Pair of Linear Equations in Two Variables
Many real-life problems involve finding the values of two unknown quantities that satisfy two conditions simultaneously. For example, finding the cost of a pencil and a pen when you know the total cost of different combinations. These situations naturally lead to a pair of linear equations in two variables. In earlier classes, you learned about a single linear equation in two variables (like 2x + 3y = 12) and how its solutions form a straight line on a graph. In Class 10, we extend this to study two such equations together. The central question is: can we find values of x and y that satisfy both equations at the same time? The answer depends on the relationship between the two equations, leading to three distinct cases: a unique solution, infinitely many solutions, or no solution. This chapter lays the foundation for solving systems of equations, which is essential in higher mathematics, physics, economics, and engineering.
What is Pair of Linear Equations in Two Variables?
A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers, and both a and b are not simultaneously zero. The variables x and y appear only in the first degree (no x^2, xy, or y^2 terms).
A pair of linear equations in two variables is a system of two such equations:
a1*x + b1*y + c1 = 0 ... (i)
a2*x + b2*y + c2 = 0 ... (ii)
A solution of this pair is an ordered pair (x, y) that satisfies both equations simultaneously. Finding such values is called solving the system or solving the pair of equations.
Three Cases:
Case 1: Consistent with a unique solution (Intersecting lines)
When a1/a2 is not equal to b1/b2, the two lines intersect at exactly one point. The system has exactly one solution.
Case 2: Consistent with infinitely many solutions (Coincident lines)
When a1/a2 = b1/b2 = c1/c2, the two equations represent the same line. Every point on this line is a solution. The system has infinitely many solutions.
Case 3: Inconsistent (Parallel lines)
When a1/a2 = b1/b2 but this ratio is not equal to c1/c2, the two lines are parallel (they never meet). The system has no solution.
A system is called consistent if it has at least one solution (Cases 1 or 2) and inconsistent if it has no solution (Case 3).
Pair of Linear Equations in Two Variables Formula
General form of a pair of linear equations:
a1*x + b1*y + c1 = 0
a2*x + b2*y + c2 = 0
Condition for unique solution (intersecting lines): a1/a2 is not equal to b1/b2.
Condition for infinitely many solutions (coincident lines): a1/a2 = b1/b2 = c1/c2.
Condition for no solution (parallel lines): a1/a2 = b1/b2 but a1/a2 is not equal to c1/c2.
Consistency check summary:
If a1/a2 =/= b1/b2 : Unique solution (consistent)
If a1/a2 = b1/b2 = c1/c2 : Infinite solutions (consistent, dependent)
If a1/a2 = b1/b2 =/= c1/c2 : No solution (inconsistent)
Derivation and Proof
The conditions for the three cases can be derived by examining the slopes and y-intercepts of the two lines.
Converting to slope-intercept form:
Line 1: a1*x + b1*y + c1 = 0 can be written as y = (-a1/b1)x + (-c1/b1), provided b1 is not 0. The slope is m1 = -a1/b1 and y-intercept is -c1/b1.
Line 2: a2*x + b2*y + c2 = 0 can be written as y = (-a2/b2)x + (-c2/b2), provided b2 is not 0. The slope is m2 = -a2/b2 and y-intercept is -c2/b2.
Case 1: Different slopes (m1 is not equal to m2)
-a1/b1 is not equal to -a2/b2, which means a1/b1 is not equal to a2/b2, or equivalently a1*b2 is not equal to a2*b1. This can be expressed as a1/a2 is not equal to b1/b2 (when a2 and b2 are non-zero).
Two lines with different slopes always intersect at exactly one point, giving a unique solution.
Case 2: Same slope, same y-intercept (identical lines)
a1/b1 = a2/b2 (same slope) AND c1/b1 = c2/b2 (same y-intercept). This simplifies to a1/a2 = b1/b2 = c1/c2.
The two equations represent the same line, so every point on the line is a common solution. There are infinitely many solutions.
Case 3: Same slope, different y-intercepts (parallel lines)
a1/b1 = a2/b2 (same slope) BUT c1/b1 is not equal to c2/b2 (different y-intercepts). This gives a1/a2 = b1/b2 but not equal to c1/c2.
