Expressions & Equations
Simplifying Expressions
Using properties of operations to generate equivalent expressions.
Prerequisites
- Writing Algebraic Expressions
Why Distributing Works
Before reading on, predict: does 3(x + 4) equal 3x + 4, or 3x + 12? Think about what the parentheses are actually saying — that 3 applies to everything inside them, not just the first term.
Definition — Distributive Property
Picture a rectangle with width 3 and a length made of two parts, x and 4, placed side by side. The rectangle's total area can be found two ways: as one rectangle, 3 times the full length (x + 4), or as two smaller rectangles added together, 3 times x plus 3 times 4. Both must give the same total area — which is exactly why 3(x + 4) = 3x + 12, not 3x + 4.
Worked Example — Distributing and Combining Like Terms
Worked Example — Distributing a Negative Number
Tip
Common Mistakes
Distributing a number to only the first term inside the parentheses, such as simplifying 3(x + 4) as 3x + 4.
The number outside the parentheses multiplies every term inside, not just the first one — check that every term received the multiplication.
Losing track of a sign when distributing a negative number, such as simplifying −2(3x − 5) as −6x − 10 instead of −6x + 10.
Multiplying a negative by a negative gives a positive — apply the sign rules carefully to every term, not just the first.
Key Takeaways
- The distributive property says a(b + c) = ab + ac — multiplying a sum is the same as multiplying each part and adding the results.
- This can be pictured as splitting one rectangle's area into two smaller rectangles that add to the same total.
- After distributing, combine any resulting like terms to fully simplify the expression.
Summary
The distributive property, understood through a split-rectangle picture, explains why distributing works rather than just how to do it. With expressions fully simplified, the next lesson uses this skill to solve equations.
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