Pair of Linear Equations in Two Variables – important questions for Boards 2027
CBSE Class 10 Maths
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, state the ratio condition that identifies intersecting lines.
Answer
Intersecting lines are identified by a₁/a₂ ≠ b₁/b₂, and have a unique solution.
A situation gives two conditions involving two unknown quantities. Which is the appropriate way to represent it as a pair of linear equations? (A) Assign variables to the unknown quantities and express each condition as an equation. (B) Assign one variable to both unknowns and write a quadratic equation. (C) Write only one equation, regardless of the number of conditions. (D) Replace the unknown quantities with fixed values before forming equations.
Answer
(A) Assign variables to the unknown quantities and express each condition as an equation. — Each condition is represented by an equation in the two variables.
For a pair of linear equations, state the type of lines and number of solutions when the lines (i) intersect, (ii) are parallel, and (iii) are coincident.
Answer
Intersecting lines have one common point and a unique solution. Parallel lines have no common point and no solution. Coincident lines have infinitely many common points and infinitely many solutions.
Solve the pair 3x + 2y = 11 and 3x − 2y = 5 by elimination. Show how you use an original equation to find the value of the other variable.
Answer
1. Add the equations: (3x + 2y) + (3x − 2y) = 11 + 5, so 6x = 16 and x = 8/3. The equal and opposite coefficients of y make addition the suitable elimination operation. 2. Substitute x = 8/3 in the original equation 3x + 2y = 11: 8 + 2y = 11, so 2y = 3 and y = 3/2. 3. Therefore, the solution is (x, y) = (8/3, 3/2), satisfying both equations.
State the coefficient-ratio conditions for two lines to be intersecting, coincident or parallel. Explain why these conditions can also be used in reverse to identify the corresponding line configuration.
Answer
The coefficient ratios determine whether two lines intersect, coincide or remain parallel. 1. For intersecting lines, a₁/a₂ ≠ b₁/b₂. The lines meet at one point, so the pair has a unique solution and is consistent. 2. For coincident lines, a₁/a₂ = b₁/b₂ = c₁/c₂. The lines overlap, so the pair has infinitely many solutions and is a dependent, consistent pair. 3. For parallel lines, a₁/a₂ = b₁/b₂ ≠ c₁/c₂. The lines never meet, so the pair has no solution and is inconsistent. These conditions can also be used in reverse to identify the configuration of any pair of lines: comparing the ratios reveals whether the lines intersect, coincide or are parallel.
For the pair 2x + 3y = 8 and 4x + 6y = 7, use elimination to obtain a statement with no variable and interpret it. Then explain what a true statement with no variable would imply for a pair of linear equations.
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In Akhila’s fair situation, x is the number of Giant Wheel rides and y is the number of Hoopla games. Which pair of equations represents the conditions that the games are half the rides and the total cost is ₹20? (A) y = 2x and 3x + 4y = 20 (B) y = x/2 and 3x + 4y = 20 (C) y = x/2 and 4x + 3y = 20 (D) x = y/2 and 3x + 4y = 20
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In the elimination method, what is done to the equations before adding or subtracting them, and why?
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After substitution, a pair of equations reduces first to 5 = 5 and, in another case, to 5 = 8. State what each result tells you about the corresponding pair of equations.
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Describe how a pair of linear equations in two variables can be solved graphically and algebraically, naming the two algebraic methods. Then state the coefficient-ratio conditions that identify a pair as (i) consistent with a unique solution, (ii) inconsistent, and (iii) dependent and consistent.
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7 more probable questions in this chapter, every one with a full answer.
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