# Systems of Equations Problems — Free Timed Practice | Quant Daily > Practise systems of equations problems free and timed: eliminate a variable, back-substitute, and check both equations. Worked solutions every time, no sign-up. Source: https://quantdaily.app/drill-types/systems-of-equations --- Linear systems ## Systems of equations problems — eliminate, then answer the right variable Systems of equations problems give two linear equations in two unknowns. Elimination is usually fastest: add or subtract the equations so one variable cancels, solve for the survivor, then substitute back for the other. A pair like x + y = 12 and x − y = 4 adds to 2x = 16, so x is 8 and y is 4. When the sum and the difference are both given, the system is already primed for elimination and a single addition solves it. What costs marks is stopping too early: you find x, and the question wants y, or the larger of the two, or their product. Both values plus their combinations sit on the option list. **Try it against the clock.** Linear Systems runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading. [Start a timed Linear Systems drill →](https://quantdaily.app/?drill=linear_systems&src=seo-systems-of-equations) ### How to solve systems of equations problems by elimination - Line the equations up Write both in the same variable order. With x + y = 12 and x − y = 4, the y terms are already set up to cancel on addition. - Add or subtract to eliminate Adding gives 2x = 16, so x = 8. If the signs match instead of opposing, subtract rather than add. - Scale first when nothing cancels For 2x + 3y = 18 and x + y = 7, multiply the second by 2 to get 2x + 2y = 14, then subtract to leave y = 4. - Substitute back and check both equations x = 8 and y = 4 satisfy both the sum and the difference. Checking against only the equation you used cannot catch an arithmetic slip. ### Choosing the move | The system | Do this | | --- | --- | | x + y = 12, x − y = 4 | Add — y cancels | | x + y = 12, x + 2y = 16 | Subtract — x cancels | | 2x + 3y = 18, x + y = 7 | Double the second, then subtract | | Neither lines up | Scale one equation to match a coefficient | ### 3 sample systems of equations problems, with worked solutions Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers. #### Question 1 The sum of two numbers is 20 and their difference is 8. Let x be the larger and y the smaller. What is the smaller number? - 6 - 5 - 14 - 20 - 3 - x + y = 20, x − y = 8. - 2x = 28 → x = 14, y = 6. - Answer: 6. #### Question 2 x + 2y = 5. 2x + 2y = 8. What is x? - 3 - 8 - 5 - 4 - 1 - Determinant Δ = 1×2 − 2×2 = -2. - x = (5×2 − 2×8) ÷ Δ = 3. - Answer: 3. #### Question 3 The sum of two numbers is 13 and their difference is 5. Let x be the larger and y the smaller. What is the smaller number? - 2 - 9 - 1 - 13 - 4 - x + y = 13, x − y = 5. - 2x = 18 → x = 9, y = 4. - Answer: 4. ### Common mistakes this drill is built from The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks. #### Adding when you should subtract Addition cancels a variable only when its signs oppose. With matching signs it doubles the term instead of removing it. #### Scaling one side of an equation only Multiplying through means every term, the constant included. Missing the right-hand side changes the equation entirely. #### Answering with the variable you happened to solve first Elimination gives you x first only because of how you set it up. The question may want y, the larger value, or the difference. ### Why your answer came out wrong The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here. #### x + y = 12 and x − y = 4 — is the answer 8? 8 is x. y is 4, found by substituting back into the first equation. Which one counts depends on the question, and some versions ask for the larger value or the product instead, so re-read before selecting. #### For x + y = 12 and x + 2y = 16, should I add the equations? Subtract instead. Both x terms have the same sign, so adding gives 2x + 3y = 28 and eliminates nothing. Subtracting leaves y = 4, and substituting back gives x = 8. Addition only cancels when the signs oppose. ### FAQ #### When is elimination better than substitution? When a variable already has matching or opposite coefficients, as in a sum-and-difference pair, elimination solves the system in a single line. Substitution is the better choice when one equation is already arranged with a variable by itself. #### What if no variable cancels directly? Scale one equation so a coefficient matches. For 2x + 3y = 18 and x + y = 7, doubling the second gives 2x + 2y = 14, and subtracting leaves y = 4. Remember to multiply every term, including the number on the right. #### How do I check my solution quickly? Substitute both values into both original equations. Checking only the equation you solved from will confirm an error rather than catch it, and the second check takes a few seconds against a system this small. ### Practise the next topic - [All 51 drill types](https://quantdaily.app/drill-types) — the shared daily cycles 36 of 51, with every published time limit. - [Algebra word problems — name the unknown before you translate](https://quantdaily.app/drill-types/algebra-word-problems) — algebra word problems, timed. - [Inequalities questions — isolate x](https://quantdaily.app/drill-types/inequalities) — inequalities questions, timed. - [Speed tricks](https://quantdaily.app/drill-types) — the seven mental-math shortcuts that make a 15-second answer possible. - [How to improve calculation speed](https://quantdaily.app/calculation-speed) — the six-week arc these drills fit into. Quant Daily is free, needs no account, and is not affiliated with ETS or GMAC. GRE/GMAT-style practice only.