Parallel lines never intersect, so there is no common solution.
Geometric interpretation: Two distinct lines in a plane can either intersect at one point, be parallel (no intersection), or be the same line (infinitely many points of intersection). There is no fourth possibility. This corresponds exactly to our three algebraic cases.
Why not three or more solutions? Two distinct lines can intersect at most once. If they share two common points, they must be the same line (since two points determine a unique line). Therefore, a system of two linear equations can never have exactly 2, 3, or any other finite number (other than 0 or 1) of solutions. It is always 0, 1, or infinitely many.
Types and Properties
Pairs of linear equations can be classified based on their solutions and their geometric relationship:
Type 1: Consistent and Independent (Unique Solution)
Example: x + y = 7 and x - y = 3. Solution: x = 5, y = 2. The lines intersect at (5, 2). Condition: a1/a2 =/= b1/b2, here 1/1 =/= 1/(-1).
Type 2: Consistent and Dependent (Infinitely Many Solutions)
Example: 2x + 3y = 6 and 4x + 6y = 12. The second equation is just 2 times the first. Every solution of one is a solution of the other. Condition: 2/4 = 3/6 = 6/12, i.e., 1/2 = 1/2 = 1/2.
Type 3: Inconsistent (No Solution)
Example: 2x + 3y = 6 and 4x + 6y = 18. The lines are parallel. Condition: 2/4 = 3/6 but 6/18 =/= 1/2, i.e., 1/2 = 1/2 =/= 1/3.
Type 4: Homogeneous System
When c1 = c2 = 0: a1*x + b1*y = 0 and a2*x + b2*y = 0. This system always has the trivial solution (0, 0). If a1/a2 =/= b1/b2, the unique solution is (0, 0). If a1/a2 = b1/b2, there are infinitely many solutions.
Based on the form of equations:
- Standard form: ax + by = c
- Slope-intercept form: y = mx + k
- Word problem form: equations derived from real-life situations (age problems, number problems, cost problems, distance-speed-time problems)
Methods
There are several methods to solve a pair of linear equations. Each method has its advantages depending on the specific equations.
Method 1: Graphical Method
Plot both lines on a coordinate plane. The point of intersection gives the solution. This method is visual but may give approximate answers if the intersection is not at integer coordinates. Covered in detail in the topic 'Graphical Method for Linear Equations'.
Method 2: Substitution Method
Solve one equation for one variable and substitute into the other. Covered in detail in the topic 'Substitution Method'.
Method 3: Elimination Method
Multiply equations by suitable constants so that one variable can be eliminated by addition or subtraction. Covered in detail in the topic 'Elimination Method'.
Method 4: Cross-Multiplication Method
A formula-based method that directly gives the solution using the coefficients: x/(b1*c2 - b2*c1) = y/(c1*a2 - c2*a1) = 1/(a1*b2 - a2*b1), provided a1*b2 - a2*b1 =/= 0.
Choosing the Right Method:
- Use graphical method when visual understanding is needed or when approximate answers suffice.
- Use substitution when one variable is easily expressible (e.g., y = 2x + 1).
- Use elimination when coefficients can be easily matched.
- Use cross-multiplication for a direct formula-based approach.
Solved Examples
Example 1: Example 1: Classify the pair of equations
Problem: Determine whether the following pair has a unique solution, no solution, or infinitely many solutions: 3x + 2y = 5 and 6x + 4y = 10.
Solution:
Write in standard form: 3x + 2y - 5 = 0 and 6x + 4y - 10 = 0.
a1 = 3, b1 = 2, c1 = -5. a2 = 6, b2 = 4, c2 = -10.
a1/a2 = 3/6 = 1/2. b1/b2 = 2/4 = 1/2. c1/c2 = -5/-10 = 1/2.
Since a1/a2 = b1/b2 = c1/c2 = 1/2, the lines are coincident.
Answer: The system has infinitely many solutions (consistent, dependent). The second equation is just 2 times the first.
Example 2: Example 2: Check consistency of the system
Problem: Check whether the following system is consistent: 5x - 3y = 11 and -10x + 6y = -22.
