Substitution & elimination — Class 10 CBSE Mathematics MCQs with Solutions
Free Class 10 CBSE Mathematics Substitution & elimination MCQs with step-by-step solutions (8 questions). Part of Pair of Linear Equations in Two Variables. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Substitution & elimination · easy · theory
When solving a system by substitution method, what is the key principle?
A. Express one variable in terms of another, then substitute to get a single equation in one variable ✓ Correct
B. Multiply both equations by constants and subtract to eliminate all variables
C. Graph both equations and find where they intersect on paper
D. Use the ratio of coefficients to find x and y directly
Solution: The substitution method works by reducing two equations in two variables to a single equation in one variable. Once we find that variable, we substitute back to find the other. This principle applies because if both equations are satisfied by the solution, expressing one variable from the first will work in the second.
Q2 — Substitution & elimination · easy · theory
Why is the elimination method sometimes preferred over substitution?
A. It always gives integer solutions
B. It directly removes one variable by adding or subtracting, avoiding fraction manipulation ✓ Correct
C. It requires fewer steps than any other method
D. It only works for consistent systems
Solution: Elimination is efficient when substitution would introduce fractions. For example, if coefficients are already suitable for multiplying and subtracting, elimination keeps calculations cleaner. The choice depends on the structure of the equations, not on the system being consistent or inconsistent.
Q3 — Substitution & elimination · medium · theory
Consider two systems: System A has one unique solution, and System B has infinitely many solutions. Which statement correctly compares their consistency?
A. System A is consistent and dependent; System B is consistent and independent
B. System A is consistent and independent; System B is consistent and dependent ✓ Correct
C. System A is inconsistent; System B is consistent
D. Both systems are inconsistent but for different reasons
Solution: A consistent system has at least one solution. It is independent if the solution is unique (two different lines, one point of intersection) or dependent if there are infinitely many solutions (same line, all points satisfy both). System A represents independent equations (different lines), while System B represents dependent equations (same line).
Q4 — Substitution & elimination · medium · theory
Why are two systems of equations considered equivalent?
A. If they use the same variables, regardless of coefficients
B. If they have the same solution set (same solution(s) for x and y) ✓ Correct
C. If both systems can be solved by the same method
D. If they have the same number of equations
Solution: Two systems are equivalent if every solution to one system is a solution to the other, and vice versa. For example, the system $x + y = 5, x - y = 1$ is equivalent to $2x + 2y = 10, x - y = 1$ (we multiplied the first equation by 2). The solution set is identical: $(3, 2)$.
Q5 — Substitution & elimination · medium · theory
When would the elimination method create fractions if we try to eliminate $x$ from a system, but not when eliminating $y$?
A. When the coefficients of $x$ are coprime and those of $y$ share a common factor ✓ Correct
B. When the system is inconsistent
C. When both variables have the same coefficient
D. This scenario never happens in real problems
Solution: In elimination, we multiply equations to make coefficients of a variable equal, then add or subtract. For example, $2x + 3y = 7$ and $5x + 4y = 13$: to eliminate $x$, we need LCM(2,5) = 10, requiring multiplication by 5 and 2. But LCM(3,4) = 12 might not require as many steps. Method selection depends on the coefficient structure to minimize fractions.
Q6 — Substitution & elimination · medium · theory
Two students solve the same system. Student A uses substitution and gets $(x, y) = (2, 3)$. Student B uses elimination and gets $(x, y) = (2, 3)$. What does this tell us?
A. Only one method is correct, and the other made an error
B. Both methods are valid and converge to the same solution for this system ✓ Correct
C. The system must have infinitely many solutions
D. The system is inconsistent despite their answers
Solution: Both substitution and elimination are equally valid algebraic methods for solving linear systems. They should always give the same solution for a given consistent system. Different methods are chosen for efficiency, not accuracy. This agreement confirms the solution is correct.
Q7 — Substitution & elimination · hard · theory
Consider two systems: System 1 has equations 2x + 3y = 11 and 3x - 2y = 4; System 2 has equations 4x + 6y = 22 and 3x - 2y = 4. Why is System 1 different from System 2 in terms of solutions?
A. System 1 has one solution; System 2 has infinitely many because the first equation of System 2 is double the first of System 1
B. Both systems have unique solutions, but the solutions are different numbers ✓ Correct
C. System 1 is inconsistent; System 2 is consistent
D. System 2 has no solution because two equations with the same slope are parallel
Solution: In System 1, the first and second equations have independent constraints with different slopes. In System 2, the first equation is exactly 2 times the first equation from System 1, so it's not a new constraint. However, it still pairs with the second equation to give a unique solution different from System 1.
Q8 — Substitution & elimination · hard · theory
A system is transformed by multiplying the first equation by a non-zero constant k while keeping the second equation unchanged. What property is preserved?
A. The solution set remains identical because multiplying by a non-zero constant does not change the line it represents ✓ Correct
B. The determinant changes by a factor of k
C. The intersection point moves but the number of solutions stays the same
D. The new system becomes inconsistent if k is negative
Solution: Multiplying an equation by a non-zero constant is an equivalent transformation that represents the same line geometrically. Therefore, the solution set is invariant. This is the theoretical basis for the elimination method.