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Pair of Linear Equations in Two Variables

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Chapter Notes

Overview
This chapter builds on the concept of linear equations in two variables from Class IX. It introduces methods to find solutions for a pair of such equations, which represent real-life situations like cost calculations, age problems, and number puzzles. The one big idea is that a pair of linear equations can be solved graphically (by plotting lines) or algebraically (by substitution or elimination), and the nature of the solution depends on the ratios of coefficients. A student must walk away knowing how to determine if a pair is consistent, inconsistent, or dependent, and how to find the solution using the most efficient method.
Key Concepts (11)
Pair of Linear Equations in Two Variables
Two linear equations in the same two variables, which can be represented as a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
e.g. x + 3y = 6 and 2x – 3y = 12 form a pair of linear equations in two variables.
Consistent Pair of Linear Equations
A pair of linear equations in two variables which has a solution (either unique or infinitely many).
e.g. x – 2y = 0 and 3x + 4y – 20 = 0 are consistent because they intersect at one point.
Inconsistent Pair of Linear Equations
A pair of linear equations which has no solution.
e.g. x + 2y – 4 = 0 and 2x + 4y – 12 = 0 are inconsistent because the lines are parallel.
Dependent Pair of Linear Equations
A pair of linear equations which are equivalent and have infinitely many distinct common solutions. Note that a dependent pair of linear equations is always consistent.
e.g. 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 are dependent because the lines coincide.
Intersecting Lines
Lines that meet at a single point, representing a pair of equations with a unique solution.
e.g. x – 2y = 0 and 3x + 4y – 20 = 0 intersect at one point.
Coincident Lines
Lines that lie exactly on top of each other, representing a pair of equations with infinitely many solutions.
e.g. 2x + 3y – 9 = 0 and 4x + 6y – 18 = 0 are coincident.
Parallel Lines
Lines that never meet, representing a pair of equations with no solution.
e.g. x + 2y – 4 = 0 and 2x + 4y – 12 = 0 are parallel.
Substitution Method
An algebraic method to solve a pair of linear equations by expressing one variable in terms of the other from one equation and substituting it into the other equation.
e.g. For 7x – 15y = 2 and x + 2y = 3, we write x = 3 – 2y and substitute into the first equation.
Elimination Method
An algebraic method to solve a pair of linear equations by making the coefficients of one variable numerically equal and then adding or subtracting the equations to eliminate that variable.
e.g. For 9x – 4y = 2000 and 7x – 3y = 2000, multiply the first by 3 and the second by 4, then subtract to eliminate y.
Graphical Method
A method to solve a pair of linear equations by plotting their graphs on the same coordinate plane and finding the point(s) of intersection.
e.g. Plotting x + 3y = 6 and 2x – 3y = 12 gives the intersection point (6, 0).
General Form of a Linear Equation
A linear equation in two variables written as a1x + b1y + c1 = 0, where a1, b1, c1 are real numbers and a1, b1 are not both zero.
e.g. 3x + 4y – 20 = 0 is in general form with a1 = 3, b1 = 4, c1 = -20.
Formulas & Equations (6)
a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
Reads as: a one x plus b one y plus c one equals zero, and a two x plus b two y plus c two equals zero
Variables: a1, b1, c1, a2, b2, c2 are real numbers; a1 and b1 are not both zero; a2 and b2 are not both zero; x and y are variables
Use when: Representing any pair of linear equations in two variables in standard form for comparison of ratios.
⚠ Forgetting to write equations in the form ax + by + c = 0 before comparing ratios, leading to wrong signs for c1 and c2.
If a1/a2 ≠ b1/b2, then lines intersect (unique solution).
Reads as: If a one over a two is not equal to b one over b two, then the lines intersect and have a unique solution.
Variables: a1, b1 from first equation; a2, b2 from second equation
Use when: Determining if a pair of linear equations is consistent with a unique solution.
⚠ Comparing only a and b without checking c, leading to misclassification when lines are coincident.
If a1/a2 = b1/b2 = c1/c2, then lines are coincident (infinitely many solutions).
Reads as: If a one over a two equals b one over b two equals c one over c two, then the lines are coincident and have infinitely many solutions.
Variables: a1, b1, c1 from first equation; a2, b2, c2 from second equation
