Unlocking Algebra: Combining Like Terms with Distribution

Welcome back to Professor Baker's Math Class! Today, we're tackling a fundamental concept in algebra: combining like terms after distribution. This skill is crucial for simplifying expressions and solving equations. It might seem daunting at first, but with a systematic approach and plenty of practice, you'll be a pro in no time!

What are Like Terms?

Like terms are terms that have the same variable raised to the same power. For example, $3x$ and $-5x$ are like terms because they both have the variable $x$ raised to the power of 1. However, $3x$ and $3x^2$ are not like terms because the variable $x$ is raised to different powers.

The Distributive Property: Spreading the Love (of Multiplication!)

The distributive property is the key to simplifying expressions with parentheses. It states that $a(b + c) = ab + ac$. Essentially, you're multiplying the term outside the parentheses by each term inside the parentheses.

For example:

$$2(x + 3) = 2 * x + 2 * 3 = 2x + 6$$

Combining Like Terms After Distribution: A Step-by-Step Guide

Here's the process we'll follow:

  1. Distribute: Apply the distributive property to eliminate any parentheses.
  2. Identify Like Terms: Look for terms with the same variable and exponent.
  3. Combine: Add or subtract the coefficients of the like terms. Remember, you're only combining the numbers in front of the variables, not changing the variables themselves.

Example Time!

Let's walk through an example:

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

  1. Distribute: $3(x + 2) = 3x + 6$. So, the expression becomes $3x + 6 + 4x - 1$.
  2. Identify Like Terms: We have two 'x' terms ($3x$ and $4x$) and two constant terms ($6$ and $-1$).
  3. Combine: $3x + 4x = 7x$ and $6 - 1 = 5$.

Therefore, the simplified expression is $7x + 5$.

Another Example:

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

  1. Distribute: $-2(y - 5) = -2 * y + (-2) * (-5) = -2y + 10$. So, the expression becomes $-2y + 10 + 3y + 2$.
  2. Identify Like Terms: We have two 'y' terms ($-2y$ and $3y$) and two constant terms ($10$ and $2$).
  3. Combine: $-2y + 3y = y$ and $10 + 2 = 12$.

Therefore, the simplified expression is $y + 12$.

Things to Watch Out For:

  • Negative Signs: Be extra careful when distributing with negative signs. Remember that multiplying a negative by a negative results in a positive.
  • Exponents: Only combine terms with the same variable and the same exponent.

Practice Makes Perfect!

The best way to master combining like terms with distribution is to practice! Work through problems on Khan Academy, in your textbook, or create your own. Don't be afraid to make mistakes – they're a learning opportunity! Keep practicing, and you'll conquer this algebraic hurdle in no time. Good luck, and see you in the next lesson!