Question

aim: you will be able to write and simplify expressions
you will be able to determine if expressions are equivalent
do now:

1. write an expression to represent the verbal phrase:

3 multiplied by the sum of a number and 8

1. write an expression to determine the perimeter of the rectangle

$x+12$
$x$

1. write an expression to represent the area of the rectangle

$x +24$

1. simplify: $1 - x + 5(5 + x)$

which of these expressions are equivalent to $4x+24$

what are equivalent expressions? how do you know they are equivalent?

Explanation:

Step1: Define variable, translate phrase

Let the number be $n$. The sum of the number and 8 is $n+8$, multiply by 3:
$3(n+8)$

Step2: Use rectangle perimeter formula

Perimeter $P=2(\text{length}+\text{width})$, substitute $l=x+12, w=x$:
$P=2[(x+12)+x] = 2(2x+12) = 4x+24$

Step3: Use rectangle area formula

Area $A=\text{length}\times\text{width}$, substitute $l=x+24, w=x$:
$A=x(x+24)=x^2+24x$

Step4: Distribute and combine like terms

First distribute 5: $1 - x + 25 + 5x$
Combine like terms: $(1+25)+(-x+5x)=26+4x$

Step5: Identify equivalent expression

Compare to $4x+24$; the perimeter expression from Step2 is $4x+24$, so it is equivalent.

Step6: Define equivalent expressions

Equivalent expressions have identical values for all valid input values of the variable(s). You can verify by simplifying each expression to the same form, substituting the same variable value into each to get matching results, or using properties (distributive, commutative, associative) to rewrite one into the other.

Answer:

1. $3(n+8)$ (where $n$ is the number)
2. $4x+24$
3. $x^2+24x$
4. $4x+26$

• The expression equivalent to $4x+24$ is the perimeter expression from question 2: $4x+24$
• Equivalent expressions are expressions that have the same value for all possible values of the variable(s). You can confirm they are equivalent by simplifying each expression to the same simplified form, substituting the same value for the variable into each expression to get equal results, or using algebraic properties to rewrite one expression exactly into the other.