Independent dMAT preparation
Mathematical Equations: simplify before calculating
Solve original equation systems with elimination, verify both values and avoid sign and constraint mistakes.
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A system is a collection of equations that must all hold at once. Your first task is to find the relationship that removes a variable cheaply. The most conspicuous equation is not always the best place to start.
The official preparation PDF specifies integer values from 1 to 20 and one valid solution for each letter.
Choose substitution or elimination
Use substitution when one equation already isolates a letter: if Y = 3X, replace Y with 3X everywhere it occurs. Use elimination when two equations share a coefficient: subtracting X + Y from 2X + Y removes Y in one step.
Before doing arithmetic, decide what will disappear. After simplifying, solve one letter, recover the other values and check the original equations. The checks should use the original statements rather than your rearranged version, because a rearrangement error can otherwise survive.
Worked original system
These are original Aptitrail teaching examples. They are independent of official exercises and are not an official difficulty benchmark.
Solve X + Y = 17 and 2X + Y = 26. What is X? Keep both equations true.
Reveal the elimination and answer
Subtract the whole first equation from the second: (2X + Y) − (X + Y) = 26 − 17, so X = 9. Then Y = 17 − 9 = 8. Verify both original equations: 9 + 8 = 17 and 2 × 9 + 8 = 26. Both values are integers within 1–20.
The useful feature is the equal coefficient of Y. Subtraction removes Y immediately, avoiding a long substitution chain.
Know what the subtraction means
The entire left side is subtracted: 2X + Y − X − Y. The right sides are also subtracted in the same order: 26 − 17. The result is X = 9. Reversing only one side would create X = −9, which fails both the arithmetic and the allowed domain.
If equal coefficients do not appear, look for a cheap scaling. For example, coefficients 1 and 2 can be matched by doubling an entire equation. Scale its right side too. Avoid multiplying several equations before deciding what variable you intend to remove.
Common mistakes
- Checking one line only: a value satisfying one equation may contradict another.
- Dropping a minus sign: subtract each term inside the parentheses, including negative terms.
- Assuming letters differ: use the stated constraints; distinct letters do not automatically imply different numbers.
- Returning every value: read which letter the question requests, even after finding the whole system.
Try a second system
P + Q = 15 and P + 2Q = 22. Find Q, then use P as a check.
Reveal the second system’s answer
Subtract the first equation from the second: Q = 22 − 15 = 7. Then P = 15 − 7 = 8. Verify 8 + 7 = 15 and 8 + 2 × 7 = 22. Both are allowed integers.
For your next session, classify each error as method selection, arithmetic, constraint checking or response selection. Rehearse the failed step on a fresh system before attempting to reduce your time.
By the Aptitrail editorial team. AI-assisted original teaching examples checked with automated tests. These checks do not constitute an independent expert review. Report a correction.