Solution:
Rewrite: 5x - 3y - 11 = 0 and -10x + 6y + 22 = 0.
a1/a2 = 5/(-10) = -1/2. b1/b2 = -3/6 = -1/2. c1/c2 = -11/22 = -1/2.
Since a1/a2 = b1/b2 = c1/c2 = -1/2, the lines are coincident.
The system is consistent with infinitely many solutions. Note that the second equation is -2 times the first equation.
Answer: Consistent with infinitely many solutions.
Example 3: Example 3: Identify parallel lines
Problem: Show that the equations 2x + 3y = 7 and 4x + 6y = 9 have no solution.
Solution:
Write as: 2x + 3y - 7 = 0 and 4x + 6y - 9 = 0.
a1/a2 = 2/4 = 1/2. b1/b2 = 3/6 = 1/2. c1/c2 = -7/(-9) = 7/9.
Since a1/a2 = b1/b2 = 1/2 but c1/c2 = 7/9 (which is not equal to 1/2), the lines are parallel.
Parallel lines never intersect, so the system has no solution.
Answer: No solution. The system is inconsistent.
Example 4: Example 4: Find the value of k for unique solution
Problem: For what value of k does the system kx + 3y = k - 3 and 12x + ky = k have a unique solution?
Solution:
For a unique solution: a1/a2 =/= b1/b2, i.e., k/12 =/= 3/k.
Cross-multiplying: k^2 =/= 36, so k =/= 6 and k =/= -6.
Additionally, a1 and a2 must not both be zero: k =/= 0 (otherwise the first equation becomes 3y = k-3, and the system might still have a unique solution, but let us check the general condition).
Actually, the condition a1/a2 =/= b1/b2 requires k/12 =/= 3/k, which gives k^2 =/= 36. So k can be any real number except 6 and -6.
Answer: The system has a unique solution for all values of k except k = 6 and k = -6.
Example 5: Example 5: Find k for no solution
Problem: Find the value of k for which the pair kx + 2y = 5 and 3x + y = 1 has no solution.
Solution:
For no solution: a1/a2 = b1/b2 =/= c1/c2.
a1/a2 = k/3. b1/b2 = 2/1 = 2. c1/c2 = -5/(-1) = 5.
For a1/a2 = b1/b2: k/3 = 2, so k = 6.
Check c1/c2: 5 =/= 2. Yes, c1/c2 =/= a1/a2. So the lines are parallel.
Answer: k = 6 gives no solution (parallel lines).
Example 6: Example 6: Word problem — forming the pair of equations
Problem: The sum of two numbers is 74 and their difference is 12. Form the pair of linear equations and find the numbers.
Solution:
Let the two numbers be x and y, where x > y.
Equation 1: x + y = 74
Equation 2: x - y = 12
Adding both equations: 2x = 86, so x = 43.
Substituting in Equation 1: 43 + y = 74, so y = 31.
Verification: 43 + 31 = 74 (correct). 43 - 31 = 12 (correct).
The pair a1/a2 = 1/1 = 1, b1/b2 = 1/(-1) = -1. Since 1 =/= -1, the system has a unique solution.
Answer: The two numbers are 43 and 31.
Example 7: Example 7: Age problem
Problem: Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?
Solution:
Let Nuri's current age be x years and Sonu's current age be y years.
Five years ago: Nuri was (x - 5), Sonu was (y - 5). Given: x - 5 = 3(y - 5). Simplifying: x - 5 = 3y - 15, so x - 3y = -10 ... (i)
Ten years later: Nuri will be (x + 10), Sonu will be (y + 10). Given: x + 10 = 2(y + 10). Simplifying: x + 10 = 2y + 20, so x - 2y = 10 ... (ii)
Subtracting (i) from (ii): (x - 2y) - (x - 3y) = 10 - (-10). y = 20.
Substituting in (ii): x - 40 = 10, so x = 50.
Verification: Five years ago: 45 = 3(15)? 45 = 45. Correct. Ten years later: 60 = 2(30)? 60 = 60. Correct.
Answer: Nuri is 50 years old and Sonu is 20 years old.
