Systems of Equations
When a problem has two unknowns, one equation is never enough — you need a second, independent relationship, and the pair together pin down the values. That is a system of equations, and the GRE loves them because they model realistic scenarios: two ticket prices, a mixture of two coins, a boat with and against a current. Two clean tools solve almost everything: substitution (solve one equation for a variable and feed it into the other) and elimination (scale the equations so one variable cancels when you add). A sharp GRE habit is realising you often need only a combination like x + y, not x and y separately — and adding or subtracting the two equations can hand you that combination in one step. This chapter covers both methods, the special cases of no or infinite solutions, and translating word problems into a solvable pair.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- If the question wants x + y or x − y, just add or subtract the two equations — you may never need x and y alone.
- Use substitution when a coefficient is 1; use elimination when it is not. Picking the right tool saves a minute.
- Line up the equations vertically before eliminating — misaligned terms cause most cancellation errors.
- One equation is a multiple of the other ⇒ infinitely many solutions; same variable side but different constant ⇒ no solution.
- In word problems, one equation is usually a count (totals) and the other a value (money, legs, weight). Write both before solving.
- Always re-read the final question — GRE answers are frequently the sum or difference, not the individual variable you solved for first.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Substituting back into the same equation you just rearranged, which collapses to a useless identity.
- Adding equations when the coefficients need subtracting (or vice versa), so nothing actually cancels.
- Forgetting to scale BOTH sides of an equation when multiplying to match coefficients.
- Declaring "no solution" when a true 0 = 0 actually means infinitely many (and vice versa).
- Solving for one variable and stopping when the question asked for the other, or for a combination.
- Mistranslating a word problem — mixing up which quantity is the count and which is the value.
📈 GRE exam insight & question patterns
Quantitative Comparison — given x + y = 10 and x − y = 2, Quantity A is x and Quantity B is y. Which is greater?
Add: 2x = 12, x = 6; then y = 4. Quantity A is greater.
A system has 2x + 4y = 10 and x + 2y = 5. A QC asks to compare x with 3. What is the answer?
The second equation is half the first — infinitely many solutions, so x is not fixed. The answer is "cannot be determined".
Numeric Entry — if a + b = 15 and a − b = 3, enter the value of ab.
a = 9, b = 6, so ab = 54. Add and subtract the equations to get a and b.
A word problem gives two ticket types and two totals. What is the reliable setup?
One equation for the number of tickets and one for total revenue; then eliminate or substitute. Two unknowns always need two independent equations.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Systems of Equations when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
The two core methods
| Substitution | solve one eq for a variable, sub into the other |
|---|---|
| Elimination | scale so one variable cancels, then add |
| Add the equations | often yields x + y directly |
| Subtract the equations | often yields x − y directly |
How many solutions
| One solution | lines cross — different slopes |
|---|---|
| No solution | parallel — same slope, different constant |
| Infinite solutions | same line — one eq is a multiple of the other |