When there is an expression involving a mixture of $+$+, $-$−, $\times$×, $\div$÷ and expanding brackets, we need to follow the correct order of operations.

1) If we see any brackets, we need to expand them first.

For example, in the expression $5\left(x+3\right)-2x$5(`x`+3)−2`x`, we need to expand $5\left(x+3\right)$5(`x`+3) before we consider the other term. We get $5x+15-2x$5`x`+15−2`x`.

2) Collect the like terms to simplify.

After expanding the brackets of $5\left(x+3\right)-2x$5(`x`+3)−2`x`, we got $5x+15-2x$5`x`+15−2`x`. Since there are now some like terms, we can combine these and simplify the expression even more. Doing this we get $3x+15$3`x`+15.

If there is more than one set of brackets, we can expand them all in the one step.

Here is an expression which involves expanding more than one set of brackets:

$4\left(2a-3b\right)+a\left(2+b\right)$4(2`a`−3`b`)+`a`(2+`b`)

We can expand both brackets at the same time. We get:

$8a-12b+2a+ab$8`a`−12`b`+2`a`+`a``b`

From here, we can collect the like terms $8a$8`a` and $2a$2`a`. Doing this, we get:

$10a-12b+ab$10`a`−12`b`+`a``b`

When expanding the brackets in an expression, we need to be careful to multiply negative terms correctly.

From our work with positive and negative numbers, we know that:

- multiplying two negatives gives us a positive result
- multiplying a positive and a negative gives us a negative result

In the expression $8y-5\left(y-3\right)-2$8`y`−5(`y`−3)−2, we need to expand the brackets first.

We get $8y-5y+15-2$8`y`−5`y`+15−2.

Notice the positive $15$15.

We can then combine like terms to get $3y+13$3`y`+13.

Remember:

Check the sign immediately before a number to determine whether it is positive or negative.

Expand and simplify:

$2y+5+3\left(y+9\right)$2`y`+5+3(`y`+9)

Expand and simplify the following:

$-10\left(x-8\right)+1$−10(`x`−8)+1

Expand and simplify:

$9\left(y-6\right)+6\left(y+2\right)$9(`y`−6)+6(`y`+2)

Generalise the properties of operations with fractional numbers and integers.