Lesson

We have previously looked at properties of operations with algebraic terms, and used substitution to evaluate algebraic expressions. Now we are going to use this knowledge to help us identify and create algebraic expressions that are equivalent.

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

Expression 1 | Value | Expression 2 | 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$10 plus $\editable{5}$5 groups of $10$10 has the same value as $\editable{6}$6 groups of $10$10 plus $\editable{2}$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.

Remember!

Equivalent expressions always have the same value.

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!

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* anywhere it is used. 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.

To verify this, we can substitute any value for $x$`x`. Let's try replacing $x$`x` with $7$7 in both expressions. For the first expression $3x+5x$3`x`+5`x` we get $3\times7+5\times7=56$3×7+5×7=56. For the second expression $6x+2x$6`x`+2`x` we get $6\times7+2\times7=56$6×7+2×7=56. When $7$7 is substituted for $x$`x` both expressions have a value of $56$56. This confirms that the two expressions are equivalent.

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

**Think: **Since $x$`x` and $y$`y` are different variables, they may represent different numbers.

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

These are not equivalent expressions.

**Reflect: **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 |

This verifies that the two expressions definitely are not equivalent!

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

**Think: **Since $ab$`a``b` and $ba$`b``a` have the same variables, both instances of $a$`a` represent the same number and both instances of $b$`b` represent the same number.

**Do:** We can use the Commutative Property of Multiplication to rewrite $3ba$3`b``a` as $3ab$3`a``b`. This makes it easier to see that $5ab$5`a``b` and $3ab$3`a``b` are like terms since they contain both of the same variables. Now we can combine like terms by subtracting the coefficients and keeping the variables the same.

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

**Reflect: **If we substitute any values for $a$`a` and $b$`b` like $a=3$`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!

Choose all expressions that are equivalent to $7k$7`k`.

$5k-2k$5

`k`−2`k`A$6+k+6k-6$6+

`k`+6`k`−6B$6k$6

`k`C$7+k$7+

`k`D$2k+5k$2

`k`+5`k`E

Consider the following:

Select the expression that is equivalent to:

$7s+2b-4s$7

`s`+2`b`−4`s`$3s-2b$3

`s`−2`b`A$11s+2b$11

`s`+2`b`B$3s+2b$3

`s`+2`b`C$3b+2w$3

`b`+2`w`DSelect all expressions that are equivalent to:

$21w+4b$21

`w`+4`b`$14w+7w-4b$14

`w`+7`w`−4`b`A$7w+7w+7w+4b$7

`w`+7`w`+7`w`+4`b`B$21w-8b$21

`w`−8`b`C$14w+7w+4b$14

`w`+7`w`+4`b`D

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of whic value is substituted into them).