Use when: Determining if a pair of linear equations is dependent and consistent.
⚠ Assuming that equal a1/a2 and b1/b2 automatically means coincident lines without checking c1/c2.
If a1/a2 = b1/b2 ≠ c1/c2, then lines are parallel (no solution).
Reads as: If a one over a two equals b one over b two but is not equal to c one over c two, then the lines are parallel and have no solution.
Variables: a1, b1, c1 from first equation; a2, b2, c2 from second equation
Use when: Determining if a pair of linear equations is inconsistent.
⚠ Confusing the condition for parallel lines with that for coincident lines by not checking c1/c2.
y = (1/2)x
Reads as: y equals half of x
Variables: x = number of rides on Giant Wheel, y = number of times played Hoopla
Use when: Representing the relationship 'number of times she played Hoopla is half the number of rides she had on the Giant Wheel'.
⚠ Writing y = 2x instead of y = (1/2)x, reversing the relationship.
3x + 4y = 20
Reads as: three x plus four y equals twenty
Variables: x = number of rides (cost Rs 3 each), y = number of Hoopla games (cost Rs 4 each), 20 = total money spent in rupees
Use when: Representing the total cost condition in the fair problem.
⚠ Forgetting to multiply the cost per ride/game by the respective number of times.
How It Works
Graphical Method of Solving a Pair of Linear Equations
  1. Step 1: Write both equations in the form y = mx + c or find at least two solutions for each equation by choosing convenient values of x and solving for y.
  2. Step 2: Plot the points for each equation on a graph paper and draw the corresponding straight lines.
  3. Step 3: Observe the intersection of the lines: if they intersect at one point, that point is the unique solution; if they coincide, every point on the line is a solution; if they are parallel, there is no solution.
💡 The graphical method is not convenient when the solution has non-integral coordinates like (3/2, 2/7) or (-1.75, 3.3), as reading such points from a graph can lead to errors.
Substitution Method
  1. Step 1: Find the value of one variable (say y) in terms of the other variable (x) from either equation, whichever is convenient.
  2. Step 2: Substitute this value of y in the other equation, reducing it to an equation in one variable (x). Solve for x. If you get a true statement with no variable, the pair has infinitely many solutions. If you get a false statement, the pair is inconsistent.
  3. Step 3: Substitute the value of x obtained in Step 2 back into the equation used in Step 1 to find the value of y.
💡 Always check if the statement obtained in Step 2 is true or false — a true statement (e.g., 18 = 18) means infinitely many solutions, while a false statement (e.g., -4 = 0) means no solution.
Elimination Method
  1. Step 1: Multiply both equations by suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.
  2. Step 2: Add or subtract the equations to eliminate that variable. If you get an equation in one variable, go to Step 3. If you get a true statement with no variable, the pair has infinitely many solutions. If you get a false statement, the pair is inconsistent.
  3. Step 3: Solve the resulting equation in one variable to get its value.
  4. Step 4: Substitute this value into either of the original equations to find the value of the other variable.
💡 When subtracting, be careful with signs — subtracting a negative term can lead to sign errors. Always double-check the arithmetic.
Determining Consistency Using Ratios of Coefficients
  1. Step 1: Write both equations in the general form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
  2. Step 2: Compute the ratios a1/a2, b1/b2, and c1/c2.
  3. Step 3: If a1/a2 ≠ b1/b2, the lines intersect (unique solution, consistent).
  4. Step 4: If a1/a2 = b1/b2 = c1/c2, the lines are coincident (infinitely many solutions, dependent and consistent).
  5. Step 5: If a1/a2 = b1/b2 ≠ c1/c2, the lines are parallel (no solution, inconsistent).
💡 The converse is also true for any pair of lines — the ratio conditions are both necessary and sufficient.
Exam Traps — Don't Make These Mistakes
⚠ Confusing 'consistent' with 'unique solution'
✗ A pair of linear equations is consistent only if it has a unique solution.
→ Consistent means the pair has at least one solution — this includes both unique and infinitely many solutions. A dependent pair (coincident lines) is also consistent.
✓ A pair of linear equations is consistent if it has a solution (either one or infinitely many).
⚠ Misapplying the ratio condition for parallel lines
✗ If a1/a2 = b1/b2, then the lines are parallel.
→ This condition alone is not sufficient. If a1/a2 = b1/b2 = c1/c2, the lines are coincident, not parallel. Parallel lines require a1/a2 = b1/b2 ≠ c1/c2.
