Add, subtract, expand and factor linear expressions

Learning goal: Generate equivalent linear expressions with rational coefficients by applying properties of operations. Keep signs and grouped terms intact, distinguish like terms and explain why a transformation preserves value.

A linear expression such as (3/4)x + 2 contains a variable term and a constant term. The coefficient 3/4 multiplies x. It does not mean x plus three fourths. A term's sign belongs to its value: in (3/4)x − 2, the constant term can be viewed as −2 when the expression is written as a sum. Rational coefficients may be positive or negative integers, fractions, or terminating or repeating decimals.

Like terms have the same variable part: the same variable raised to the same power. In this lesson's linear expressions every variable term is a multiple of x, so (3/4)x and (1/2)x can combine, while (3/4)x and a constant two cannot become a single multiple of x without changing the expression. Terms such as x and x² are not like terms, because their powers differ. Using distribution, ax + bx = (a + b)x. For (3/4)x + (1/2)x, add the coefficients in fourths: (3/4 + 2/4)x = (5/4)x. The variable represents the same quantity in both terms.

Subtraction also combines signed coefficients. (3/4)x − (1/2)x = (3/4 − 2/4)x = (1/4)x. For a positive quantity x, a bar divided into four equal parts can show three fourths compared with two fourths, leaving one fourth. The whole is all four regions. The algebraic equality remains valid for zero or negative x even though a physical length bar only models positive quantities directly.

Add complete expressions by combining like terms: [(3/4)x + 2] + [(1/2)x − 3] = (5/4)x − 1. Grouping the x-terms and constants is justified by associative and commutative properties of addition. Reordering must preserve the negative three; changing it to positive three would no longer represent the supplied sum.

Subtracting a grouped expression changes the sign of every term being subtracted. [(3/4)x + 2] − [(1/2)x − 3] becomes (3/4)x + 2 − (1/2)x + 3, because the opposite of the whole second expression is −(1/2)x + 3. The result is (1/4)x + 5. Only changing the first sign inside the subtracted group would leave an incorrect constant.

Distribution expands a factor across every term in a sum. For −2(3x − 4), multiply −2 by 3x and by −4 to obtain −6x + 8. The positive eight follows from the product of two negatives. For (2/3)(x + 6), the result is (2/3)x + 4. A fractional multiplier applies to the constant term as well as the variable term.

Factoring reverses distribution by identifying a common factor. In (2/3)x + 4/3, both terms have a factor of 2/3, so the expression is (2/3)(x + 2). Check by expansion: (2/3)x + (2/3)2 = (2/3)x + 4/3. A factor outside parentheses multiplies the entire sum, not only its first term.

More than one factored form can be equivalent. For 6x + 9, both 3(2x + 3) and (3/2)(4x + 6) expand to the same expression. A common convenient choice uses a simple shared factor, but equivalence depends on preserving value rather than producing one visually preferred form. The best form may depend on the question or context.

Substitution provides a useful error check, though one matching value does not prove equivalence for every x. At x = 4, [(3/4)x + 2] − [(1/2)x − 3] equals five minus negative one, or six. The simplified form (1/4)x + 5 also gives six. Test another value to catch additional mistakes, then use the properties to justify the equality generally.

An expression is not an equation to solve until an equality or other condition is supplied. Simplifying 2x + 4 does not determine x. Factoring it as 2(x + 2) gives a different form with the same value for each x. Do not set an expression equal to zero merely because a familiar equation-solving step is available.

Worked decimal expansion: 1.5(2x − 0.4) − 0.5x = 3x − 0.6 − 0.5x = 2.5x − 0.6. Distribution applies to both terms, then like variable terms combine. At x = 2, the original gives 1.5(3.6) − 1 = 4.4, agreeing with five minus 0.6.

Worked rational subtraction: (5/6)x − [(1/3)x + 1/2] = (5/6)x − (2/6)x − 1/2 = (1/2)x − 1/2. The constant becomes negative because the entire bracketed expression is subtracted. Factoring gives (1/2)(x − 1).

Worked error diagnosis: A student writes −3(x − 2) = −3x − 6. Expand term by term: −3 times x is −3x, and −3 times −2 is +6. At x = 0, the original is six while the student's result is negative six, so the mismatch exposes the sign error.

Structure practice: Mark terms and coefficients in four expressions with rational coefficients. Sort like terms before combining them. For each subtraction of a group, rewrite it as addition of the group's opposite. For each factored expression, expand to check that every original term is recovered.

Independent performance: Create and simplify six linear-expression problems: a sum, a difference, a fractional expansion, a negative-factor expansion, a common-factor expression and a combination of operations. Include at least one negative decimal coefficient. Annotate the property used at each main step. Check two answers at positive and negative inputs, then explain why those numerical checks support error detection while property-based reasoning establishes general equivalence.