Consistency of Linear Equations in Two Variables
When we are given a pair of linear equations in two variables, one of the first and most important questions to ask is whether the system actually has a solution. This question leads us to the concept of consistency. A system of linear equations is said to be consistent if it has at least one solution and inconsistent if it has no solution at all. For CBSE Class 10 students, understanding consistency is fundamental because it determines whether the effort of solving the system will yield meaningful results. The concept is deeply connected to the graphical interpretation of linear equations — two straight lines in a plane can either intersect at exactly one point (unique solution), coincide completely (infinitely many solutions), or be parallel and never meet (no solution). By comparing the ratios of the coefficients, students can determine the consistency of a system without actually solving it or drawing its graph. This analytical approach is both elegant and practical, and it appears frequently in board examinations. In this comprehensive topic, we will explore the conditions for consistency, understand the graphical meaning, study the algebraic conditions using coefficient ratios, and work through numerous examples.
What is Consistency of Linear Equations?
A system (or pair) of linear equations in two variables is said to be consistent if it possesses at least one solution. This means there exists at least one pair of values (x, y) that satisfies both equations simultaneously. A consistent system may have:
- Exactly one solution (unique solution) — the two lines intersect at precisely one point, or
- Infinitely many solutions — the two lines are coincident (they are the same line).
A system is said to be inconsistent if it has no solution at all. This happens when the two lines are parallel — they never intersect, so no pair (x, y) can satisfy both equations.
Consider the general pair of linear equations:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
The consistency of this system depends entirely on the relationship between the ratios a1/a2, b1/b2, and c1/c2. These ratios encode information about the slopes and intercepts of the two lines, which in turn determine whether and how the lines interact.
A system that is consistent with a unique solution is sometimes called an independent consistent system. A system that is consistent with infinitely many solutions is called a dependent consistent system. These distinctions help clarify the nature of the solution set.
Consistency of Linear Equations in Two Variables Formula
Conditions for Consistency:
For the system a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:
| Condition on Ratios | Type of System | Number of Solutions | Graphical Meaning |
|---|---|---|---|
| a1/a2 ≠ b1/b2 | Consistent (Independent) | Exactly one (unique) | Lines intersect at one point |
| a1/a2 = b1/b2 = c1/c2 | Consistent (Dependent) | Infinitely many | Lines are coincident (same line) |
| a1/a2 = b1/b2 ≠ c1/c2 | Inconsistent | No solution | Lines are parallel |
Quick Summary:
- If the ratio of x-coefficients is NOT equal to the ratio of y-coefficients → Unique solution (consistent).
- If all three ratios (x-coefficients, y-coefficients, constants) are equal → Infinitely many solutions (consistent).
- If the first two ratios are equal but the third is different → No solution (inconsistent).
Derivation and Proof
The conditions for consistency can be understood through the slope-intercept form of linear equations.
Step 1: Consider two lines:
a1x + b1y + c1 = 0 → y = -(a1/b1)x - c1/b1 (slope = -a1/b1)
a2x + b2y + c2 = 0 → y = -(a2/b2)x - c2/b2 (slope = -a2/b2)
Step 2 — Case 1 (Intersecting lines): Two lines intersect if and only if they have different slopes:
-a1/b1 ≠ -a2/b2 → a1/b1 ≠ a2/b2 → a1/a2 ≠ b1/b2
When the slopes are different, the lines must cross at exactly one point, giving a unique solution.
Step 3 — Case 2 (Coincident lines): Two lines are the same line if they have the same slope AND the same y-intercept. Same slope means a1/a2 = b1/b2. Same y-intercept means c1/b1 = c2/b2, which simplifies to b1/b2 = c1/c2. Combined: a1/a2 = b1/b2 = c1/c2.
In this case, every point on one line is also on the other, giving infinitely many solutions.
Step 4 — Case 3 (Parallel lines): Two lines are parallel if they have the same slope but different y-intercepts. Same slope means a1/a2 = b1/b2. Different y-intercepts means c1/b1 ≠ c2/b2, which gives b1/b2 ≠ c1/c2. Combined: a1/a2 = b1/b2 ≠ c1/c2.
Parallel lines never meet, so the system has no solution. The system is inconsistent.
Methods
Step-by-Step Procedure to Check Consistency:
Step 1: Write both equations in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. Move all terms to one side.
Step 2: Identify the coefficients a1, b1, c1, a2, b2, c2.
Step 3: Compute the three ratios: a1/a2, b1/b2, and c1/c2.
Step 4: Compare the ratios:
- If a1/a2 ≠ b1/b2 → The system is consistent with a unique solution. The lines intersect.
- If a1/a2 = b1/b2 = c1/c2 → The system is consistent with infinitely many solutions. The lines coincide.
- If a1/a2 = b1/b2 ≠ c1/c2 → The system is inconsistent. The lines are parallel.