✓ Lines are parallel if a1/a2 = b1/b2 ≠ c1/c2.
⚠ Sign errors when writing equations in general form
✗ For the equation 3x + 4y = 20, students write a1 = 3, b1 = 4, c1 = 20.
→ The general form is a1x + b1y + c1 = 0. So 3x + 4y = 20 must be rewritten as 3x + 4y - 20 = 0, giving c1 = -20, not 20.
✓ Rewrite the equation as 3x + 4y - 20 = 0, so a1 = 3, b1 = 4, c1 = -20.
⚠ Forgetting to check the third ratio for coincident lines
✗ For 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0, students say a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2, so lines are parallel.
→ They forgot to check c1/c2 = -9/-18 = 1/2. Since all three ratios are equal, the lines are coincident, not parallel.
✓ Check all three ratios: a1/a2 = 1/2, b1/b2 = 1/2, c1/c2 = 1/2. Since all are equal, the lines are coincident.
⚠ Incorrect substitution in the substitution method
✗ For equations x + 2y = 3 and 7x - 15y = 2, students write x = 3 + 2y.
→ When solving x + 2y = 3 for x, we subtract 2y from both sides, giving x = 3 - 2y, not 3 + 2y.
✓ x = 3 - 2y (correct transposition).
⚠ Not reducing equations to simplest form before comparing ratios
✗ For 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0, students say a1/a2 = 2/4 = 1/2, b1/b2 = 3/6 = 1/2, c1/c2 = -9/-18 = 1/2, so they are parallel.
→ They correctly computed the ratios but then misidentified the relationship. Since all three ratios are equal, the lines are coincident, not parallel.
✓ Since a1/a2 = b1/b2 = c1/c2, the lines are coincident (infinitely many solutions).
Compare & Contrast
Existence of solution
Consistent Pair
Inconsistent Pair
A consistent pair has at least one solution (unique or infinitely many), while an inconsistent pair has no solution.
Procedure to eliminate a variable
Substitution Method
Elimination Method
In substitution, one variable is expressed in terms of the other and substituted into the second equation. In elimination, equations are multiplied by constants and added/subtracted to cancel one variable.
Number of solutions
Intersecting Lines
Coincident Lines
Intersecting lines have a unique solution (one point of intersection), while coincident lines have infinitely many solutions (every point on the line is common).
Ratio condition
Parallel Lines
Coincident Lines
Parallel lines satisfy a1/a2 = b1/b2 ≠ c1/c2, while coincident lines satisfy a1/a2 = b1/b2 = c1/c2.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA-I (2 marks) on ratio conditions, 1 SA-II (3 marks) on solving by substitution or elimination, 1 LA (5 marks) on a word problem involving pair of linear equations. Total: 12 marks.
Most asked: Determining consistency/inconsistency using ratio conditions · Solving pairs of linear equations by substitution and elimination methods · Word problems involving ages, numbers, costs, and geometry · Graphical representation and interpretation of solutions
Don't skip: Word problems that require forming and solving a pair of linear equations (e.g., age problems, number problems, cost problems) — typically 5 marks.; Questions asking to determine whether a pair is consistent, inconsistent, or dependent using ratio conditions — often 2-3 marks.
One-Line Revision (read aloud before the exam)
  1. A pair of linear equations in two variables is consistent if it has at least one solution.
  2. A pair of linear equations in two variables is inconsistent if it has no solution.
  3. A dependent pair of linear equations has infinitely many solutions and is always consistent.
  4. If a1/a2 ≠ b1/b2, the lines intersect and the pair has a unique solution.
  5. If a1/a2 = b1/b2 = c1/c2, the lines are coincident and the pair has infinitely many solutions.
  6. If a1/a2 = b1/b2 ≠ c1/c2, the lines are parallel and the pair has no solution.
  7. In the substitution method, express one variable in terms of the other from one equation and substitute into the other equation.
  8. In the elimination method, multiply equations to make coefficients equal, then add or subtract to eliminate a variable.
  9. Always rewrite equations in the form ax + by + c = 0 before comparing ratios.
  10. The graphical method is not suitable when the solution has non-integral coordinates.
  11. A pair of linear equations can represent intersecting, coincident, or parallel lines.
  12. Word problems on ages, numbers, and costs are commonly solved using pair of linear equations.
  13. The sum of a two-digit number and its reverse is always a multiple of 11.
  14. If a pair of equations reduces to a true statement (e.g., 18 = 18), it has infinitely many solutions.
  15. If a pair of equations reduces to a false statement (e.g., -4 = 0), it has no solution.

