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Pair of Linear Equations in Two Variables — Class 10 CBSE Mathematics MCQs with Solutions
Free Class 10 CBSE Mathematics Pair of Linear Equations in Two Variables MCQs with step-by-step solutions covering Graphical method, Substitution & elimination, Cross-multiplication, Word problems. Practise online on Prepizo — no login needed.
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Sample questions with solutions
Q1 — Graphical method · easy · theory
If two linear equations have the condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, what does this tell us geometrically?
A. The lines are parallel and never intersect
B. The lines are coincident (the same line) and have infinite solutions ✓ Correct
C. The lines intersect at exactly one point
D. The lines are perpendicular to each other
Solution: When all three ratios are equal, the equations represent the same line (coincident). Every point on this line satisfies both equations, giving infinitely many solutions. Geometrically, the two equations are just scalar multiples of each other.
Q2 — Graphical method · easy · theory
A system of two linear equations has a unique solution. What must be true about the geometric representation?
A. The two lines are parallel
B. The two lines are the same (coincident)
C. The two lines intersect at exactly one point ✓ Correct
D. The lines are perpendicular at the origin
Solution: A unique solution corresponds to the point where the two lines intersect. This occurs when $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, meaning the slopes are different and the lines are neither parallel nor coincident.
Q3 — 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.
Q4 — Graphical method · easy · theory
For the system $2x + 3y = 7$ and $4x + 6y = 14$, which method would be MOST appropriate to recognize the relationship without full solving?
A. Graphical method to plot both lines
B. Observe the ratio condition to recognize these are proportional equations (coincident lines) ✓ Correct
C. Substitution method to find x first
D. Cross-multiplication to get the unique solution
Solution: Notice that the second equation is exactly 2 times the first equation: $4x + 6y = 2(2x + 3y)$ and $14 = 2(7)$. The ratio condition $\frac{2}{4} = \frac{3}{6} = \frac{7}{14} = \frac{1}{2}$ immediately tells us this system has infinitely many solutions without solving anything.
Q5 — 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.
Q6 — Graphical method · easy · theory
If the ratio $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, what can we conclude about the system without solving?
A. The system has no solution (inconsistent)
B. The system has infinitely many solutions
C. The system has exactly one unique solution ✓ Correct
D. We cannot determine without knowing the constant terms
Solution: When the ratios of coefficients of $x$ and $y$ are different ($\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$), the lines have different slopes, meaning they are not parallel and not coincident. Therefore, they must intersect at exactly one point, guaranteeing a unique solution.
Q7 — Word problems · easy · theory
A student sets up a word problem and gets the equations $3x + 2y = 12$ and $6x + 4y = 24$. Before solving, what does the student notice?
A. The equations are independent and will give different values
B. One equation is a multiple of the other, so they represent the same constraint ✓ Correct
C. The equations are contradictory and have no solution
D. A unique solution exists but must be found by cross-multiplication
Solution: The second equation is exactly $2 \times$ the first equation. This means they represent the same linear relationship, not two different constraints. In the context of a word problem, this often indicates the problem has been stated redundantly, and there are infinitely many solutions (or the problem is under-determined).
Q8 — Cross-multiplication · hard · theory
Prove why the cross-multiplication method works conceptually. Which principle underlies this technique?
A. Cramer's rule, using determinants of the coefficient and augmented matrices ✓ Correct
B. The graphical intersection of two lines always gives integer coordinates
C. Substitution is equivalent to multiplication of equation coefficients
D. The order of equations determines the sign of the solution
Solution: Cross-multiplication is an application of Cramer's rule, which states that for a system with non-zero determinant, the solution can be expressed using determinants. This determinant-based approach is both theoretically sound and practically efficient for hand calculation.
Q9 — 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.
Q10 — 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.
Q11 — Graphical method · hard · theory
Given a system where the ratio of a coefficients equals the ratio of b coefficients but not the ratio of c coefficients, explain why checking these ratios (without solving) guarantees no solution exists.
A. Because equal slopes with different y-intercepts mean parallel lines that never intersect, making the system inconsistent ✓ Correct
B. Because the determinant is always positive, which precludes solutions
C. Because the system is always dependent when ratios are equal
D. Because constant terms with unequal ratios cancel out both variables
Solution: When the ratios of x and y coefficients are equal, the lines have the same slope. But if the constant ratio is different, the y-intercepts differ. This means parallel lines with no intersection, making the system inconsistent.
Q12 — Graphical method · hard · theory
A student claims: 'If a system has infinitely many solutions, then one equation must be a scalar multiple of the other.' Is this claim true and why?
A. True. Infinitely many solutions occur when both equations represent the same line, which means one must be a scalar multiple of the other ✓ Correct
B. False. Infinitely many solutions can occur even when equations are completely different
C. True, but only if the constant term is zero in both equations
D. False, because infinitely many solutions means the lines are perpendicular
Solution: For infinitely many solutions, all three ratios (coefficients of x, coefficients of y, and constants) must be equal. This means the first equation equals k times the second equation for some non-zero k, representing the same line.
