Part 1: Combining Like Terms
Think of this as sorting. You can only combine terms that are the "same kind of item."
Rule: Like terms must have the exact same variable(s) and the exact same exponents. The coefficients (numbers in front) can be different.
Examples: Step-by-Step
Easy: $7x + 2 + 3x + 5$
Step 1: Identify and group like terms.
$7x + 3x$ + $2 + 5$
Step 2: Combine the coefficients of each group.
$(7 + 3)x = 10x$
$2 + 5 = 7$
Result: Write the new, simplified expression.
$10x + 7$
Medium: $10 - 5d + 2 - 9d$
Step 1: Identify and group like terms (watch the signs!).
$-5d - 9d$ + $10 + 2$
Step 2: Combine each group.
$(-5 - 9)d = -14d$
$10 + 2 = 12$
Result:
$-14d + 12$
Hard: $a^2 + 5a + 3a^2 - 2a$
Step 1: Group like terms. Remember $a^2$ and $a$ are NOT alike!
$a^2 + 3a^2$ + $5a - 2a$
Step 2: Combine each group.
$(1 + 3)a^2 = 4a^2$
$(5 - 2)a = 3a$
Result:
$4a^2 + 3a$
Part 2: The Distributive Property
This is how we get rid of parentheses. You "distribute" the outside term to everything inside.
Rule: $a(b + c) = ab + ac$. Multiply the term outside the parentheses by every term inside.
Examples: Step-by-Step
Easy: $4(x + 5)$
Step 1: Distribute the 4 to the first term (x).
$4 \cdot x = 4x$
Step 2: Distribute the 4 to the second term (5).
$4 \cdot 5 = 20$
Result: Combine the results.
$4x + 20$
Medium: $-3(a - 6)$
Step 1: Distribute the -3 to the 'a'.
$-3 \cdot a = -3a$
Step 2: Distribute the -3 to the '-6'. Be careful with signs!
$(-3) \cdot (-6) = +18$
Result: Combine them. The signs changed!
$-3a + 18$
Hard: $-(x - 8)$
Hint: A negative sign outside parentheses is the same as distributing a -1.
Step 1: Distribute the -1 to the 'x'.
$-1 \cdot x = -x$
Step 2: Distribute the -1 to the '-8'.
$(-1) \cdot (-8) = +8$
Result:
$-x + 8$
Part 3: Putting It All Together
Now, let's use both skills. The order is important!
Order of Operations: 1st, Distribute to remove parentheses. 2nd, Combine Like Terms.
Example: $3(x + 2) + 5x$
Step 1: Distribute the 3.
$3x + 6$ $+ 5x$
Step 2: Identify and combine like terms.
$3x + 5x$ $+ 6$
$(3+5)x + 6 = 8x + 6$
Final Result:
$8x + 6$
Harder Example: $4(d - 3) - (d - 5)$
Step 1: Distribute the 4 and the -1.
$4d - 12$ $-d + 5$
Step 2: Identify and combine like terms.
$4d - d$ $- 12 + 5$
$(4-1)d + (-12+5) = 3d - 7$
Final Result:
$3d - 7$