Simplifying Algebraic Expressions

Simplifying an algebraic expression means making it as short and neat as possible. We do this by grouping similar parts together and getting rid of parentheses. The two main tools you need are combining like terms and the distributive property.

What Are Like Terms?

Like terms are terms that have the exact same variables and exponents.

• $3x$ and $2x$ are like terms.
• $4y$ and $7y$ are like terms.
• Plain numbers (constants) like $5$ and $-4$ are also like terms.
• $3x$ and $3x^2$ are not like terms because the exponents are different.

Combining Like Terms

To combine like terms, you simply add or subtract their coefficients (the numbers in front of the variables).

Example: Simplify $3x + 2x - 4$

1. Identify the like terms: $3x$ and $2x$.
2. Add their coefficients: $3 + 2 = 5$, so $3x + 2x = 5x$.
3. The $-4$ has no like terms, so it stays exactly as it is.

Answer: $5x - 4$

Using the Distributive Property

When an expression has parentheses, you usually need to remove them first using the distributive property. This means multiplying the number outside the parentheses by every term inside.

Example: Simplify $2(3x + 1) + 4x$

1. Distribute the $2$ to everything inside the parentheses: $2 \cdot 3x + 2 \cdot 1 = 6x + 2$
2. Rewrite the full expression: $6x + 2 + 4x$
3. Combine the like terms ($6x$ and $4x$): $6x + 4x = 10x$

Answer: $10x + 2$

Watch Out for Negative Signs!

Be extra careful when distributing a negative number. Remember that a negative times a negative makes a positive.

Example: Simplify $-3(2y - 5) + y$

1. Distribute the $-3$: $-3(2y) - (-3)(5) = -6y + 15$
2. Rewrite the full expression: $-6y + 15 + y$ (Note: $y$ is the same as $1y$)
3. Combine the like terms ($-6y$ and $1y$): $-6y + 1y = -5y$

Answer: $-5y + 15$