Algebra Basics: Manipulation & Simplification
1. What is a "Term"?
In algebra, an expression is made up of parts called terms. A term can be a number, a variable, or numbers and variables multiplied together.
Consider the term: $$ -5x^2 $$
- -5 is the Coefficient (The number part, including the sign).
- x is the Variable (The unknown letter).
- 2 is the Power/Index.
2. Multiplying and Dividing
When multiplying or dividing terms, you operate on the coefficients (numbers) and the variables (letters) separately.
Rules for Multiplication
1. Numbers multiply numbers: \( 2a \times 3 = 6a \)
2. Letters multiply letters: \( a \times b = ab \)
3. Same letters add powers: \( a \times a = a^2 \)
4. Watch the signs:
- Positive \(\times\) Positive = Positive
- Negative \(\times\) Negative = Positive
- Positive \(\times\) Negative = Negative
Rules for Division
Cancel out common factors: Treat division like a fraction.
$$ \frac{12x^2}{4x} $$
1. Divide numbers: \(12 \div 4 = 3\)
2. Divide letters: \(x^2 \div x = x\)
Result: \(3x\)
3. Collecting Like Terms
Like Terms are terms that have exactly the same variable and the same power.
- \(2x\) and \(5x\) are like terms (Both are \(x\)).
- \(3a\) and \(3b\) are NOT like terms (Different letters).
- \(x^2\) and \(x\) are NOT like terms (Different powers).
The Sign Belongs to the Term!
When you move a term, take the \(+\) or \(-\) sign directly in front of it with you.
In \(3x - 2y\), the \(y\) term is negative (\(-2y\)).
Practice Questions & Solutions
Try these yourself, then click to reveal the solution.
Q1: Simplify \( 5x + 3x \)
These are like terms (both \(x\)).
Just add the coefficients: \(5 + 3 = 8\).
Answer: \( 8x \)
Q2: Simplify \( 7y - 2y + y \)
Remember that \(y\) on its own counts as \(1y\).
\(7 - 2 + 1 = 6\)
Answer: \( 6y \)
Q3: Collect like terms: \( 4a + 3b - a + 2b \)
Group the \(a\)'s: \(4a - 1a = 3a\)
Group the \(b\)'s: \(+3b + 2b = +5b\)
Answer: \( 3a + 5b \)
Q4: Simplify \( 3 \times 4y \)
Multiply the numbers: \(3 \times 4 = 12\).
The variable \(y\) stays attached.
Answer: \( 12y \)
Q5: Multiply \( 2a \times 5a \)
Numbers: \(2 \times 5 = 10\)
Letters: \(a \times a = a^2\)
Answer: \( 10a^2 \)
Q6: Simplify \( -3x \times 4x \)
Check the signs: Negative \(\times\) Positive = Negative.
Numbers: \(3 \times 4 = 12\).
Letters: \(x \times x = x^2\).
Answer: \( -12x^2 \)
Q7: Simplify \( \frac{15x}{3} \)
Divide the coefficients: \(15 \div 3 = 5\).
The \(x\) remains.
Answer: \( 5x \)
Q8: Simplify \( \frac{20x^2}{4x} \)
Numbers: \(20 \div 4 = 5\).
Letters: \(x^2\) means \(x \times x\).
So, \(\frac{x \times x}{x}\) leaves just \(x\).
Answer: \( 5x \)
Q9: Complicated Collection: \( 2x^2 + 5x - x^2 + 3 \)
Group \(x^2\): \(2x^2 - 1x^2 = x^2\)
Group \(x\): \(+5x\) (no other \(x\) terms)
Group Numbers: \(+3\) (no other numbers)
Answer: \( x^2 + 5x + 3 \)
Q10: Challenge - \( 3(2x) - 4(x) \)
First, multiply the brackets out (Multiplication comes before Addition/Subtraction).
\(3 \times 2x = 6x\)
\(-4 \times x = -4x\)
Now collect: \(6x - 4x\)
Answer: \( 2x \)