Equivalent Expressions

We have previously looked at combining like terms, learnt how to build algebraic expressions from word problems, and used substitution to evaluate algebraic expressions. Now we are going to use this knowledge to help us identify when two algebraic expressions are equivalent.

 

Regrouping Numbers

Let's start by looking at what happens with numbers:

Expression	Value	Expression	Value	Equivalent Expressions
$7+7+7+7$7+7+7+7	$28$28	$4\times7$4×7	$28$28	$7+7+7+7=4\times7$7+7+7+7=4×7
$\editable{3}\times10+\editable{5}\times10$3×10+5×10	$80$80	$\editable{6}\times10+\editable{2}\times10$6×10+2×10	$80$80	$3\times10+5\times10=6\times10+2\times10$3×10+5×10=6×10+2×10

 

What we are doing here is grouping like numbers. 

$\editable{3}$3 groups of $10+\editable{5}$10+5 groups of $10$10 has the same value as $\editable{6}$6 groups of $10+\editable{2}$10+2 groups of $10$10.

So $3\times10+5\times10$3×10+5×10 and $6\times10+2\times10$6×10+2×10 are equivalent expressions.

 

 

There is nothing special about the numbers used above; this would work with groups of $4$4, or groups of $9.8$9.8, or any other number we choose!

 

Regrouping Variables

Let's use the variable $x$x to represent "an unknown number".

Expression		Expression
$\editable{3}x+\editable{5}x$3x+5x		$\editable{6}x+\editable{2}x$6x+2x

Although $x$x represents an unknown number, it represents the same number in both terms $3x$3x and $5x$5x. So we can think of this as $3$3 groups of $x$x plus another $5$5 groups of $x$x, or $8$8 groups of $x$x altogether.

Similarly, we can think of the other expression as $6$6 groups of $x$x plus another $2$2 groups of $x$x, or $8$8 groups of $x$x altogether.

If we substitute any value of $x$x, we'll see that $3x+5x$3x+5x has the same value as $6x+2x$6x+2x.  

So these two expressions are equivalent.

 

Example

Is $3x+2y$3x+2y equivalent to $5x$5x?

Think: 

Since $x$x and $y$y are different variables, they may represent different numbers.

So we can't say that $3x+2y$3x+2y always has the same value as $5x$5x. 

These are not equivalent expressions.

Check:

If we substitute the particular values $x=10$x=10 and $y=3$y=3, we get:

Expression	$3x+2y$3x+2y	$5x$5x
Value	$36$36	$50$50

So the two expressions definitely are not equivalent!

 

Example

What is $5ab-3ba$5ab−3ba equivalent to?

Think: 

Since $ab$ab and $ba$ba are the same variables, they represent the same number. So we can group $5ab$5ab and $3ba$3ba together.

$5\editable{ab}-3\editable{ba}=2\editable{ab}$5ab−3ba=2ab    OR     $2\editable{ba}$2ba

 

Check:

If we substitute the particular values $a=3$a=3 and $b=4$b=4, we get:

Expression	$5ab-3ba$5ab−3ba	$2ab$2ab
Value	$24$24	$24$24

The two expressions have the same value, as they should!

 

Examples

Question 1

Choose all expressions that are equivalent to $6h+6$6h+6.

A) $36h$36h     B) $3h+3h+2+2+2$3h+3h+2+2+2     C) $6h+6$6h+6     D) $6h+10-4$6h+10−4     E) $6h-6$6h−6

Think: Notice that the value of the expression is made up of $6$6 groups of the value of $h$h and $6$6 'units'. Which other expression(s) is made up of the same values?

Do:

A) We only have $6h$6h in our expression, so this is not equivalent.

B) When we add the like terms, this expression simplifies to $6h+6$6h+6, so this is equivalent.

C) This expression is exactly the same as the question, so this is equivalent.

D) When we simplify this expression by collecting the like terms, we also get $6h+6$6h+6.

E) $6h+6$6h+6 is not the same as $6h-6$6h−6, so these are not equivalent.

So, the equivalent expressions are B), C) and D).

 

QUESTION 2

Which of the following expressions is equivalent to $7rt^2$7rt2?

A) $7+r+t+t$7+r+t+t     B) $4tr^2+3tr^2$4tr2+3tr2     C) $7r+7t^2$7r+7t2     D) $2rt^2+5t^2r+0rt^2$2rt2+5t2r+0rt2

Think: The expression $7rt^2$7rt2 has only one term, which is equal to $7$7 groups of $r$r times $t^2$t2. Which of the options simplifies to the same thing?

Do:

A) The expression $7+r+t+t$7+r+t+t is adding each of the numbers and variables, rather than multiplying.

B) These two are like terms, which we can add to give $7tr^2$7tr2. Notice that $r$r is squared instead of $t$t, however, so this is not equivalent!

C) These two terms have different variable parts, so they are not like terms and we cannot combine them.

D) These terms all have the same variable parts (note that the order of the variables doesn't matter). So they are like terms and we can add them to give $7rt^2$7rt2, which is equivalent.

So the equivalent expression is D).

 

Question 3

Questions are coming soon!

 

QUESTION 4

Questions are coming soon!

 

QUESTION 5

Tom has to pay the price of his meal, $\$b$$b, plus a $10%$10% tip.

Which of the following show the price he will have to pay for his meal, including the tip? Select all correct options.