Applying the Distributive Property with Rational Coefficients - Seventh Grade Mathematics

The distributive property bridges multiplication and addition across algebraic expressions, systematically scaling multi-term groupings with integer, decimal, and fractional coefficients.

The Geometric Foundation of Distribution

The distributive property states that multiplying a sum by a factor is identical to multiplying each addend individually by that factor and then summing the products: a times the quantity b plus c equals a times b plus a times c.

               THE GEOMETRIC AREA MODEL
                     b             c
              +-------------+-------------+
            a |    a * b    |    a * c    |
              +-------------+-------------+
              Total Area = a * (b + c) = ab + ac

Geometrically, consider a rectangle of height a whose base is partitioned into two segments of lengths b and c. The total area of the large rectangle can be calculated either by multiplying the height by the entire base length (a times the quantity b plus c), or by calculating the areas of the two smaller sub-rectangles independently and adding them together (a times b plus a times c).

Distributing Negative Factors

When distributing a factor with a negative sign, the sign of every term inside the parentheses reverses. The negative factor must be multiplied by both the variable term and the constant term, strictly observing signed integer multiplication rules.

               DISTRIBUTING A NEGATIVE FACTOR
                    -a * (bx - c) = -abx + ac

Let us trace the expansion of negative four times the quantity three x minus five: Step 1: Multiply negative four by positive three x: negative four times three x equals negative twelve x. Step 2: Multiply negative four by negative five: negative four times negative five equals positive twenty. Result: negative twelve x plus twenty.

Example 1: -5 * (2x + 7)  = (-5 * 2x) + (-5 * 7)   = -10x - 35
Example 2: -3 * (4x - 8)  = (-3 * 4x) - (-3 * 8)   = -12x + 24
Example 3: -(6x - 11)     = -1 * (6x - 11)         = -6x + 11

Notice in Example 3 that a solitary negative sign in front of parentheses represents multiplication by negative one, inverting every term inside.

Distributing Rational Coefficients: Fractions and Decimals

In seventh grade, the multiplier outside the parentheses is frequently a fraction or decimal. The distributive mechanics remain identical: multiply the rational factor by each term inside.

Fractions Example:
  (2/3) * (6x - 15)
  = [ (2/3) * 6x ] - [ (2/3) * 15 ]
  = [ 12/3 x ] - [ 30/3 ]
  = 4x - 10

Decimals Example:
  0.4 * (5x + 12)
  = (0.4 * 5x) + (0.4 * 12)
  = 2x + 4.8

Cross-canceling fractions before multiplying significantly accelerates calculations: sixty-thirds reduces directly to four, and thirty-thirds reduces to ten.

Multi-Step Simplification: Distribute then Combine

Complex algebraic expressions require applying the distributive property first, followed by combining all resultant like terms.

Expression:   5x - 2 * (3x - 4) + 9

Step 1 (Distribute -2):
              -2 * (3x) = -6x
              -2 * (-4) = +8
              --> 5x - 6x + 8 + 9

Step 2 (Combine Like Terms):
              (5x - 6x) = -1x (or -x)
              (8 + 9)   = +17
              --> -x + 17

Do not subtract two from five x before distributing. The order of operations strictly dictates that multiplication (distribution) takes precedence over subtraction.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Expand the expression negative seven times the quantity four x minus three using the distributive property.

Exercise 2. Apply the distributive property with a fractional coefficient: expand three-fifths times the quantity ten x plus twenty-five.

Exercise 3. Expand and simplify the decimal expression: zero point six times the quantity three y minus four plus one point two y.

Exercise 4. Simplify the multi-step algebraic expression completely: eight minus three times the quantity two x minus five plus four x.

Solutions

Solution to Exercise 1. Distribute negative seven to both terms inside the parentheses: negative seven times four x equals negative twenty-eight x. Next, negative seven times negative three equals positive twenty-one. Combining these products gives negative twenty-eight x plus twenty-one.

Solution to Exercise 2. Distribute three-fifths to each term: three-fifths times ten x plus three-fifths times twenty-five. For the first term, ten divided by five is two, and two times three is six, producing six x. For the second term, twenty-five divided by five is five, and five times three is fifteen. The expanded expression is six x plus fifteen.

Solution to Exercise 3. First distribute zero point six to the terms inside the parentheses: zero point six times three y equals one point eight y; zero point six times negative four equals negative two point four. The expression becomes one point eight y minus two point four plus one point two y. Now combine the like variable terms: one point eight y plus one point two y equals three point zero y (or three y). The fully simplified expression is three y minus two point four.

Solution to Exercise 4. First identify that negative three must be distributed across the quantity two x minus five. Negative three times two x equals negative six x; negative three times negative five equals positive fifteen. Substitute these back into the expression: eight minus six x plus fifteen plus four x. Now group like terms: group the variable terms negative six x plus four x to obtain negative two x. Group the constants: eight plus fifteen equals positive twenty-three. The final simplified expression is negative two x plus twenty-three.