Introduction to Two-Step Equations - Sixth Grade Mathematics

Two-step equations involve two distinct mathematical operations, requiring a disciplined sequence of inverse operations executed in reverse order of operations to isolate the variable.

The Reverse Order of Operations (SADMEP)

In standard arithmetic evaluation, PEMDAS directs you to compute parentheses, exponents, multiplication and division, and finally addition and subtraction. When solving equations, you are unwrapping or deconstructing the operations that were applied to the variable. Therefore, you apply inverse operations in REVERSE order.

Evaluating an Expression:        Parentheses -> Exponents -> Mult/Div -> Add/Sub
Solving an Equation (Unwrapping): Add/Sub FIRST  -->  Mult/Div SECOND (SADMEP)

The Two-Step Equation Protocol:
                     Form:  ax + b = c

Step 1: Undo Addition or Subtraction first.
        Use the Addition or Subtraction Property of Equality
        to eliminate the constant term (+ b or - b).

Step 2: Undo Multiplication or Division second.
        Use the Multiplication or Division Property of Equality
        to eliminate the coefficient (a).

Step-by-Step Two-Step Executions

Studying two core prototypes illustrates the unwrapping procedure.

Prototype A: Solve  3x + 7 = 22

Step 1: Undo addition (+ 7) by subtracting 7 from both sides:
        3x + 7  =  22
           - 7     - 7
        --------------
        3x      =  15

Step 2: Undo multiplication (* 3) by dividing both sides by 3:
        3x / 3  =  15 / 3
        x       =   5

Check: Substitute x = 5:  3(5) + 7 = 15 + 7 = 22. (Confirmed!)

Prototype B: Solve  (m / 4) - 5 = 3

Step 1: Undo subtraction (- 5) by adding 5 to both sides:
        m / 4 - 5  =  3
              + 5   + 5
        ---------------
        m / 4      =  8

Step 2: Undo division (/ 4) by multiplying both sides by 4:
        (m / 4) * 4 = 8 * 4
        m           = 32

Check: Substitute m = 32: (32 / 4) - 5 = 8 - 5 = 3. (Confirmed!)

Real-World Two-Step Problem Scenarios

Real-world scenarios with a flat starting fee plus a variable rate naturally form two-step equations: A car rental costs $40 flat fee plus $0.20 per mile. If the total bill is $76: Equation: 0.20m + 40 = 76. Step 1: Subtract 40: 0.20m = 36. Step 2: Divide by 0.20: m = 36 / 0.20 = 180 miles driven.

Exercises

Exercise 1. Solve each two-step equation and verify using substitution: (a) 4x + 9 = 37; (b) 5y - 8 = 32; (c) (w / 3) + 6 = 11; (d) (k / 5) - 4 = 8.

Exercise 2. A plumber charges a $50 house call fee plus $35 per hour of work. If the total bill was $190, write a two-step equation to find the hours worked h, and solve it.

Exercise 3. Identify and correct the error in the following solution: Equation: 2x + 6 = 20 Student Step 1: Divide by 2: x + 6 = 10 Student Step 2: Subtract 6: x = 4.

Solutions

Solution to Exercise 1. For (a) 4x + 9 = 37: subtract 9: 4x = 28; divide by 4: x = 7. Check: 4(7) + 9 = 28 + 9 = 37. For (b) 5y - 8 = 32: add 8: 5y = 40; divide by 5: y = 8. Check: 5(8) - 8 = 40 - 8 = 32. For (c) (w / 3) + 6 = 11: subtract 6: w / 3 = 5; multiply by 3: w = 15. Check: (15 / 3) + 6 = 5 + 6 = 11. For (d) (k / 5) - 4 = 8: add 4: k / 5 = 12; multiply by 5: k = 60. Check: (60 / 5) - 4 = 12 - 4 = 8.

Solution to Exercise 2. Let h = number of hours worked. The house call fee is $50, and hourly labor is 35h. The two-step equation is: 35h + 50 = 190. Step 1: Subtract 50 from both sides: 35h = 140. Step 2: Divide both sides by 35: h = 140 / 35 = 4. The plumber worked for 4 hours.

Solution to Exercise 3. While the student's final answer happened to be 7 in another context, in Step 1 the student divided only 2x and 20 by 2, failing to divide the term 6 by 2! When dividing an entire equation by 2, EVERY term must be divided: (2x + 6)/2 = x + 3. The standard, error-free method is to undo addition FIRST: subtract 6 from both sides to get 2x = 14, then divide by 2 to get x = 7.