MCQs — 10 Questions with Answers

Q1. A pair of linear equations in two variables is represented graphically by two lines that intersect at exactly one point. Which statement correctly describes the relationship between the coefficients of these two equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0?

  • A. a₁/a₂ ≠ b₁/b₂ ✓
  • B. a₁/a₂ = b₁/b₂ = c₁/c₂
  • C. a₁/a₂ = b₁/b₂ ≠ c₁/c₂
  • D. a₁/a₂ = c₁/c₂ but b₁/b₂ is undefined

Answer: A — Intersecting lines have unique solution only when a₁/a₂ ≠ b₁/b₂; option B describes coincident lines (infinitely many solutions) and option C describes parallel lines (no solution).

Q2. Assertion (A): A pair of linear equations representing two parallel lines has no solution. Reason (R): Parallel lines never meet at any point, so there is no common point of intersection that satisfies both equations simultaneously. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both statements are true, and the reason correctly explains why parallel lines lead to no solution: the absence of intersection means no point satisfies both equations.

Q3. A student claims: 'If two linear equations have the same coefficients for x and y (that is, a₁ = a₂ and b₁ = b₂), then the lines must be coincident and have infinitely many solutions.' Is this claim always true, sometimes true, or false? Why?

  • A. Always true, because equal coefficients guarantee identical slopes and y-intercepts
  • B. Sometimes true; it is true only when c₁/c₂ is also equal to a₁/a₂ and b₁/b₂ ✓
  • C. False, because two equations with the same x and y coefficients could represent parallel distinct lines
  • D. False, because coefficients alone determine neither slope nor intersection without knowing the constant terms

Answer: B — Equal coefficients for x and y mean equal slopes, but the lines are coincident (infinitely many solutions) only if a₁/a₂ = b₁/b₂ = c₁/c₂; if the constant terms differ, the lines are parallel and distinct.

Q4. In the context of solving a pair of linear equations graphically, a teacher states that 'the graphical method always gives the exact solution to any pair of linear equations.' Which of the following identifies a flaw in this reasoning?

  • A. The statement is entirely correct; graphical methods always provide exact solutions
  • B. The statement ignores that graphical methods depend on accurate plotting and reading of coordinates, which may introduce errors in identifying intersection points ✓
  • C. The statement is false because pairs of equations with no solution cannot be solved graphically
  • D. The statement overlooks that parallel lines have no solution, so the graphical method cannot provide a unique answer in those cases

Answer: B — While graphical methods can show solution types, they depend on measurement accuracy; reading coordinates from a graph introduces practical limitations, unlike algebraic methods which give exact solutions.

Q5. Assertion (A): A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is consistent if and only if the lines either intersect at one point or are coincident. Reason (R): Consistent pairs of equations are those that have at least one common solution, and only intersecting and coincident lines share such points. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both A and R are true; the reason directly explains the assertion by defining consistency as having at least one solution, which occurs only for intersecting or coincident lines.

