Combining Like Terms with Distribution

Welcome to today's math adventure! We explored the powerful combination of the distributive property and combining like terms. This is a crucial skill for simplifying expressions and solving equations. Let's break it down:

Key Concepts

  • Term: A term is a product of a number (coefficient) and a variable. For example, in the term $3x$, 3 is the coefficient and $x$ is the variable.
  • Like Terms: Like terms have the same variable raised to the same power. Constant terms (numbers without variables) are also like terms. Examples: $3x$ and $-5x$ are like terms; $7$ and $-2$ are like terms.
  • Simplified Expression: An expression is simplified when it has no grouping symbols (parentheses) and all like terms have been combined.

The Distributive Property

The distributive property allows us to multiply a single term by two or more terms inside parentheses. Remember the formula: $a(b + c) = ab + ac$.

Example: Simplify $-2(y - 3)$.

We distribute the -2 to both terms inside the parentheses:

$$ -2(y - 3) = (-2)(y) - (-2)(3) = -2y + 6 $$

Combining Like Terms

Combining like terms involves adding or subtracting the coefficients of terms with the same variable and exponent.

Example: Simplify $17s - 8 - 12s$.

First, group the like terms together:

$$ 17s - 12s - 8 $$

Then, combine the coefficients of the 's' terms:

$$ (17 - 12)s - 8 = 5s - 8 $$

Putting it All Together

Now let's tackle expressions that require both distribution and combining like terms.

Example: Simplify $7 - 3(2 + z)$.

  1. Distribute: Distribute the $-3$ to both terms inside the parentheses: $7 + (-3)(2 + z) = 7 + (-3)(2) + (-3)(z) = 7 - 6 - 3z$
  2. Combine Like Terms: Combine the constant terms: $7 - 6 - 3z = 1 - 3z$

Therefore, $7 - 3(2 + z)$ simplifies to $1 - 3z$.

Practice Makes Perfect!

Remember, the key to mastering these skills is practice! Work through plenty of examples, and don't be afraid to ask questions. You've got this!

Homework

Complete the assigned homework to reinforce these concepts: First 3 pages do everything, and the last page do only the odd-numbered problems. Good luck!