Solving One-Step Multiplication & Division Equations - Sixth Grade Mathematics

Solving one-step multiplication and division equations isolates the variable using multiplicative inverses, governed by the Multiplication and Division Properties of Equality.

The Multiplicative Properties of Equality

Just as with addition and subtraction, performing identical multiplication or division operations on both sides of an equation preserves balance.

Division Property of Equality:
If  a = b,  then  a / c = b / c  (where c is not 0).
Dividing both sides of an equation by the same non-zero number preserves equality.

Multiplication Property of Equality:
If  a = b,  then  a * c = b * c.
Multiplying both sides of an equation by the same number preserves equality.

Inverse Operations for Multiplication and Division

Multiplication and division are inverse operations that undo each other: To undo multiplication: DIVIDE both sides by the variable's coefficient. To undo division: MULTIPLY both sides by the denominator (divisor).

The Core Strategies:
Case 1: Variable is multiplied (4x = 28)
        Divide both sides by 4:  4x / 4 = 28 / 4  -->  x = 7

Case 2: Variable is divided (x / 5 = 9)
        Multiply both sides by 5: (x / 5) * 5 = 9 * 5  -->  x = 45

Step-by-Step Formal Executions

Executing one-step equations requires clear vertical mathematical presentation.

Example A: Solve  6x = 72
Step 1: Identify operation: x is multiplied by 6.
Step 2: Apply Division Property of Equality: Divide both sides by 6:
        6x / 6 = 72 / 6
Step 3: Simplify:
        x = 12.
Check: 6(12) = 72. (True!)

Example B: Solve  y / 8 = 7
Step 1: Identify operation: y is divided by 8.
Step 2: Apply Multiplication Property of Equality: Multiply both sides by 8:
        (y / 8) * 8 = 7 * 8
Step 3: Simplify:
        y = 56.
Check: 56 / 8 = 7. (True!)

Managing Equations with Negative Coefficients

When the coefficient of the variable is negative, divide by the entire signed number including the negative sign. Solve: -5x = 35. Divide both sides by -5: (-5x) / (-5) = 35 / (-5). Since a positive divided by a negative is negative: x = -7. Check: -5(-7) = +35. (Confirmed!)

Exercises

Exercise 1. Solve each equation and verify using substitution: (a) 7x = 56; (b) m / 6 = 9; (c) 12p = 108; (d) w / 4 = 15.

Exercise 2. Solve the equation with a negative coefficient: -8k = 72.

Exercise 3. Six identical tickets to a music concert cost a total of $216. Write a multiplication equation with variable t for the cost of one ticket, and solve it.

Solutions

Solution to Exercise 1. For (a) 7x = 56: divide both sides by 7: x = 56 / 7 = 8. Check: 7(8) = 56. For (b) m / 6 = 9: multiply both sides by 6: m = 9 * 6 = 54. Check: 54 / 6 = 9. For (c) 12p = 108: divide both sides by 12: p = 108 / 12 = 9. Check: 12(9) = 108. For (d) w / 4 = 15: multiply both sides by 4: w = 15 * 4 = 60. Check: 60 / 4 = 15.

Solution to Exercise 2. For -8k = 72: divide both sides by -8: (-8k) / (-8) = 72 / (-8). Different signs produce a negative quotient: k = -9. Check: -8(-9) = 72.

Solution to Exercise 3. Let t = cost of one ticket. The equation modeling 6 tickets totaling $216 is: 6t = 216. Divide both sides by 6: 6t / 6 = 216 / 6. Perform division: 216 / 6 = 36. Each ticket costs $36.