# Manipulating Linear Expressions

## The commutative law of addition

In each row of this table, do both arithmetic problems. The problems will be illustrated on the grid to the left.

As you can see, both answers are the same in each row of the table, which illustrates the following rule:

The commutative law of addition: for two numbers \$a\$ and \$b\$,
\$\$ a + b = b + a \$\$

You can use the commutative law of addition to simplify expressions. For example, in the expression

\$\$ 2x+3+4x \$\$

you can use it to switch the \$3\$ and the \$4x\$, to get

\$\$ 2x+4x+3 \$\$

and then combine like terms, to get

\$\$ 6x+3 \$\$

Simplify each of these expressions by using the commutative law of addition and then combining like terms.

## Simplifying expressions that use subtraction

If you’re given an expression that uses subtraction, like \$4x+5-2x\$, you can simplify it in the same way as in commSimp, after turning the subtraction into addition. In this case, you would start by turning the expression into \$4x+5+(-2)x\$.

Simplify each of these expressions by converting subtraction into addition, using the commutative law, and combining like terms.

If you’re given an expression like \$2-(3x+5)\$, where you’re subtracting a combination of terms, you can simplify it by writing it as \$2+[-(3x+5)]\$ and then using the distributive law on the negated terms. (The square brackets ‘\$[]\$’ around \$-(3x+5)\$ mean the same thing as parentheses, and are used whenever they look clearer.)

Simplify these expressions by converting subtraction into addition and using the distributive law.

## The commutative law of multiplication

In each row of this table, do both arithmetic problems. The problems will be illustrated on the grids to the left.

As you can see, both answers are the same in each row of the table, which illustrates the following rule:

The commutative law of multiplication: for two numbers \$a\$ and \$b\$,
\$\$ a(b)=b(a) \$\$

Because of the commutative law of multiplication, you can rewrite any product of a variable and a number with the number first. For example, \$x(3)\$ is the same as \$3x\$. This is illustrated to the left, as you can see by sliding the slider for \$x\$. Notice that when you slide \$x\$ to 0, each grid’s rectangle collapses to a heavy black line of length 3, which we think of as a rectangle whose sides have lengths 0 and 3.

Rewrite each of the expressions in the table below in this way.

You can use this fact to simplify certain expressions. For example, \$3(x(2))\$ is the same as \$3(2x)\$, which you’ve learned can be simplified to \$6x\$.

Simplify these expressions.

## Zero and one laws

Notice that when you add \$0\$ to any number, you get the same number back. For example, \$3+0=0+3=3\$. You can also use this to simplify expressions; for example, \$3(x+0)\$ is the same as \$3x\$.

Simplify these expressions involving the addition of \$0\$.

Similarly, multiplying any number by \$1\$ gives you the same number back. For example, \$5(1)=1(5)=5\$. You can also use this to simplify expressions; for example, \$1(3x+5)\$ can be simplified to \$3x+5\$.

Simplify these expressions involving multiplication by \$1\$.

It’s also true that multiplying any number by \$0\$ gives you \$0\$. For example, \$4(0)=0\$. You can again use this to simplify expressions. For example, \$0(2x+5)=0\$, and \$0x+7=7\$.

Simplify these expressions involving multiplication by \$0\$.

## More than one simplification

If you put together everything you’ve learned, there are a lot of expressions you can simplify!

Simplify each of these expressions. Your final answer should look like (some number)\$x\$ + (some other number).