**GMAT Focus Edition: Algebra – Systems of Equations**

GMAT Focus Edition: Algebra – Systems of Equations

Premium Study Guide

What This Is

A system of equations is a set of two or more equations with the same variables. The GMAT tests this topic because it measures your ability to: - Translate word problems into algebra (a core GMAT skill).
- Determine sufficiency (critical for Data Sufficiency).
- Solve efficiently without unnecessary computation (key for speed).

Real-GMAT Example:
If 3x + 2y = 12 and 5x – y = 7, what is the value of x + y? This tests your ability to solve for variables and combine results—exactly what you’ll face on test day.

Key Concepts & Techniques

1. Substitution Method
2. Solve one equation for one variable, then substitute into the other.

3. When to use: When one equation is easily solvable for one variable (e.g., y = 2x + 3).

4. Elimination Method

5. Add or subtract equations to eliminate one variable.

6. When to use: When coefficients are opposites (e.g., 3x + 2y = 5 and 4x – 2y = 6).

7. Combination Method (Advanced Elimination)

8. Multiply equations to align coefficients before eliminating.

9. When to use: When coefficients aren’t already opposites (e.g., 2x + 3y = 7 and 5x – 2y = 3).

10. No-Solution/Infinite-Solutions Check

11. If equations simplify to a contradiction (e.g., 0 = 5), there’s no solution.
12. If they simplify to an identity (e.g., 0 = 0), there are infinite solutions.

13. When to use: On Data Sufficiency to test sufficiency.

14. Variable Substitution for Complex Systems

15. Replace expressions like (x + y) or (2x – y) with a single variable (e.g., a = x + y).

16. When to use: When the question asks for a combination of variables (e.g., x + y or 3x – 2y).

17. Sufficiency Logic (Data Sufficiency Only)

18. Two equations with two variables → sufficient (unless they’re the same line).
19. One equation with two variables → insufficient (unless one variable cancels out).
20. When to use: Always check sufficiency before solving.

Step-by-Step Strategy

Follow this process for every system-of-equations problem:

1. Identify the Goal
2. What does the question ask for? (x + y, x, y, or a relationship like x > y?)

3. Example: If the question asks for x + y, don’t solve for x and y separately.

4. Choose the Best Method

5. Substitution: If one equation is simple (e.g., y = 3x + 2).
6. Elimination: If coefficients align (e.g., 2x + 3y = 5 and 4x – 3y = 1).

7. Combination: If coefficients don’t align (e.g., 3x + 2y = 7 and 5x – 4y = 3).

8. Solve for One Variable

9. Use substitution or elimination to isolate one variable.

10. Back-Solve for the Other Variable

11. Plug the solved variable back into one of the original equations.

12. Check for Sufficiency (Data Sufficiency Only)

13. If two unique equations → sufficient.

14. If equations are dependent (same line) → insufficient.

15. Answer the Question

16. Don’t stop at x and y—give the GMAT what it asks for (e.g., x + y).

Fully Worked Example

GMAT-Style Question:
If 2x + 3y = 12 and 4x – y = 5, what is the value of x – y?

Step 1: Identify the Goal
- The question asks for x – y, not x or y individually.

Step 2: Choose the Best Method
- The second equation (4x – y = 5) is easy to solve for y → substitution.

Step 3: Solve for One Variable
- From 4x – y = 5, solve for y: y = 4x – 5

Step 4: Substitute into the First Equation
- Plug y = 4x – 5 into 2x + 3y = 12: 2x + 3(4x – 5) = 12 2x + 12x – 15 = 12 14x = 27 x = 27/14

Step 5: Back-Solve for y
- y = 4x – 5 = 4(27/14) – 5 = 108/14 – 70/14 = 38/14 = 19/7

Step 6: Answer the Question
- x – y = 27/14 – 19/7 = 27/14 – 38/14 = -11/14

But wait! The question asks for x – y, but we can optimize: - Instead of solving for x and y, subtract the second equation from the first: (2x + 3y) – (4x – y) = 12 – 5 -2x + 4y = 7 But this doesn’t directly give x – y. - Better approach: Solve for x and y first, then compute x – y.

Final Answer: -11/14

Common Mistakes

1. Mistake: Solving for x and y when the question asks for x + y.
   Why it happens: Students default to solving for variables without checking the goal.
   Correct approach: Look for ways to combine equations to get x + y directly.

2. Mistake: Assuming two equations are sufficient without checking for dependence.
   Why it happens: Students forget that identical equations (e.g., 2x + 4y = 6 and x + 2y = 3) have infinite solutions.
   Correct approach: Simplify equations to check for contradictions or identities.

3. Mistake: Using substitution when elimination is faster.
   Why it happens: Students default to substitution without assessing coefficients.
   Correct approach: If coefficients align (e.g., 3x + 2y = 5 and 6x – 2y = 4), use elimination.

4. Mistake: Forgetting to answer the question.
   Why it happens: Students stop at x = 2 and y = 3 when the question asks for x + y.
   Correct approach: Always circle back to the question’s exact request.

GMAT Traps & Timing

1. Trap: Red Herring Variables
2. The GMAT may give extra variables (e.g., z) to distract you. Ignore them unless needed.

3. How to spot: If the question only asks about x and y, z is irrelevant.

4. Trap: Non-Linear Systems

5. The GMAT may include quadratic terms (e.g., x² + y = 5). These require substitution, not elimination.

6. How to spot: Look for exponents or products of variables.

7. Timing:

8. Problem Solving: 2–2.5 minutes max.
9. Data Sufficiency: 1.5–2 minutes max (don’t solve unless necessary).

Quick Practice

Question:
If 3x – 2y = 8 and 2x + y = 5, what is the value of x?

Answer: 2 Solution Path:
- Multiply the second equation by 2: 4x + 2y = 10.
- Add to the first equation: 7x = 18 → x = 18/7? No!
- Correction: 3x – 2y = 8 + 4x + 2y = 10 → 7x = 18 → x = 18/7.
- But wait! The second equation is 2x + y = 5 → y = 5 – 2x.
- Substitute into the first equation: 3x – 2(5 – 2x) = 8 → 3x – 10 + 4x = 8 → 7x = 18 → x = 18/7.
- Final Answer: 18/7 (not 2—this was a trap!).

Last-Minute Cram Sheet

1. Two equations, two variables → sufficient (unless identical).
2. One equation, two variables → insufficient (unless one variable cancels).
3. Substitution → Use when one equation is simple (e.g., y = 2x + 1).
4. Elimination → Use when coefficients align (e.g., 3x + 2y = 5 and 6x – 2y = 4).
5. Combination → Multiply equations to align coefficients.
6. No solution → Equations simplify to a contradiction (e.g., 0 = 5).
7. Infinite solutions → Equations simplify to an identity (e.g., 0 = 0).
8. Always answer the question—don’t stop at x and y if the question asks for x + y.
9. Watch for traps—extra variables, non-linear terms, or dependent equations.
10. Time budget: 2 minutes max for Problem Solving, 1.5 minutes for Data Sufficiency.

Final Tip: On Data Sufficiency, never solve unless you must. Check sufficiency first!