Q6. A student solves a pair of linear equations by substitution and finds x = 2, y = 3. When she checks this solution in both original equations, it satisfies equation 1 but not equation 2. What does this outcome tell us about the pair of equations?

  • A. The pair is consistent and the solution is x = 2, y = 3
  • B. The pair is inconsistent, and the student made a computational error in the substitution method
  • C. The pair represents two intersecting lines, so the solution should satisfy both equations
  • D. The pair is inconsistent because a solution must satisfy both equations in a solvable pair ✓

Answer: D — A pair of equations is solvable (consistent) only if any solution satisfies both equations simultaneously; if a point satisfies only one equation, the pair has no common solution and is inconsistent.

Q7. Consider two equations: 4x + 6y = 12 and 2x + 3y = 6. Without graphing or algebraic solving, determine the nature of this pair using only the coefficient ratios.

  • A. The lines intersect at one point because the ratios of coefficients are different
  • B. The lines are coincident because 4/2 = 6/3 = 12/6 = 2 ✓
  • C. The lines are parallel because 4/2 = 6/3 but 12 ≠ 6
  • D. The relationship cannot be determined without graphing

Answer: B — When a₁/a₂ = b₁/b₂ = c₁/c₂, the lines are coincident; here 4/2 = 6/3 = 12/6 = 2, so infinitely many solutions exist.

Q8. Assertion (A): The graphical method and algebraic methods (substitution/elimination) always give the same conclusion about whether a pair of linear equations is consistent or inconsistent. Reason (R): Both methods are based on the same underlying principle that a solution exists if and only if the corresponding lines intersect or are coincident. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both are true; both methods rely on the same geometric principle that solutions exist when lines share points, ensuring identical conclusions about consistency.

Q9. A pair of linear equations has the property that a₁/a₂ = b₁/b₂ but c₁/c₂ is not equal to this common ratio. What graphical configuration does this represent, and what does it imply about solutions?

  • A. Intersecting lines with a unique solution
  • B. Coincident lines with infinitely many solutions
  • C. Parallel lines with no solution ✓
  • D. The configuration cannot be determined from these ratios alone

Answer: C — When a₁/a₂ = b₁/b₂ but c₁/c₂ differs, the slopes are equal (parallel) but y-intercepts differ (distinct), so the lines never meet and no solution exists.

Q10. Assertion (A): When graphically solving a pair of linear equations, if the two lines appear to be the same line, the pair has infinitely many solutions. Reason (R): Two distinct equations that represent the same line must satisfy the condition a₁/a₂ = b₁/b₂ = c₁/c₂. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both are true; coincident lines (same line) have infinitely many solutions, and this occurs exactly when the coefficient ratios are all equal, so R correctly explains A.

Flashcards

Define: Consistent pair of linear equations

A pair of linear equations in two variables that has at least one solution.

Define: Inconsistent pair of linear equations

A pair of linear equations in two variables that has no solution.

What does it mean graphically when two linear equations are inconsistent?

The lines representing the two equations are parallel and do not intersect at any point.

Condition for unique solution using ratios

If a₁/a₂ ≠ b₁/b₂, the pair has a unique solution and the lines intersect at one point.

Condition for infinitely many solutions using ratios

If a₁/a₂ = b₁/b₂ = c₁/c₂, the pair has infinitely many solutions and the lines are coincident.

Condition for no solution using ratios

If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the pair has no solution and the lines are parallel.

What is a dependent pair of linear equations?

A pair of linear equations that are equivalent and represent the same line, having infinitely many solutions.

How many points do you need to draw a line on a graph?

You need at least two points to draw a line accurately on a graph.

When solving graphically, how do you find the solution of a pair of equations?

Find the coordinates of the point where the two lines intersect; that point is the unique solution.

What is the general form of a linear equation in two variables?

The general form is a₁x + b₁y + c₁ = 0, where a₁, b₁, and c₁ are constants and a₁, b₁ are not both zero.

