5.06 Equivalent expressions
Introduction
We have previously looked at properties of operations with numeric terms, and used substitution to evaluate algebraic expressions. We have also applied distributive property to express a sum of two whole numbers with a commn factor. Now we are going to use this knowledge to help us identify and create algebraic expressions that are equivalent.
Ideas
Equivalent expressions by combining like terms
Equivalent expressions are expressions that have the same value or worth but do not look the same.
In the following expressions, we have been given the variable x to represent "an unknown number".
| Expression 1 | Expression 2 |
|---|---|
| 3x+5x | 6x+2x |
Although x represents an unknown number, it represents the same number anywhere it is used in an expression. So we can think of Expression 1 as 3 groups of x plus another 5 groups of x, or 8 groups of x altogether.
We can combine the two terms because they are "like terms", which means they have the same makeup of coefficients and variables. Sometimes we have terms that are not the same makeup, and we are not able to combine them. For example 3x + 5xy cannot be combined because they are not like terms. The term 5xy has the additional variable of y, which means 5 \times x \times y.
We can think of the Expression 2 as 6 groups of x plus another 2 groups of x. Again we can combine these to have 8 groups of x because they are like terms.
Now let's check if the two expressions are equivalent. To do this, we can substitute any value for x. Let's try replacing x with 7 in both expressions.
Expression 1:
| \displaystyle 3x + 5x | \displaystyle = | \displaystyle 3\times 7+5\times 7 | Substitute the given value of 7 for x |
| \displaystyle = | \displaystyle 21 + 35 | ||
| \displaystyle = | \displaystyle 56 |
Expression 2:
| \displaystyle 6x + 2x | \displaystyle = | \displaystyle 6\times 7+2\times 7 | Substitute the given value of 7 for x |
| \displaystyle = | \displaystyle 42 + 14 | ||
| \displaystyle = | \displaystyle 56 |
When 7 is substituted for x, both expressions have a value of 56. This confirms that the two expressions are equivalent.
Examples
Example 1
Choose all expressions that are equivalent to 7k.
Example 2
Consider the following:
Select the expression that is equivalent to: 7s+2b-4s
Select the expression that is equivalent to: 21w+4b
To determine whether two expressions are equivalent we can simplify the expressions and compare the simplified versions.
Equivalent expressions using distributive property
Equivalent expressions can also be produced by applying the distributive property. If we were to think about 3(m+2), using the idea that multiplication relates to groups, we can say that 3(m+2) is 3 groups of (m+2). So we have, 3\times m and 3\times 2. This gives us 3m and 6.
| \displaystyle 3(m+2) | \displaystyle = | \displaystyle 3m+6 |
Similarly, given an algebraic expression, we can use our knowledge of the distributive property and greatest common factor (GCF), to factor the expression.
To find an equivalent expression for 16x+24, use the GCF of 16 and 24, which is 8.
| \displaystyle 16x+24 | \displaystyle = | \displaystyle 8\times 2 + 8\times 3 | 16x is 8 groups of 2x and 24 is 8 groups of 3 |
| \displaystyle 16x+24 | \displaystyle = | \displaystyle 8(2x+3) | Factor out the GCF of 8 |
Examples
Example 3
Use distributive property to write an equivalent expression for 10(m+4)
We can also apply the distributive property and greatest common factor to find equivalent expressions.