Q13 — Word problems · hard · theory
In a word problem, a student solves the system x + y = 50 and 2x + y = 80, finding x = 30, y = 20. What is the critical reasoning step before accepting this solution?
A. Check that both x and y are positive and contextually meaningful (30 items and 20 items make sense) ✓ Correct
B. Verify that the system has infinitely many solutions
C. Confirm that the two equations are parallel
D. Ensure the solution uses only integer coordinates
Solution: Mathematical solutions must be checked against the real-world context. A negative number of items is algebraically valid but contextually impossible. The reasoning connects the abstract solution back to the problem requirements.
Q14 — Cross-multiplication · hard · theory
Why does the condition that the determinant of the coefficient matrix is non-zero guarantee that a linear system has a unique solution?
A. Because a non-zero determinant ensures the coefficient matrix is invertible, allowing unique solution ✓ Correct
B. Because it ensures the lines are always perpendicular
C. Because it makes all solutions positive
D. Because it prevents constant terms from affecting the solution
Solution: A non-zero determinant means the coefficient matrix is invertible in linear algebra terms. This is equivalent to the slopes of the lines being different, guaranteeing exactly one intersection point.
Q15 — Graphical method · hard · theory
Two lines intersect at point (2, 3). Based solely on this geometric fact, can we determine whether the system is consistent or inconsistent?
A. No, because knowing the intersection point tells us nothing about consistency
B. Yes, it is consistent because the lines have at least one common point at (2, 3) ✓ Correct
C. It depends on whether the lines are horizontal or vertical
D. Inconsistent systems cannot have intersecting lines, so this is impossible
Solution: A consistent system has at least one solution. If two lines intersect at a point, that point is a solution. Therefore, the system is consistent. This demonstrates the connection between geometry and algebra.
Q16 — 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).
Q17 — Cross-multiplication · medium · theory
Two lines are described as $L_1: a_1x + b_1y = c_1$ and $L_2: a_2x + b_2y = c_2$. If the determinant $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \neq 0$, what is guaranteed about the system?
A. The lines are parallel and have no solution
B. The lines are coincident with infinitely many solutions
C. The lines intersect at exactly one unique point ✓ Correct
D. The solution involves only positive integers
Solution: A non-zero determinant of the coefficient matrix means the coefficient matrix is invertible, which is equivalent to the condition $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$. This guarantees the lines are not parallel and not coincident, so they intersect at exactly one point (unique solution).
Q18 — 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: $(3, 2)$.
Q19 — Graphical method · medium · theory
A system of equations is said to be inconsistent. What does this mean geometrically?
A. The two lines intersect at the origin
B. The two lines have no point in common (they are parallel) ✓ Correct
C. The two lines are perpendicular
D. The two lines pass through the same three points
Solution: An inconsistent system has no solution, meaning the equations cannot be satisfied simultaneously. Geometrically, this occurs when the lines are parallel: they run side by side but never meet. Algebraically, this is detected by the ratio condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
Q20 — 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.
Q21 — Graphical method · medium · theory
Consider the system $kx + 3y = 7$ and $6x + 9y = 21$. For what value of $k$ does this system have infinitely many solutions?
A. $k = 2$ ✓ Correct
B. $k = 3$
C. $k = 6$
D. $k = 9$
Solution: For infinitely many solutions, the equations must be proportional: $\frac{k}{6} = \frac{3}{9} = \frac{7}{21}$. Simplifying: $\frac{3}{9} = \frac{1}{3}$ and $\frac{7}{21} = \frac{1}{3}$. So $\frac{k}{6} = \frac{1}{3}$, giving $k = 2$. This makes the second equation exactly 3 times the first.
Q22 — 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.
Q23 — Cross-multiplication · medium · theory
The cross-multiplication formula for solving $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is: $\frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{1}{a_1b_2 - a_2b_1}$. What does the denominator $a_1b_2 - a_2b_1$ represent?
A. The determinant of the coefficient matrix, which must be non-zero for a unique solution ✓ Correct
B. The product of all coefficients in the system
C. The sum of the constant terms divided by 2
D. A measure of how large the coefficients are
Solution: The denominator $a_1b_2 - a_2b_1$ is the determinant of the coefficient matrix $\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}$. If this is zero, the system does not have a unique solution (lines are parallel or coincident). This is why cross-multiplication only works when the determinant is non-zero.
Q24 — Graphical method · medium · theory
For what condition on the parameters does the system $x + ky = 5$ and $2x + 4y = 10$ have infinitely many solutions?
A. $k = 1$
B. $k = 2$ ✓ Correct
C. $k = \frac{1}{2}$
D. $k = 5$
Solution: For infinitely many solutions: $\frac{1}{2} = \frac{k}{4} = \frac{5}{10}$. From $\frac{5}{10} = \frac{1}{2}$ and $\frac{1}{2} = \frac{k}{4}$, we get $k = 2$. The second equation becomes $2(x + 2y = 5)$, so the equations are proportional and represent the same line.