Lesson 2: Generate Equivalent Expressions

Property

Equivalent expressions have the same value when we substitute a number for the variable.
We cannot combine unlike terms. If the variable parts of the terms are not identical, they are not like terms, and they cannot be combined.

Examples

• The expressions $6x + 2x$ and $8x$ are equivalent. If $x=3$, both expressions equal 24.
• The expressions $6 + 2x$ and $8x$ are not equivalent. If $x=3$, $6+2x$ is 12, while $8x$ is 24.
• In the expression $9a + 4b - 3a$, we can combine the like terms to get the equivalent expression $6a + 4b$.

Explanation

Equivalent expressions are different ways to write the same mathematical idea. For example, $2+3$ and $5$ are equivalent. Simplifying an expression means finding a shorter, equivalent version, which makes it easier to work with.

Property

Like terms are terms that involve the same variable raised to the same exponent. Constants are also like terms.
The importance of distinguishing like terms is that they can be combined to make the expression easier to read and compute, a process called simplifying.

Examples

• To simplify $7x + 5 + 3x$, combine the like terms $7x$ and $3x$ to get $10x$. The simplified expression is $10x + 5$.
• To simplify $9a + 4b + 2a - b$, combine the $a$ terms ($9a+2a=11a$) and the $b$ terms ($4b-b=3b$). The result is $11a + 3b$.
• To simplify $12 + 5y - 8 + 2y$, combine the constants ($12-8=4$) and the $y$ terms ($5y+2y=7y$). The result is $7y + 4$.

Explanation

Combining like terms is like organizing your toys. You can put all the toy cars together and all the building blocks together, but you can't mix them. In math, you add or subtract terms with the same variable part.