Practice Questions
Concept Explanation
Simultaneous equations are two or more equations with the same variables, solved together for a unique (or consistent) solution set.
Methods
- Substitution: Express one variable in terms of the other and substitute.
- Elimination: Add or subtract equations (or multiply first) to eliminate one variable.
Example
x + y = 10 and x − y = 2. Add: 2x = 12, x = 6. Then y = 4.
Common Trap
Check the solution in both original equations. Arithmetic slips in elimination are common.
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Q1 EasyIf x + y = 10 and x − y = 2, what is the value of x?ExplanationAdding: 2x = 12, so x = 6.
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Q2 EasyIf x + y = 10 and x − y = 2, what is the value of y?Explanationx = 6, so y = 10 − 6 = 4.
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Q3 EasyFor the simultaneous equations 2x + y = 11 and x + y = 7, what is the value of x?ExplanationSubtract: x = 4.
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Q4 EasyFor the simultaneous equations 2x + y = 11 and x + y = 7, what is the value of y?Explanationx = 4, so y = 7 − 4 = 3.
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Q5 EasyIf x + y = 12 and 2x − y = 3, what is the value of 2x + 2y?Explanation2x + 2y = 2(x + y) = 2 × 12 = 24.
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Q6 MediumIf x + 2y = 8 and 2x + y = 7, what is the value of x + y?ExplanationFrom elimination: x = 2, y = 3. x + y = 5.
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Q7 MediumFor the system 3x + 2y = 16 and 2x + 3y = 14, what is the value of x?ExplanationElimination gives x = 4, y = 2.
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Q8 MediumFor the system 3x + 2y = 16 and 2x + 3y = 14, what is the value of y?ExplanationElimination gives x = 4, y = 2.
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Q9 MediumIf 2x + 3y = 13 and x + y = 5, what is the value of y?ExplanationSubstitute x = 5 − y: 2(5 − y) + 3y = 13 gives y = 3.
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Q10 MediumFor the system 5x + 2y = 19 and 3x + 2y = 13, what is the value of x?ExplanationSubtract: 2x = 6, so x = 3.
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Q11 HardThere are two examinations rooms A and B. If 10 students are sent from A to B, then the number of students in each room is the same. If 20 candidates are sent from B to A, then the number of students in A is double the number of students in B. The number of students in room A is:Explanationx − 10 = y + 10 and x + 20 = 2(y − 20) give x = 100, y = 80.
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Q12 HardIf x/3 + y/2 = 5 and x + y = 12, what is the value of x?ExplanationMultiply first by 6: 2x + 3y = 30. With x + y = 12: x = 6, y = 6.
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Q13 HardIf 3x + 4y = 20 and 2x + 3y = 14, what is the value of xy?ExplanationSolution: x = 4, y = 2. xy = 8.
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Q14 HardIf x − y = 4 and x + y = 12, what is the value of x² − y²?Explanationx² − y² = (x + y)(x − y) = 12 × 4 = 48.
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DS1 Easy
What are the values of x and y that satisfy the system x + y = 10 and x − y = 2?
Statement Ix + y = 10.
Statement IIx − y = 2.
ExplanationNeed both equations to determine x = 6 and y = 4 uniquely. -
DS2 Easy
What is the value of x in the system of equations 2x + y = 11 and x + y = 7?
Statement I2x + y = 11.
Statement IIx + y = 7.
ExplanationNeed both: subtract to get x = 4.
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DS3 Medium
What is the value of y in the system 3x + 2y = 16 and 2x + 3y = 14?
Statement I3x + 2y = 16.
Statement II2x + 3y = 14.
ExplanationTwo independent linear equations in two unknowns; need both. -
DS4 Medium
What is the value of x + y if x + 2y = 8 and 2x + y = 7?
Statement Ix + 2y = 8.
Statement II2x + y = 7.
ExplanationNeed both: x = 2, y = 3, so x + y = 5.