Example 8: Example 8: Speed, distance, and time problem
Problem: A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Find the speed of the boat in still water and the speed of the stream.
Solution:
Let the speed of the boat in still water = x km/h and speed of the stream = y km/h.
Upstream speed = (x - y) km/h. Downstream speed = (x + y) km/h.
Using time = distance/speed:
30/(x - y) + 44/(x + y) = 10 ... (i)
40/(x - y) + 55/(x + y) = 13 ... (ii)
Let u = 1/(x - y) and v = 1/(x + y). Then:
30u + 44v = 10 ... (iii)
40u + 55v = 13 ... (iv)
Multiply (iii) by 4: 120u + 176v = 40. Multiply (iv) by 3: 120u + 165v = 39.
Subtracting: 11v = 1, so v = 1/11. Substituting in (iii): 30u + 4 = 10, so 30u = 6, u = 1/5.
u = 1/(x-y) = 1/5, so x - y = 5. v = 1/(x+y) = 1/11, so x + y = 11.
Adding: 2x = 16, x = 8. Subtracting: 2y = 6, y = 3.
Answer: Speed of boat = 8 km/h, speed of stream = 3 km/h.
Example 9: Example 9: Find k for infinitely many solutions
Problem: For what value of k does the system 2x + 3y = 7 and (k - 1)x + (k + 2)y = 3k have infinitely many solutions?
Solution:
For infinitely many solutions: a1/a2 = b1/b2 = c1/c2.
Rewriting: 2x + 3y - 7 = 0 and (k-1)x + (k+2)y - 3k = 0.
a1/a2 = 2/(k-1). b1/b2 = 3/(k+2). c1/c2 = -7/(-3k) = 7/(3k).
Setting a1/a2 = b1/b2: 2/(k-1) = 3/(k+2). Cross-multiply: 2(k+2) = 3(k-1). 2k + 4 = 3k - 3. k = 7.
Check c1/c2: 7/(3*7) = 7/21 = 1/3. And a1/a2 = 2/(7-1) = 2/6 = 1/3. Yes, all ratios equal 1/3.
Answer: k = 7.
Example 10: Example 10: Determine the nature without solving
Problem: Without solving, determine the nature of solutions for: (a) x + 2y = 8 and 2x + 4y = 16. (b) 2x - 3y = 8 and 4x - 6y = 9. (c) 3x - y = 2 and 6x + 2y = 4.
Solution:
(a) a1/a2 = 1/2, b1/b2 = 2/4 = 1/2, c1/c2 = -8/(-16) = 1/2. All ratios equal. Infinitely many solutions (coincident lines).
(b) a1/a2 = 2/4 = 1/2, b1/b2 = -3/(-6) = 1/2, c1/c2 = -8/(-9) = 8/9. First two ratios equal but third is different. No solution (parallel lines).
(c) Rewrite: 3x - y - 2 = 0 and 6x + 2y - 4 = 0. a1/a2 = 3/6 = 1/2, b1/b2 = -1/2. Since a1/a2 = 1/2 = b1/b2 ... wait, -1/2 = -1/2? Let me check: b1 = -1, b2 = 2, so b1/b2 = -1/2. And a1/a2 = 1/2. Since 1/2 =/= -1/2, the lines are intersecting. Unique solution.
Answer: (a) Infinitely many solutions. (b) No solution. (c) Unique solution.
Real-World Applications
Pairs of linear equations are used to model and solve countless real-world problems involving two unknown quantities.
Business and Commerce: Determining the cost of individual items when the total costs of different combinations are known. For example, finding the price of one pen and one pencil given the total cost of 3 pens and 2 pencils, and the total cost of 2 pens and 3 pencils.
Age Problems: Finding the present ages of two people given relationships between their ages at different times (past or future).
Distance-Speed-Time: Finding the speed of a vehicle and the speed of the wind/current when travel times for upstream/downstream or with/against wind journeys are known.
Geometry: Finding the dimensions of geometric figures when relationships between length and breadth are given, such as perimeter and area conditions.
Mixture Problems: Determining the quantities of two solutions to mix to get a desired concentration. For example, mixing two acid solutions of different concentrations to get a specific concentration.
