Use Properties to Rewrite the Given Equation

Use Properties to Rewrite the Given Equation.

Learning Outcomes

  • Place the associative and commutative properties of addition and multiplication
  • Use the associative and commutative properties of add-on and multiplication to rewrite algebraic expressions

Utilize the Commutative and Associative Properties

Recollect near adding two numbers, such every bit [latex]five[/latex] and [latex]3[/latex].

[latex]\begin{array}{cccc}\hfill 5+3\hfill & & & \hfill 3+five\hfill \\ \hfill viii\hfill & & & \hfill viii\hfill \stop{array}[/latex]

The results are the same. [latex]5+3=iii+5[/latex]

Observe, the order in which we add does not matter. The same is true when multiplying [latex]5[/latex] and [latex]three[/latex].

[latex]\brainstorm{array}{cccc}\hfill 5\cdot 3\hfill & & & \hfill 3\cdot 5\hfill \\ \hfill fifteen\hfill & & & \hfill 15\hfill \cease{array}[/latex]

Again, the results are the same! [latex]v\cdot 3=3\cdot v[/latex]. The order in which nosotros multiply does not matter.

These examples illustrate the commutative backdrop of addition and multiplication.

Commutative Properties

Commutative Holding of Addition: if [latex]a[/latex] and [latex]b[/latex] are existent numbers, so

[latex]a+b=b+a[/latex]

Commutative Property of Multiplication: if [latex]a[/latex] and [latex]b[/latex] are real numbers, and so

[latex]a\cdot b=b\cdot a[/latex]

The commutative properties have to do with order. If you lot change the guild of the numbers when calculation or multiplying, the result is the aforementioned.

example

Use the commutative backdrop to rewrite the following expressions:
1. [latex]-1+three=[/latex]
2. [latex]iv\cdot ix=[/latex]

Solution:

ane.
[latex]-i+3=[/latex]
Use the commutative holding of add-on to change the guild. [latex]-1+iii=three+\left(-1\correct)[/latex]
2.
[latex]four\cdot 9=[/latex]
Use the commutative property of multiplication to change the order. [latex]4\cdot 9=ix\cdot 4[/latex]

try it

What almost subtraction? Does order matter when we subtract numbers? Does [latex]7 – 3[/latex] give the same result as [latex]3 – vii?[/latex]

[latex]\brainstorm{assortment}{ccc}\hfill 7 – three\hfill & & \hfill three – vii\hfill \\ \hfill 4\hfill & & \hfill -4\hfill \\ & \hfill four\ne -4\hfill & \end{assortment}[/latex]
The results are non the same. [latex]vii – 3\ne 3 – 7[/latex]

Since changing the gild of the subtraction did not give the same result, nosotros can say that subtraction is not commutative.

Let’s come across what happens when we divide two numbers. Is division commutative?

[latex]\brainstorm{array}{ccc}\hfill 12\div 4\hfill & & \hfill four\div 12\hfill \\ \hfill \frac{12}{iv}\hfill & & \hfill \frac{4}{12}\hfill \\ \hfill 3\hfill & & \hfill \frac{1}{3}\hfill \\ & \hfill three\ne \frac{i}{3}\hfill & \stop{array}[/latex]

The results are not the same. So [latex]12\div 4\ne 4\div 12[/latex]

Since changing the order of the division did not requite the same upshot, partition is not commutative.

Addition and multiplication are commutative. Subtraction and sectionalization are not commutative.

Suppose you were asked to simplify this expression.

[latex]vii+8+2[/latex]

How would you do it and what would your answer be?

Some people would think [latex]seven+8\text{ is }15[/latex] and so [latex]15+2\text{ is }17[/latex]. Others might start with [latex]8+2\text{ makes }10[/latex] and then [latex]7+10\text{ makes }17[/latex].

Both means requite the same result, every bit shown beneath. (Call up that parentheses are grouping symbols that indicate which operations should be done beginning.)

The image shows an equation. The left side of the equation shows the quantity 7 plus 8 in parentheses plus 2. The right side of the equation show 7 plus the quantity 8 plus 2. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 17 below.

When adding three numbers, irresolute the grouping of the numbers does not change the upshot. This is known as the Associative Property of Addition.

The same principle holds true for multiplication besides. Suppose we want to find the value of the following expression:

[latex]5\cdot \frac{1}{3}\cdot 3[/latex]

Changing the group of the numbers gives the aforementioned result.

The image shows an equation. The left side of the equation shows the quantity 5 times 1 third in parentheses times 3. The right side of the equation show 5 times the quantity 1 third times 3. Each side of the equation is boxed separately in red. Each box has an arrow pointing from the box to the number 5 below.

When multiplying three numbers, irresolute the group of the numbers does non alter the result. This is known as the Associative Property of Multiplication.

If we multiply iii numbers, irresolute the grouping does not affect the product.

You lot probably know this, just the terminology may exist new to you. These examples illustrate the
Associative Properties.

Associative Properties

Associative Holding of Addition: if [latex]a,b[/latex], and [latex]c[/latex] are real numbers, then

[latex]\left(a+b\correct)+c=a+\left(b+c\right)[/latex]

Associative Belongings of Multiplication: if [latex]a,b[/latex], and [latex]c[/latex] are existent numbers, then

[latex]\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)[/latex]

case

Utilise the associative properties to rewrite the following:

i. [latex]\left(3+0.six\right)+0.four=[/latex]
2. [latex]\left(-four\cdot \frac{two}{5}\right)\cdot 15=[/latex]

try it

Also using the associative backdrop to brand calculations easier, we volition ofttimes use it to simplify expressions with variables.

example

Use the Associative Holding of Multiplication to simplify: [latex]6\left(3x\right)[/latex].

attempt it

The post-obit video provides more examples of how to simplify expressions using the commutative and associative backdrop of multiplication and addition.

Use Properties to Rewrite the Given Equation

Source: https://courses.lumenlearning.com/mathforliberalartscorequisite/chapter/rewriting-expressions-using-the-commutative-and-associative-properties/