Common Mistakes to Avoid:
- Not writing both equations in the same form. If one equation is 2x + 3y = 5 and the other is 4x + 6y + 10 = 0, convert both to the standard form with zero on the right.
- Confusing the sign of c. For 2x + 3y = 5, rewriting gives 2x + 3y - 5 = 0, so c = -5, not +5.
- Using cross-multiplication ratios to compare. Ensure you compare a1/a2 with b1/b2 (not a1/b1 with a2/b2).
Finding Parameter Values for Specific Consistency:
A common exam question asks: 'Find the value of k for which the system has no solution / infinitely many solutions / a unique solution.' For such problems, set up the ratio conditions and solve the resulting equation(s) for k.
Solved Examples
Example 1: Consistent System with Unique Solution
Problem: Check the consistency of the system: 3x + 2y = 5 and x - y = 2.
Solution:
Step 1: Standard form:
3x + 2y - 5 = 0
x - y - 2 = 0
Step 2: a1 = 3, b1 = 2, c1 = -5, a2 = 1, b2 = -1, c2 = -2
Step 3: Ratios:
a1/a2 = 3/1 = 3
b1/b2 = 2/(-1) = -2
Step 4: Since 3 ≠ -2, i.e., a1/a2 ≠ b1/b2
Answer: The system is consistent with a unique solution. The lines intersect at exactly one point.
Example 2: Inconsistent System (Parallel Lines)
Problem: Check the consistency of: 2x + 3y = 7 and 4x + 6y = 9.
Solution:
Step 1: Standard form:
2x + 3y - 7 = 0
4x + 6y - 9 = 0
Step 2: a1 = 2, b1 = 3, c1 = -7, a2 = 4, b2 = 6, c2 = -9
Step 3: Ratios:
a1/a2 = 2/4 = 1/2
b1/b2 = 3/6 = 1/2
c1/c2 = -7/-9 = 7/9
Step 4: a1/a2 = b1/b2 = 1/2, but c1/c2 = 7/9 ≠ 1/2
Answer: The system is inconsistent. The lines are parallel and have no solution. The equations represent two parallel lines with the same slope but different y-intercepts.
Example 3: Consistent System with Infinitely Many Solutions (Coincident Lines)
Problem: Check the consistency of: 4x - 2y = 6 and 6x - 3y = 9.
Solution:
Step 1: Standard form:
4x - 2y - 6 = 0
6x - 3y - 9 = 0
Step 2: a1 = 4, b1 = -2, c1 = -6, a2 = 6, b2 = -3, c2 = -9
Step 3: Ratios:
a1/a2 = 4/6 = 2/3
b1/b2 = (-2)/(-3) = 2/3
c1/c2 = (-6)/(-9) = 2/3
Step 4: a1/a2 = b1/b2 = c1/c2 = 2/3
Answer: The system is consistent with infinitely many solutions. The two equations represent the same line (coincident lines). Any point on 2x - y = 3 is a solution.
Example 4: Finding k for Inconsistency
Problem: For what value of k does the system kx + 3y = k - 3 and 12x + ky = k have no solution?
Solution:
Step 1: Standard form:
kx + 3y - (k - 3) = 0
12x + ky - k = 0
Step 2: a1 = k, b1 = 3, c1 = -(k - 3) = 3 - k, a2 = 12, b2 = k, c2 = -k
Step 3: For no solution: a1/a2 = b1/b2 ≠ c1/c2
Step 4: a1/a2 = b1/b2 gives:
k/12 = 3/k
k2 = 36
k = 6 or k = -6
Step 5: Check c1/c2 for each value:
For k = 6: c1/c2 = (3 - 6)/(-6) = -3/-6 = 1/2. And a1/a2 = 6/12 = 1/2. Since c1/c2 = a1/a2, all three ratios are equal. This gives infinitely many solutions, not no solution.
For k = -6: c1/c2 = (3-(-6))/(-(-6)) = 9/6 = 3/2. And a1/a2 = -6/12 = -1/2. Since 3/2 ≠ -1/2, this gives no solution.
Answer: k = -6. The system has no solution when k = -6.
Example 5: Finding k for Infinitely Many Solutions
Problem: Find k for which the system 2x + 3y = 9 and 6x + 9y = k has infinitely many solutions.
Solution:
Step 1: Standard form:
2x + 3y - 9 = 0
6x + 9y - k = 0
Step 2: a1/a2 = 2/6 = 1/3, b1/b2 = 3/9 = 1/3
Step 3: Already a1/a2 = b1/b2 = 1/3. For infinitely many solutions, we also need c1/c2 = 1/3.
Step 4: c1/c2 = (-9)/(-k) = 9/k. Setting 9/k = 1/3 gives k = 27.
Answer: k = 27. The system has infinitely many solutions when k = 27.
