Solving One-Step Addition & Subtraction Equations - Sixth Grade Mathematics

Solving one-step addition and subtraction equations isolates the variable using inverse operations, governed by the Addition and Subtraction Properties of Equality.

The Properties of Equality

To maintain balance in an algebraic equation, whatever operation is performed on one side must be performed on the opposite side.

Addition Property of Equality:
If  a = b,  then  a + c = b + c.
Adding the same number to both sides of an equation preserves equality.

Subtraction Property of Equality:
If  a = b,  then  a - c = b - c.
Subtracting the same number from both sides of an equation preserves equality.

The Principle of Inverse Operations

Inverse operations are operations that undo each other: Addition and Subtraction are inverse operations. To undo addition of a number, SUBTRACT that number from both sides. To undo subtraction of a number, ADD that number to both sides.

The Goal: Isolate the Variable
                x + 8 = 15   --> Undo +8 by subtracting 8:  x = 15 - 8 = 7
                x - 6 = 11   --> Undo -6 by adding 6:       x = 11 + 6 = 17

The Formal Two-Column Solution Format

Presenting solutions in a vertical, two-column layout ensures mathematical rigor and clear error tracing.

Case A: Solving an Addition Equation (x + 14 = 39)
        x + 14 =  39
          - 14   -14    <-- Subtraction Property of Equality
        ------------
        x      =  25

Check: Substitute x = 25: 25 + 14 = 39. (True!)

Case B: Solving a Subtraction Equation (y - 9 = 24)
        y -  9 =  24
          +  9   + 9    <-- Addition Property of Equality
        ------------
        y      =  33

Check: Substitute y = 33: 33 - 9 = 24. (True!)

Equations with the Variable on the Right

Equations often position the variable on the right-hand side (e.g., 28 = m + 12). The exact same properties of equality apply: Subtract 12 from both sides: 28 - 12 = m + 12 - 12. This simplifies to: 16 = m. By the Symmetric Property of Equality, if 16 = m, then m = 16.

Exercises

Exercise 1. Solve each equation using the appropriate property of equality, and verify your answer with a check: (a) x + 18 = 42; (b) y - 15 = 28; (c) 54 = k + 19; (d) w - 37 = 63.

Exercise 2. A book was discounted by $7, bringing its sale price to $18. Write an addition or subtraction equation to find the original price p, and solve it.

Exercise 3. Identify and correct the error in the student's work: Equation: x + 9 = 21 Student step: Add 9 to both sides: x = 21 + 9 = 30.

Solutions

Solution to Exercise 1. For (a) x + 18 = 42: undo addition by subtracting 18 from both sides: x = 42 - 18 = 24. Check: 24 + 18 = 42. For (b) y - 15 = 28: undo subtraction by adding 15 to both sides: y = 28 + 15 = 43. Check: 43 - 15 = 28. For (c) 54 = k + 19: subtract 19 from both sides: 54 - 19 = k, so k = 35. Check: 54 = 35 + 19. For (d) w - 37 = 63: add 37 to both sides: w = 63 + 37 = 100. Check: 100 - 37 = 63.

Solution to Exercise 2. Let p = original price of the book. The discount of $7 means p - 7 = 18. To solve, add 7 to both sides: p = 18 + 7 = 25. The original price of the book was $25.

Solution to Exercise 3. The student's error was applying the wrong operation. In the equation x + 9 = 21, the number 9 is being added to x. The inverse operation that undoes addition is subtraction, not addition. Adding 9 made the equation x + 18 = 30. The correct step is to subtract 9 from both sides: x = 21 - 9 = 12.