Important Board Questions

Define a consistent pair of linear equations in two variables. Give one example of a consistent pair. [2 marks]

State that a consistent pair has at least one solution. Provide any pair where lines intersect or coincide, such as x + y = 5 and 2x – y = 4.

For the pair of equations 6x – 3y + 10 = 0 and 2x – y + 9 = 0, determine whether the lines are parallel, intersecting, or coincident by comparing the ratios a₁/a₂, b₁/b₂, and c₁/c₂. Justify your answer. [3 marks]

Calculate a₁/a₂ = 6/2 = 3, b₁/b₂ = -3/(-1) = 3, c₁/c₂ = 10/9. Since a₁/a₂ = b₁/b₂ but not equal to c₁/c₂, the lines are parallel and the pair is inconsistent.

Champa purchased pants and skirts. The number of skirts is two less than twice the number of pants, and also the number of skirts is four less than four times the number of pants. Form the pair of linear equations, represent them graphically, and determine how many pants and skirts she bought. Verify your solution. [5 marks]

Let x = pants, y = skirts. Form equations y = 2x – 2 and y = 4x – 4. Plot both lines by finding two points each; they intersect at (1, 0). Verify: y = 2(1) – 2 = 0 ✓ and y = 4(1) – 4 = 0 ✓, so 1 pant and 0 skirts.

Assertion-Reason Questions for Pair of Linear Equations in Two Variables

Q1. Assertion (A):

If two lines represented by a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel, then a₁/a₂ = b₁/b₂.

Reason (R):

Parallel lines have the same slope, and the ratio of coefficients of x to coefficients of y determines the slope.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. Parallel lines do have the property that a₁/a₂ = b₁/b₂, and the reason correctly explains that this is because parallel lines have identical slopes. The ratio a/b is directly related to the slope of the line, so equal ratios ensure equal slopes (parallel lines).

Q2. Assertion (A):

A pair of linear equations with a₁/a₂ = b₁/b₂ = c₁/c₂ represents coincident lines and has infinitely many solutions.

Reason (R):

Coincident lines are two representations of the same line, and every point on the line satisfies both equations.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true, and the reason correctly explains the assertion. When all three ratios are equal, the equations represent the same line (coincident), so infinitely many points satisfy both equations. This is the correct explanation for why coincident lines have infinitely many solutions.

Q3. Assertion (A):

The graphical method of solving a pair of linear equations involves finding the coordinates of the intersection point of two lines.

Reason (R):

The intersection point coordinates satisfy both linear equations simultaneously.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true statements. The graphical method does involve plotting lines and finding their intersection point, and those coordinates do satisfy both equations. However, the reason is not the correct explanation of the assertion. The assertion describes the method itself (finding intersection), while the reason explains a property of the solution, not why we use the graphical method. The graphical method works because the intersection point represents the common solution.

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Case-Based Questions (CBQ) for Pair of Linear Equations in Two Variables

Case Study:

Rajesh is planning a birthday party for his friend. He needs to arrange tables and chairs. He knows that the total number of items (tables and chairs) is 30, and the number of chairs is 6 more than twice the number of tables. Rajesh wants to use the pair of linear equations to find how many tables and chairs he should arrange. Let x denote the number of tables and y denote the number of chairs. The equations formed are: x + y = 30 and y = 2x + 6.

Q1. What is the value of x (number of tables) from the given pair of equations?

  • A. 8 ✓
  • B. 10
  • C. 12
  • D. 15

Answer: A

Q2. What is the value of y (number of chairs) from the given pair of equations?

  • A. 18
  • B. 20
  • C. 22 ✓
  • D. 24

Answer: C

Q3. Comparing the ratios a₁/a₂ and b₁/b₂ for the equations x + y = 30 and y = 2x + 6 (rewritten as 2x - y + 6 = 0), the lines will:

  • A. Be parallel
  • B. Be coincident
  • C. Intersect at one point ✓
  • D. Be perpendicular

Answer: C

Q4. If Rajesh actually needs the number of chairs to be 4 more than the number of tables, and the total is still 30, what would be the new equation?