Example 6: Finding k for a Unique Solution
Problem: For what values of k does the system kx + 2y = 5 and 3x + 4y = 10 have a unique solution?
Solution:
Step 1: For a unique solution: a1/a2 ≠ b1/b2
Step 2: a1/a2 = k/3, b1/b2 = 2/4 = 1/2
Step 3: For unique solution: k/3 ≠ 1/2, which gives k ≠ 3/2
Answer: The system has a unique solution for all real values of k except k = 3/2. When k = 3/2, the system is either inconsistent or dependent.
Example 7: Checking Consistency of a System with Fractional Coefficients
Problem: Check the consistency of: x/2 + y/3 = 7 and x/3 + y/2 = 7.
Solution:
Step 1: Multiply the first equation by 6: 3x + 2y = 42 → 3x + 2y - 42 = 0
Multiply the second equation by 6: 2x + 3y = 42 → 2x + 3y - 42 = 0
Step 2: a1/a2 = 3/2, b1/b2 = 2/3
Step 3: 3/2 ≠ 2/3, so a1/a2 ≠ b1/b2
Answer: The system is consistent with a unique solution.
Example 8: Determining Consistency from Word Problem Equations
Problem: A student forms two equations from a problem: 5x - 15y = 10 and x - 3y = 2. Is the system consistent?
Solution:
Step 1: Standard form:
5x - 15y - 10 = 0
x - 3y - 2 = 0
Step 2: a1/a2 = 5/1 = 5, b1/b2 = (-15)/(-3) = 5, c1/c2 = (-10)/(-2) = 5
Step 3: a1/a2 = b1/b2 = c1/c2 = 5
Answer: The system is consistent with infinitely many solutions. The first equation is simply 5 times the second, so they represent the same line. The student actually has only one independent equation, not two, and cannot find unique values for x and y.
Example 9: Finding Two Values of k for Different Consistency Types
Problem: Consider the system (k - 1)x + y = 5 and (k + 1)x + 3y = 9. Find the value of k for which the system is inconsistent.
Solution:
Step 1: Standard form:
(k-1)x + y - 5 = 0
(k+1)x + 3y - 9 = 0
Step 2: For inconsistency: a1/a2 = b1/b2 ≠ c1/c2
Step 3: a1/a2 = (k-1)/(k+1), b1/b2 = 1/3, c1/c2 = (-5)/(-9) = 5/9
Step 4: Setting a1/a2 = b1/b2:
(k-1)/(k+1) = 1/3
3(k-1) = k+1
3k - 3 = k + 1
2k = 4
k = 2
Step 5: Check: When k = 2, a1/a2 = 1/3, b1/b2 = 1/3, c1/c2 = 5/9. Since 1/3 ≠ 5/9, the system is indeed inconsistent.
Answer: The system is inconsistent when k = 2.
Example 10: Graphical Interpretation of Consistency
Problem: Without solving, classify each system as consistent (unique), consistent (infinite), or inconsistent, and describe the graph:
(a) x + y = 4 and 2x + 2y = 8
(b) x + y = 4 and 2x + 2y = 10
(c) x + y = 4 and x - y = 2
Solution:
(a) Standard form: x + y - 4 = 0 and 2x + 2y - 8 = 0
Ratios: a1/a2 = 1/2, b1/b2 = 1/2, c1/c2 = (-4)/(-8) = 1/2
All ratios equal → Consistent (infinitely many solutions). The two lines are coincident — they are the same line x + y = 4.
(b) Standard form: x + y - 4 = 0 and 2x + 2y - 10 = 0
Ratios: a1/a2 = 1/2, b1/b2 = 1/2, c1/c2 = (-4)/(-10) = 2/5
a1/a2 = b1/b2 but ≠ c1/c2 → Inconsistent. The lines are parallel. Both have slope -1, but different y-intercepts (4 and 5).
(c) Standard form: x + y - 4 = 0 and x - y - 2 = 0
Ratios: a1/a2 = 1/1 = 1, b1/b2 = 1/(-1) = -1
1 ≠ -1 → Consistent (unique solution). The lines intersect at one point. (Solving: x = 3, y = 1.)
Real-World Applications
Understanding the consistency of linear equations has important real-world applications:
System Design and Feasibility: In engineering and business, systems of equations model constraints (budget limits, resource availability, time constraints). Checking consistency tells us whether the set of constraints can all be satisfied simultaneously. An inconsistent system means the requirements are contradictory and cannot all be met.
Data Validation: When measurements or data points lead to a system of equations, checking consistency helps identify whether the data is self-consistent or contains errors. Parallel lines (inconsistency) suggest contradictory data.
Navigation and Positioning: Determining a position from distance measurements (triangulation) involves solving simultaneous equations. If the system is inconsistent, the measurements contain errors. If dependent, more measurements are needed.
