Answer: After applying the distributive property, Ezra's expression was .
After he used the commutative property, his expression was.
Step-by-step explanation:
An expression is in simplest form when it has no grouping symbols and no like terms. Therefore, when an expression contains grouping symbols, such a parentheses, apply the distributive property to eliminate the parentheses, and then combine any like terms.
To simplify 4(x+2y)+x
4 open parentheses x plus 2 y close parentheses plus x, first apply the distributive property: 4x + 8y + x.
Then combine the like terms of 4x and x: 5x + 8y.
The expressions 4(x+2y)+x
4 open parentheses x plus 2 y close parentheses plus x, 4x + 8y + x, and 5x + 8y are all equivalent expressions. 5x+4
4x+15+x+5
2(x+4)+3(x+4)
Simplify 5(x+4)
.
5(x+4)=5x+20
How can you use the distributive property to eliminate the parentheses?
Use the distributive property to eliminate the parentheses. Multiply 5 by each term inside the parentheses. Therefore, multiply 5 by x to get 5x
, and multiply 5 by 4 to get 20.
Simplify and compare all of the remaining expressions.
5x+4
is not equivalent to 5x+20
, so it is not equivalent to 5(x+4)
.
Is 5x+4
in simplest form? Is 5x+4
equivalent to 5x+20
?
5x+4
is already in simplest form because it has no parentheses and no like terms.
Since each term of 5x+4
is not identical to each term of 5x+20
, 5x+4
is not equivalent to 5(x+4)
.
4x+15+x+5=4x+x+15+5
=5x+20
4x+15+x+5
is equivalent to 5(x+4)
.
Is 4x+15+x+5
in simplest form? Is 4x+15+x+5
equivalent to 5x+20
?
4x+15+x+5
is not in simplest form because it has like terms 4x
and x, and like terms 15 and 5.
Use the commutative property of addition to let 15 and x swap positions. Then, add the coefficients of the like variable terms, and add the constants. Since the coefficient of x is understood to be 1, 4x+x
is 5x
and 15+5
is 20.
Since 4x+15+x+5
simplifies to 5x+20
, 4x+15+x+5
is equivalent to 5(x+4)
.
2(x+4)+3(x+4)=2x+8+3x+12
=2x+3x+8+12
=5x+20
2(x+4)+3(x+4)
is equivalent to 5(x+4)
.
Is 2(x+4)+3(x+4)
in simplest form? Is 2(x+4)+3(x+4)
equivalent to 5x+20
?
2(x+4)+3(x+4)
is not in simplest form because it has grouping symbols. Use the distributive property to eliminate the parentheses. 2 times the sum x+4
equals 2x+8
, and 3 times the sum x+4
equals 3x+12
.
Use the commutative property of addition to let 8 and 3x
swap positions.
Add the coefficients of the like variable terms, and add the constants. 2x+3x
is 5x
, and 8+12
is 20.
Since 2(x+4)+3(x+4)
simplifies to 5x+20
, 2(x+4)+3(x+4)
is equivalent to 5(x+4)
.
Determine whether the two expressions are equivalent.
6(5a+8)+2(a+15)
and 2(16a+20)+58
Think About It:The plan is to simplify each expression and then compare the simplified expressions.
Simplify the first expression.
6(5a+8)+2(a+15)=30a+48+2a+30
The distributive property is used twice.
Multiply 5a and 8 by 6. Then multiply a and 15 by 2.
=32a+78
Which terms are like terms?
30a and 2a are like terms.
48 and 30 are like terms.
Do you have to write out the commutative property that is being used?
No, you do not need to rearrange the terms on paper so that like terms are next to each other. You can perform that step mentally.
Simplify the second expression.
2(16a+20)+58=32a+40+58
Apply the distributive property.
Don’t forget to multiply 2 by 20! It’s a very common mistake to forget to multiply the second term inside the parentheses by the term outside the parentheses.
=32a+98
Which terms are like terms?
40 and 58 are like terms.
Compare the simplified expressions.
32a+78
is not equivalent to 32a+98
.
The expressions are not equivalent. Their constant terms are different.
Think About It:If you evaluate each simplified expression when a=1
, you would get a value of 110 for the first expression and a value of 130 for the second expression. If the expressions were equivalent, you would get the same values when evaluated for the same value of a. I HOPE THIS A GREAT ANSWER