  • A. y = x + 4 ✓
  • B. y = 2x + 4
  • C. y = x - 4
  • D. x = y + 4

Answer: A

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Previous Year Questions (PYQ) for Pair of Linear Equations in Two Variables

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 31/4/1 · 5 marks

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. (i) Representing the above situation, frame two linear equations in two variables. (ii) Show that the equations have unique solution. (iii) Solve the equations and hence find the numbers.

CBSE 2026 · Set 31/4/1 · 1 mark

Observe the figure given above. It shows six identical rectangular enclosures made by using fencing wire mesh. These enclosures are used to protect baby animals in a zoo. Dimensions of each enclosure is x feet × y feet. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet.

Observe the figure given above. It shows six identical rectangular enclosures made by using fencing wire mesh. These enclosures are used to protect baby animals in a zoo. Dimensions of each enclosure is x feet × y feet. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet. Write an expression for length of fencing required in terms of x and y.

CBSE 2026 · Set 31/4/2 · 1 mark

Observe the figure given above. It shows six identical rectangular enclosures made by using fencing wire mesh. These enclosures are used to protect baby animals in a zoo. Dimensions of each enclosure is x feet × y feet. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet.

Write an expression for length of fencing required in terms of x and y.

CBSE 2026 · Set 31/5/1 · 1 mark

Seema daily goes to a park to exercise on machines available there. When Seema spent 15 minutes on exercise bicycle and 30 minutes on double cross walker, she received a message of burning 435 calories on her fitness watch. When she spent 30 minutes on exercise bicycle and 40 minutes on double cross walker, she received a message of burning 690 calories.

Represent the above situation in terms of a pair of linear equations in two variables.

CBSE 2026 · Set 31/5/2 · 4 marks

Seema daily goes to a park to exercise on machines available there. When Seema spent 15 minutes on exercise bicycle and 30 minutes on double cross walker, she received a message of burning 435 calories on her fitness watch. When she spent 30 minutes on exercise bicycle and 40 minutes on double cross walker, she received a message of burning 690 calories.

(i) Represent the above situation in terms of a pair of linear equations in two variables.

CBSE 2026 · Set 31/1164/30 · 1 mark

If pair of linear equations given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 has infinitely many solutions, then which of the following is definitely true ?

  • a1/a2 = c1/c2
  • a1a2 ≠ b1b2
  • a1/b1 = a2/b2
  • a1b2 = a2b1

CBSE 2026 · Set 31/1167/430 · 1 mark

The value of k for which the pair of linear equations kx – 3y = 5, 4x – 6y = 10 has infinitely many solutions, is:

  • 4
  • 3
  • 2
  • 1

CBSE 2026 · Set 31/1167/430 · 3 marks

Solve the following pair of linear equations: 31x + 43y – 117 = 0; 43x + 31y = 105

CBSE 2026 · Set 31/5/1 · 1 mark

Assertion (A): The system of linear equations 3x – 5y + 7 = 0 and –6x + 10y + 14 = 0 is inconsistent. Reason (R): When two linear equations don't have unique solution, they always represent parallel lines.

  • Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
  • Assertion (A) is true, but Reason (R) is false.
  • Assertion (A) is false, but Reason (R) is true.

CBSE 2026 · Set 31/5/1 · 3 marks

Solve the system of linear equations: x = 4 and 3x – 2y = 6 graphically.

CBSE 2026 · Set 31/5/1 · 5 marks

Venkat can row a boat in still water at the speed of 12 km/h. He ferries tourists 15 km upstream and 18 km downstream in 3 hours. Find the speed of the stream.

CBSE 2026 · Set 31/1/1 · 1 mark

Assertion (A): The value of p for which the system of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 is consistent is 4. Reason (R): The system of equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ is consistent with infinitely many solutions, if a₁/a₂ = b₁/b₂ = c₁/c₂.

  • Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
  • Assertion (A) is true, but Reason (R) is false.
  • Assertion (A) is false, but Reason (R) is true.

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