Economics: Supply and demand curves are modelled as linear equations. If the supply and demand lines are parallel, there is no equilibrium price — the market model is flawed or the scenario is infeasible.
Algebraic Problem Solving: Before spending time solving a system, checking the coefficient ratios can immediately reveal whether a unique solution exists. This saves effort and prevents futile calculations on inconsistent systems.
Key Points to Remember
- A system of linear equations is consistent if it has at least one solution (one unique solution or infinitely many).
- A system is inconsistent if it has no solution at all.
- For a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, compare the ratios a1/a2, b1/b2, and c1/c2.
- a1/a2 ≠ b1/b2 → unique solution (intersecting lines).
- a1/a2 = b1/b2 = c1/c2 → infinitely many solutions (coincident lines).
- a1/a2 = b1/b2 ≠ c1/c2 → no solution (parallel lines).
- Always write equations in standard form (ax + by + c = 0) before comparing ratios.
- When asked to find k for specific consistency, set up the ratio equation and solve for k.
- Always verify the c-ratio condition after finding k from the a and b ratios.
- Graphically: intersecting lines → one point, coincident lines → overlap completely, parallel lines → never meet.
Practice Problems
- Check the consistency: 3x + 4y = 12 and 6x + 8y = 24.
- Check the consistency: 5x - 2y = 7 and 10x - 4y = 11.
- For what value of k is the system 2x + ky = 1 and 3x - 5y = 7 inconsistent?
- Determine whether the following system has a unique solution, infinitely many solutions, or no solution: x + 2y = 3 and 3x + 6y = 12.
- Find the value of p for which the system px + 3y = p - 3 and 12x + py = p has infinitely many solutions.
- Without solving, classify the system: 7x - 3y = 11 and 14x + 6y = 22.
Frequently Asked Questions
Q1. What does consistency of linear equations mean?
Consistency refers to whether a system of linear equations has a solution. A consistent system has at least one solution (either exactly one or infinitely many), while an inconsistent system has no solution at all.
Q2. How do I check if a system of equations is consistent?
Write both equations in the form ax + by + c = 0. Compute the ratios a₁/a₂, b₁/b₂, and c₁/c₂. If a₁/a₂ ≠ b₁/b₂, the system is consistent with a unique solution. If all three ratios are equal, it's consistent with infinitely many solutions. If the first two are equal but different from the third, it's inconsistent.
Q3. What is the graphical meaning of an inconsistent system?
An inconsistent system means the two lines are parallel — they have the same slope but different y-intercepts. Since parallel lines never intersect, there is no point (x, y) that lies on both lines, and hence no solution.
Q4. Can a system with infinitely many solutions be called consistent?
Yes. A system with infinitely many solutions is consistent (specifically, it is called a 'dependent' consistent system). Consistent simply means 'has at least one solution', and infinitely many is certainly 'at least one'.
Q5. What is the difference between dependent and independent consistent systems?
An independent consistent system has exactly one unique solution (the lines intersect at one point). A dependent consistent system has infinitely many solutions (the lines are coincident, meaning they are the same line). Both are consistent because they have solutions.
Q6. How do I find the value of k for a system to have no solution?
Set a₁/a₂ = b₁/b₂ and solve for k. Then verify that c₁/c₂ is NOT equal to a₁/a₂ for that value of k. If c₁/c₂ also equals a₁/a₂, the system has infinitely many solutions (not no solution), and you must reject that value of k.
Q7. Why is checking consistency important before solving a system?
Checking consistency tells you whether a solution exists before you spend time trying to find one. If the system is inconsistent, no amount of algebraic manipulation will produce a valid answer. Checking consistency first saves time and prevents frustration.
Q8. What if one equation is a multiple of the other?
If one equation is a constant multiple of the other (e.g., 2x + 3y = 5 and 6x + 9y = 15), then all three ratios a₁/a₂ = b₁/b₂ = c₁/c₂ are equal. The system has infinitely many solutions — the two equations are really just one equation written in two different forms.
Q9. Is consistency a topic in CBSE Class 10 board exams?
Yes, checking the consistency of a pair of linear equations is a frequently tested topic in CBSE Class 10 board exams. Questions may ask you to determine the type of system, find parameter values for specific consistency types, or interpret the graphical meaning.
Q10. Can three or more equations be checked for consistency?
Yes, though the method becomes more complex. For three equations in two unknowns, you check consistency pairwise and also verify that all three are mutually consistent. Systems of three or more equations are studied in detail in higher classes using matrix methods.
Related Topics
- Pair of Linear Equations in Two Variables
- Graphical Method for Linear Equations
- Substitution Method
- Elimination Method
- Cross-Multiplication Method
- Word Problems on Pair of Linear Equations
- Word Problems Solved Graphically
- Reducing Equations to Linear Form
- Conditions for Solvability
- Linear Equations in Real Life
