Linear Expressions: Adding, Subtracting, and Combining Like Terms - Seventh Grade Mathematics
Algebraic expressions model relationships through terms, coefficients, variables, and constants, simplified by unifying like terms through the distributive property.
The Anatomy of an Algebraic Expression
An algebraic expression is a mathematical phrase containing numbers, variables, and operation symbols. Unlike an equation, an expression does not contain an equals sign.
ANATOMY OF AN EXPRESSION
7x - 4y + 12
Terms: 7x, -4y, 12 (Separated by + or - signs)
Variables: x, y (Letters representing unknown values)
Coefficients: 7 (for x), -4 (for y) (Numerical factor multiplying variable)
Constant: 12 (A fixed numeric term with no variable)
In any term with a variable, the coefficient dictates how many multiples of that variable exist. If a variable appears alone, such as x, its coefficient is an understood positive one (1x). If it appears as negative x, its coefficient is negative one (-1x).
Defining Like Terms
Like terms are terms that contain the exact same variables raised to the exact same exponents. The numerical coefficients do not need to be equal for terms to be like terms.
+-----------------------+-----------------------+-----------------------------+
| Pair of Terms | Are they Like Terms? | Reason |
+-----------------------+-----------------------+-----------------------------+
| 5x and -8x | YES | Same variable (x) to power 1|
| 3a and 3b | NO | Different variables (a vs b)|
| 4y^2 and 9y | NO | Different exponents (2 vs 1)|
| -14 and 22 | YES | Both are pure constants |
| 2xy and 7yx | YES | Same variables (xy = yx) |
+-----------------------+-----------------------+-----------------------------+
Only like terms can be combined into a single term. Attempting to add five x and three y into eight xy is a fundamental algebraic error; different variables represent different categories of quantities that cannot merge into a single quantity.
The Mechanics of Combining Like Terms
Combining like terms is an application of the distributive property in reverse: a times x plus b times x equals the quantity a plus b times x. We add or subtract the numerical coefficients while leaving the variable and exponent completely unchanged.
COMBINING LIKE TERMS PROTOCOL
1. Identify and group terms sharing the same variable part.
2. Keep the sign (+ or -) immediately preceding each term with it.
3. Add the numerical coefficients using integer rules.
4. Attach the common variable factor.
Consider the expression: eight x minus five plus three x plus twelve. Group the x-terms: eight x plus three x. Group the constants: negative five plus twelve. Add coefficients: eight plus three equals eleven, yielding eleven x. Add constants: negative five plus twelve equals positive seven. Simplified form: eleven x plus seven.
Visual Representation with Algebra Tiles:
8x - 5 + 3x + 12
[x][x][x][x][x][x][x][x] + [x][x][x] --> 11 tiles of [x] = 11x
[-][-][-][-][-] + [+][+][+][+][+][+][+][+][+][+][+][+]
Five zero pairs cancel out, leaving 7 positive unit tiles = +7
Total: 11x + 7
Adding and Subtracting Linear Expressions
When adding linear expressions, grouping parentheses can simply be dropped because the associative and commutative properties allow full rearrangement of terms.
Adding Expressions:
(4x - 7) + (3x + 11)
= 4x - 7 + 3x + 11
= (4x + 3x) + (-7 + 11)
= 7x + 4
However, when subtracting linear expressions, the subtraction sign in front of the second expression applies to every term inside that parenthetical group. You must distribute the negative sign (multiplying each term by negative one) to change every sign in the subtrahend to its opposite.
Subtracting Expressions:
(6x - 5) - (2x - 9)
= 6x - 5 - 2x - (-9)
= 6x - 5 - 2x + 9
= (6x - 2x) + (-5 + 9)
= 4x + 4
Notice that subtracting negative nine transformed into positive nine. Forgetting to distribute the negative sign to the constant term is the single most common error in expression subtraction.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. Simplify the following linear expression by identifying and combining all like terms: nine a minus four plus six b minus two a plus seven minus nine b.
Exercise 2. Find the sum of the two linear expressions: the quantity five x minus eight plus the quantity negative nine x plus fourteen.
Exercise 3. Subtract the linear expression four x minus seven from the linear expression seven x minus three.
Exercise 4. The three side lengths of a scalene triangle are represented by the expressions two x plus five, three x minus two, and x plus four. Write and simplify an expression representing the perimeter of the triangle.
Solutions
Solution to Exercise 1. Identify and group like terms: group the a-terms: nine a minus two a; group the b-terms: six b minus nine b; group the constants: negative four plus seven. Combine the coefficients for each group: nine minus two equals seven, producing seven a. Six minus nine equals negative three, producing negative three b. Negative four plus seven equals positive three. Combining these parts gives the simplified expression: seven a minus three b plus three.
Solution to Exercise 2. Write the addition without parentheses: five x minus eight minus nine x plus fourteen. Group the variable terms: five x minus nine x equals negative four x. Group the constants: negative eight plus fourteen equals positive six. The combined sum is negative four x plus six.
Solution to Exercise 3. The phrase subtract A from B means B minus A. Write the subtraction expression: the quantity seven x minus three minus the quantity four x minus seven. Distribute the negative sign across the second expression: seven x minus three minus four x plus seven. Group like terms: seven x minus four x equals three x; negative three plus seven equals positive four. The resulting simplified expression is three x plus four.
Solution to Exercise 4. The perimeter of a triangle is the sum of its three side lengths: the quantity two x plus five plus the quantity three x minus two plus the quantity x plus four. Remove parentheses and group the variable terms: two x plus three x plus one x equals six x. Group the constants: five minus two plus four; five minus two is three, and three plus four is seven. The simplified expression representing the perimeter of the triangle is six x plus seven.