Simplifying by Combining Like Terms - Sixth Grade Mathematics
Combining like terms simplifies algebraic expressions by grouping terms with identical variable structures, reducing complex multi-term expressions to their simplest form.
The Principle of Like Terms
Like terms are terms that contain the exact same variables raised to the exact same exponents. Constants (pure numbers) are also like terms to each other. When combining like terms, you add or subtract their coefficients while leaving the variable part unchanged.
The Core Algebraic Rule:
ax + bx = (a + b)x
Concrete Analogy:
3 apples + 5 apples = 8 apples --> 3a + 5a = 8a
4 cats + 2 dogs cannot be combined! --> 4c + 2d remains 4c + 2d!
The Four-Step Simplification Protocol
Follow this protocol to simplify any multi-term expression.
+-------------------------------------------------------------------------+
| COMBINING LIKE TERMS PROTOCOL |
| |
| Step 1: Identify all terms and their signs. Remember that the sign in |
| front of a term (+ or -) belongs to that term! |
| Step 2: Group like terms together using shapes (boxes, circles, |
| underlines) or by applying the commutative property. |
| Step 3: Add or subtract the coefficients of each group of like terms. |
| Step 4: Write the final simplified expression, traditionally placing |
| variable terms in alphabetical order, followed by constants. |
+-------------------------------------------------------------------------+
Worked Example: Simplify 7x + 5y - 3x + 8 + 2y - 11
Step 1: Group x-terms: +7x and -3x --> (7 - 3)x = 4x
Step 2: Group y-terms: +5y and +2y --> (5 + 2)y = 7y
Step 3: Group constants: +8 and -11 --> 8 - 11 = -3
Step 4: Combine the simplified components:
Result: 4x + 7y - 3.
Managing "Invisible" Coefficients
Remember that a variable without a written number in front has an implicit coefficient of 1. In the expression x + 4x: x means 1x, so 1x + 4x = (1 + 4)x = 5x. In the expression 6y - y: y means 1y, so 6y - 1y = (6 - 1)y = 5y. Forgetting the coefficient of 1 is the most frequent mistake when combining lone variables.
Geometric Perimeter Applications
Combining like terms is the primary tool for finding the perimeter of polygons with algebraic side lengths.
Perimeter of a Triangle with sides: (2x + 3), (4x - 1), and (3x + 5)
Perimeter = Side 1 + Side 2 + Side 3
P = (2x + 3) + (4x - 1) + (3x + 5)
Group x-terms: 2x + 4x + 3x = 9x.
Group constants: 3 - 1 + 5 = 7.
Simplified Perimeter: P = 9x + 7.
Exercises
Exercise 1. Simplify each expression by combining like terms: (a) 8m + 5m; (b) 14y - 6y; (c) 9x + 4 + 3x + 8; (d) 12k - k.
Exercise 2. Simplify the multi-term expression: 6a + 8b - 2a + 3 + 5b - 9.
Exercise 3. A rectangle has a length of (3x + 4) units and a width of (2x - 1) units. Write and simplify an algebraic expression for the perimeter of the rectangle.
Solutions
Solution to Exercise 1. For (a) 8m + 5m: add coefficients: (8 + 5)m = 13m. For (b) 14y - 6y: subtract coefficients: (14 - 6)y = 8y. For (c) 9x + 4 + 3x + 8: group x-terms: 9x + 3x = 12x; group constants: 4 + 8 = 12; result is 12x + 12. For (d) 12k - k: k has coefficient 1: (12 - 1)k = 11k.
Solution to Exercise 2. Group like terms with their signs: a-terms: 6a - 2a = 4a. b-terms: +8b + 5b = +13b. Constants: +3 - 9 = -6. Combine all parts: 4a + 13b - 6.
Solution to Exercise 3. Perimeter of a rectangle is P = 2(length) + 2(width). Write expression: P = 2(3x + 4) + 2(2x - 1). Apply distributive property: P = 6x + 8 + 4x - 2. Group like terms: (6x + 4x) + (8 - 2). Combine: P = 10x + 6 units.