Economics: Supply and demand analysis uses systems of linear equations to find equilibrium price and quantity.
Key Points to Remember
- A pair of linear equations in two variables has the form a1*x + b1*y + c1 = 0 and a2*x + b2*y + c2 = 0.
- The solution is an ordered pair (x, y) that satisfies both equations simultaneously.
- If a1/a2 =/= b1/b2: unique solution (intersecting lines, consistent).
- If a1/a2 = b1/b2 = c1/c2: infinitely many solutions (coincident lines, consistent and dependent).
- If a1/a2 = b1/b2 =/= c1/c2: no solution (parallel lines, inconsistent).
- Graphically, the solution is the point of intersection of the two lines.
- A system of two linear equations can never have exactly 2 or any finite number (other than 0 or 1) of solutions.
- The methods for solving include: graphical, substitution, elimination, and cross-multiplication.
- Word problems must be carefully translated into equations, defining variables clearly.
- Always verify your solution by substituting back into both original equations.
Practice Problems
- Determine whether the system 3x + 4y = 12 and 6x + 8y = 24 has a unique solution, no solution, or infinitely many solutions.
- For what values of p and q does the system px + 3y = p - 3 and 12x + py = p have infinitely many solutions?
- The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.
- A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to the denominator. Find the fraction.
- Determine whether the following system is consistent or inconsistent: 5x - 4y + 8 = 0 and 7x + 6y - 9 = 0.
- Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If they travel in the same direction, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. Find their speeds.
Frequently Asked Questions
Q1. What is a pair of linear equations in two variables?
It is a system of two equations, each of the form ax + by + c = 0, where a and b are not both zero. The goal is to find values of x and y that satisfy both equations at the same time.
Q2. How many solutions can a pair of linear equations have?
Exactly three possibilities: (1) A unique solution (one ordered pair satisfies both). (2) Infinitely many solutions (both equations represent the same line). (3) No solution (the lines are parallel and never intersect). A pair of linear equations can never have exactly 2 or 3 solutions.
Q3. What does it mean for a system to be consistent?
A system is consistent if it has at least one solution. This means either the lines intersect at one point (unique solution) or the lines coincide (infinitely many solutions). A system with no solution is called inconsistent.
Q4. What does it mean for a system to be dependent?
A dependent system is one where both equations represent the same line (a1/a2 = b1/b2 = c1/c2). The equations are scalar multiples of each other, and every solution of one is also a solution of the other. There are infinitely many solutions.
Q5. How do you determine the type of solution without solving?
Compare the ratios a1/a2, b1/b2, and c1/c2. If a1/a2 is not equal to b1/b2, there is a unique solution. If all three ratios are equal, there are infinitely many solutions. If the first two ratios are equal but the third is different, there is no solution.
Q6. What is the geometric interpretation of a pair of linear equations?
Each linear equation represents a straight line in the coordinate plane. The solution of the pair is the point(s) of intersection. If the lines intersect at one point, there is one solution. If they are the same line, every point on it is a solution. If they are parallel, they never meet, so there is no solution.
Q7. Can a pair of linear equations have exactly two solutions?
No. Two distinct straight lines can intersect at most at one point. If they share two common points, they must be the same line (since two points uniquely determine a line). Therefore, the number of solutions is always 0, 1, or infinity.
Q8. What are the different methods to solve a pair of linear equations?
The four main methods are: (1) Graphical method — plot both lines and find the intersection point. (2) Substitution method — express one variable in terms of the other and substitute. (3) Elimination method — add or subtract equations to eliminate one variable. (4) Cross-multiplication method — use a direct formula based on the coefficients.
Q9. What is a homogeneous system of linear equations?
A homogeneous system is one where both equations have zero constant terms: a1*x + b1*y = 0 and a2*x + b2*y = 0. This system always has at least the trivial solution (0, 0). It has infinitely many solutions if a1/a2 = b1/b2.
Q10. How is this topic tested in board exams?
Common question types include: determining the nature of solutions using coefficient ratios, finding values of k for specific types of solutions, forming equations from word problems, and solving the system using substitution, elimination, or graphical methods. This topic typically carries 3-5 marks in CBSE Class 10 board exams.
