K-12 Math (US): 9-12 Algebra K-12 Math Linear Equations Variables on both sides

Study Guide: Linear Equations — Variables on Both Sides

Grade 9–12 | Algebra

1. The Driving Question

You’re playing a video game where two characters start with different amounts of health, and each second, they lose (or gain) health at different rates. How do you figure out exactly when they’ll have the same health—so you can time your attack perfectly? And why can’t you just guess and check your way to the answer?

2. The Core Idea — Built, Not Listed

Imagine two friends, Jamal and Priya, are saving money for concert tickets. Jamal starts with $50 and adds $10 every week. Priya starts with $80 but spends $5 every week on snacks, so her savings shrink. Right now, Jamal has less money, but he’s adding to it while Priya’s savings are draining. At some point, their savings will cross—they’ll have the same amount. That crossing point is what we’re solving for.

To find it, we write two expressions: - Jamal’s savings after x weeks: 50 + 10x
- Priya’s savings after x weeks: 80 – 5x

Set them equal: 50 + 10x = 80 – 5x. Now, we need to isolate x—the number of weeks—by moving all the x terms to one side and the constants to the other. This is like balancing a seesaw: whatever you do to one side, you must do to the other. Subtract 50 from both sides, add 5x to both sides, and you’ll find x = 2. That means in 2 weeks, both will have $70.

Key Vocabulary:
- Variable term: A term with a variable (e.g., 10x or -5x). Example: In the equation 3y + 7 = 2y – 4, 3y and 2y are variable terms.
- Constant term: A term without a variable (e.g., 50 or 80). Example: In 4a – 9 = 15, -9 and 15 are constants.
- College note: In linear algebra, constants become coefficients in systems of equations, and "variable" can refer to entire vectors, not just single letters.
- Solution: The value of the variable that makes the equation true. Example: For 2x + 3 = 7, the solution is x = 2 because plugging it in gives 7 = 7.
- Inverse operation: An operation that "undoes" another (e.g., addition undoes subtraction). Example: To solve x + 5 = 12, subtract 5 from both sides—the inverse of addition.

3. Assessment Translation

How this appears on assessments:
- SAT/ACT: Multiple-choice questions with equations like 3x + 7 = 2x – 5, where you solve for x and match it to an answer choice. Distractors often include: - Sign errors (e.g., x = -12 instead of x = -12/5).
- Forgetting to move all variable terms (e.g., leaving x on both sides).
- Misapplying inverse operations (e.g., dividing before combining like terms).
- AP Algebra/Precalculus: Free-response questions where you must show all steps and justify each move (e.g., "Explain why adding 5x to both sides is necessary"). Rubrics reward: - Correct algebraic manipulation.
- Clear, logical progression (no skipped steps).
- Verification by plugging the solution back into the original equation.

Proficient vs. Developing Responses:
- Proficient: Solves 4x – 3 = 2x + 7 by: 1. Subtracting 2x from both sides: 2x – 3 = 7.
2. Adding 3 to both sides: 2x = 10.
3. Dividing by 2: x = 5.
4. Verifies: 4(5) – 3 = 17 and 2(5) + 7 = 17.
- Developing: Solves 4x – 3 = 2x + 7 but: - Forgets to subtract 2x from both sides, leaving 4x – 3 = 7.
- Or subtracts 2x only from the right side, writing 4x – 3 = 7 (loses credit for imbalance).

Model Proficient Response (Short Answer):
Prompt: Solve for x: 5x + 12 = 3x – 6. Show all steps.
Response: 1. Subtract 3x from both sides: 2x + 12 = -6.
2. Subtract 12 from both sides: 2x = -18.
3. Divide by 2: x = -9.
Check: 5(-9) + 12 = -33 and 3(-9) – 6 = -33. ✔️

4. Mistake Taxonomy

Mistake 1: The "One-Sided Move"
- Prompt: Solve 6x + 4 = 2x + 20.
- Wrong Response: 6x = 2x + 16 (subtracted 4 only from the left side).
- Why It Loses Credit: The equation is no longer balanced. The student ignored the golden rule: whatever you do to one side, do to the other.
- Correct Approach: 1. Subtract 2x from both sides: 4x + 4 = 20.
2. Subtract 4 from both sides: 4x = 16.
3. Divide by 4: x = 4.

Mistake 2: The "Sign Flip"
- Prompt: Solve -3x + 5 = 7x – 15.
- Wrong Response: -3x – 7x = -15 – 5 → -10x = -20 → x = 2.
- Why It Loses Credit: The student moved 7x to the left but forgot to change its sign (should be -7x). This flips the solution’s sign.
- Correct Approach: 1. Add 3x to both sides: 5 = 10x – 15.
2. Add 15 to both sides: 20 = 10x.
3. Divide by 10: x = 2.

Mistake 3: The "Premature Division"
- Prompt: Solve 4x + 8 = 2x + 10.
- Wrong Response: 4x = 2x + 2 → 2x = 2 → x = 1.
- Why It Loses Credit: The student divided 4x and 2x by 2 before moving terms, which is unnecessary and can lead to errors. It’s clearer to combine like terms first.
- Correct Approach: 1. Subtract 2x from both sides: 2x + 8 = 10.
2. Subtract 8 from both sides: 2x = 2.
3. Divide by 2: x = 1.

5. Connection Layer

1. Within Math: Linear equations → Systems of equations.
   Why it matters: Solving y = 2x + 3 and y = -x + 1 for their intersection point is just solving a linear equation with variables on both sides (2x + 3 = -x + 1). The logic is identical—you’re finding where two lines cross.

2. Across Subjects: Algebra → Physics (kinematics).
   Why it matters: In physics, you might set two distance equations equal to find when two objects meet (e.g., d₁ = 5t + 20 and d₂ = 10t). The algebra is the same as solving 5t + 20 = 10t—just with t (time) instead of x.

3. Outside School: Budgeting → Break-even points.
   Why it matters: If you’re comparing two phone plans—one with a high upfront cost but low monthly fees (50 + 10x) and another with no upfront cost but higher fees (30x)—solving 50 + 10x = 30x tells you how many months it takes for the plans to cost the same. Now you can decide which is cheaper for you.

6. The Stretch Question

What if the equation has variables on both sides and parentheses—like 3(x + 2) = 2(x – 5) + 4? Does the order in which you distribute vs. move terms change the answer? Try solving it two ways: (1) distribute first, then move terms, and (2) move terms first, then distribute. Do you get the same solution? Why or why not?

Pointer Toward the Answer: The order doesn’t change the solution, but it can change how messy the steps get. Distributing first often simplifies the equation faster (e.g., 3x + 6 = 2x – 10 + 4 → 3x + 6 = 2x – 6), but moving terms first (e.g., 3(x + 2) – 2x = -10 + 4) can sometimes avoid negative numbers. The key is that algebraic operations are reversible—as long as you follow the rules, you’ll end up in the same place. Try it with 4(2x – 1) = 3(x + 5) to see for yourself.