K-12 Math (US): 6-8 Algebra K-12 Math Equations Two-step equations

Study Guide: Two-Step Equations (Grade 6–8, Algebra)

1. The Driving Question

You’re playing a video game where your score starts at 50 points, and every level you complete adds 10 points—but every time you lose, you get a 5-point penalty. After a few rounds, your score is 75. How do you figure out exactly how many levels you’ve completed and how many times you’ve lost, just from the final score? And why can’t you just guess and check forever—what’s the rule that always works?

2. The Core Idea — Built, Not Listed

Imagine you’re at a carnival with a $20 bill. You buy a $3 lemonade, and then you play a game that costs $2 per try. After all that, you have $9 left. How many times did you play the game?

Here’s the puzzle: you started with $20, lost $3 right away, and then lost some unknown amount ($2 × number of games). The key is that the order matters—you can’t just subtract $3 and $2 at the same time. Instead, you undo the steps in reverse: first, you add back the $3 to find out how much you spent on games ($9 + $3 = $12). Then, you divide by $2 to find out how many games you played ($12 ÷ $2 = 6 games).

Two-step equations work the same way: they’re like a locked box with two layers. You have to undo the last operation first (usually addition/subtraction), then the first operation (usually multiplication/division), to find the hidden number inside.

Key Vocabulary:
- Equation: A math sentence that says two expressions are equal (e.g., 3x + 5 = 20).
Example: If a pizza costs $12 plus $1.50 per topping, the equation 12 + 1.5t = 18 tells you how many toppings (t) you can get for $18.
- Inverse operation: The "opposite" operation that undoes another (e.g., subtraction undoes addition).
Example: If you multiply a number by 4 and get 28, the inverse operation is dividing by 4 to get 7.
- Solution: The value that makes the equation true.
Example: In 2x – 3 = 7, the solution is x = 5 because 2(5) – 3 = 7.
- Isolate the variable: Getting the variable alone on one side of the equation.
Example: In 4y + 8 = 24, you isolate y by first subtracting 8, then dividing by 4.
Grade 9–12 note: In college algebra, you’ll see equations where isolating the variable isn’t enough (e.g., quadratic equations), and you’ll need factoring or the quadratic formula.

3. Assessment Translation

How this appears on state tests (Grade 6–8):
- Multiple choice: Questions like "Solve for x: 3x – 7 = 14" with distractors that show common errors (e.g., x = 7 from forgetting to divide by 3).
Distractor patterns: - Forgetting to do the inverse operation in reverse (e.g., dividing first, then subtracting).
- Sign errors (e.g., subtracting 7 instead of adding).
- Misapplying the order of operations (e.g., trying to divide 3x – 7 by 3 as a single step).
- Short answer/constructed response: Problems like "A gym membership costs $25 to join and $10 per month. If you’ve paid $85 total, how many months have you been a member? Write an equation and solve it." Proficient response:

Let m = number of months.
Equation: 25 + 10m = 85 Subtract 25 from both sides: 10m = 60 Divide by 10: m = 6 Answer: 6 months.
Developing response: 25 + 10m = 85 10m = 85 – 25 10m = 60 (missing the final step of dividing by 10).
What the teacher looks for: - Correct equation setup.
- One operation at a time, in reverse order.
- Final answer with units (if applicable).

Model proficient response (short answer):
Prompt: Solve for x: 5x + 12 = 47 Response:

Subtract 12 from both sides: 5x = 35 Divide both sides by 5: x = 7 Check: 5(7) + 12 = 35 + 12 = 47 ✓

4. Mistake Taxonomy

Mistake 1: Wrong order of operations
Prompt: Solve 4x – 9 = 15.
Common wrong response:

4x = 15 – 9 4x = 6 x = 6 ÷ 4 → x = 1.5 Why it loses credit: The student divided before adding 9, which doesn’t isolate x correctly.
Correct approach: Add 9 first: 4x = 24 Then divide: x = 6

Mistake 2: Sign errors
Prompt: Solve –2x + 5 = 11.
Common wrong response:

–2x = 11 + 5 –2x = 16 x = –8 (correct answer, but the student wrote x = 8 and forgot the negative sign).
Why it loses credit: The student misapplied the negative sign when dividing.
Correct approach: Subtract 5: –2x = 6 Divide by –2: x = –3

Mistake 3: Misreading the equation
Prompt: A taxi charges a $3 base fee plus $2 per mile. If your fare is $19, how many miles did you travel? Write an equation and solve.
Common wrong response:

Equation: 3 + 2 = 19m Solution: 5 = 19m → m ≈ 0.26 Why it loses credit: The student misplaced the variable (m) and treated the equation as if the $2 per mile was a one-time fee.
Correct approach: Equation: 3 + 2m = 19 Subtract 3: 2m = 16 Divide by 2: m = 8

5. Connection Layer

1. Within math: Two-step equations → Inequalities
   Why? Solving 3x + 4 > 10 uses the same steps as an equation, but the solution is a range of numbers (e.g., x > 2), not just one answer.

2. Across subjects: Two-step equations → Chemistry (balancing reactions)
   Why? In chemistry, you balance equations like 2H₂ + O₂ → 2H₂O by solving for coefficients—just like isolating x, but with molecules instead of numbers.

3. Outside school: Two-step equations → Budgeting apps
   Why? Apps like Mint or YNAB use equations to track spending: Starting balance – expenses + income = new balance. If you know three of those, you can solve for the fourth (e.g., "How much can I spend this month?").

6. The Stretch Question

If you solve 2(x + 3) = 14 two different ways—first by distributing, then by dividing first—which method is "better," and why? Does it depend on the numbers?

Pointer toward the answer: - Distributing first: 2x + 6 = 14 → 2x = 8 → x = 4.
- Dividing first: x + 3 = 7 → x = 4.
Both work, but dividing first is faster here because 14 is divisible by 2. However, if the equation were 2(x + 3) = 15, distributing first would avoid fractions. The "better" method depends on the numbers—but the rule (undoing operations in reverse) stays the same.