MP Board Class 10 Maths Ch 3: Pair of Linear Equations Notes
MP Board Class 10 Maths Chapter 3: Pair of Linear Equations in Two Variables (दो चर वाले रैखिक समीकरण युग्म) — This chapter carries significant weight in the MP Board Class 10 board exam (लगभग 8–10 अंक). You will learn to solve pairs of linear equations using graphical, substitution, elimination, and cross-multiplication methods. Real-life applications of linear equations and their consistency conditions are also covered. Mastery of this chapter is essential for higher studies in algebra, coordinate geometry, and economics.
📑 Table of Contents
📐 1. Introduction to Pair of Linear Equations in Two Variables
A linear equation in two variables is of the form ax + by + c = 0, where a, b, c are real numbers and a, b ≠ 0. When we have two such equations, they form a pair of linear equations in two variables:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0
Here, x and y are the variables, and a₁, b₁, c₁, a₂, b₂, c₂ are real numbers such that a₁² + b₁² ≠ 0 and a₂² + b₂² ≠ 0.
Standard Forms and Examples
📊 2. Graphical Method of Solution
Each linear equation in two variables represents a straight line when plotted on a coordinate plane. The solution of a pair of linear equations is the point of intersection of the two lines (if they intersect).
Three Possible Cases
Step-by-Step Graphical Solution
- Step 1: Write both equations in the form y = mx + c (slope-intercept form)
- Step 2: Make a table of values for each equation (take x = -3, 0, 3 and find y)
- Step 3: Plot the points on graph paper and draw the straight lines
- Step 4: Find the coordinates of the intersection point
- Step 5: Verify by substituting the coordinates back into both equations
Solution: For x + y = 5 → y = 5 – x. Points: (0,5), (2,3), (5,0)
For 2x – y = 4 → y = 2x – 4. Points: (0,-4), (2,0), (4,4)
Intersection point: (3, 2) — Check: 3+2=5 ✓, 2(3)-2=4 ✓
🧮 3. Algebraic Methods of Solution
Method 1: Substitution Method
- Step 1: Express one variable (say y) in terms of the other (x) from one equation
- Step 2: Substitute this expression into the other equation
- Step 3: Solve the resulting single-variable equation to find x
- Step 4: Substitute the value of x back to find y
- Step 5: Verify the solution in both original equations
Solution using substitution:
From equation 2: 2x = 4 – 3y → x = (4 – 3y)/2
Substitute into eq 1: 3(4 – 3y)/2 + 2y = 11
Multiply by 2: 3(4 – 3y) + 4y = 22 → 12 – 9y + 4y = 22 → -5y = 10 → y = -2
Then x = (4 – 3(-2))/2 = (4 + 6)/2 = 5
Solution: x = 5, y = -2 — Verify: 3(5)+2(-2)=15-4=11 ✓, 2(5)+3(-2)=10-6=4 ✓
Method 2: Elimination Method
- Step 1: Multiply equations by suitable constants so that coefficients of one variable become equal (or opposites)
- Step 2: Add or subtract the equations to eliminate that variable
- Step 3: Solve the resulting equation for the remaining variable
- Step 4: Substitute back to find the eliminated variable
Solution using elimination:
Multiply eq 1 by 3: 9x – 15y = -12
Subtract eq 2 from this: (9x – 15y) – (9x + 2y) = -12 – 7
→ -17y = -19 → y = 19/17
Substitute back: 3x – 5(19/17) = -4 → 3x = -4 + 95/17 = (-68+95)/17 = 27/17 → x = 9/17
Solution: x = 9/17, y = 19/17
Method 3: Cross-Multiplication Method
For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the solution is:
x / (b₁c₂ – b₂c₁) = y / (c₁a₂ – c₂a₁) = 1 / (a₁b₂ – a₂b₁)
- Then, x = (b₁c₂ – b₂c₁) / (a₁b₂ – a₂b₁)
- And, y = (c₁a₂ – c₂a₁) / (a₁b₂ – a₂b₁)
- This works only if a₁b₂ – a₂b₁ ≠ 0 (i.e., equations are not parallel or coincident)
🔍 4. Consistency and Inconsistency of Equations
The relationship between the coefficients a₁, b₁, c₁, a₂, b₂, c₂ determines whether the system has a unique solution, infinite solutions, or no solution at all.
📘 Quick Memory Trick:
Intersecting: a₁/a₂ ≠ b₁/b₂ → Unique solution (think “different ratios” = “different slopes” = “meet”)
Parallel: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → No solution (think “same slope, different intercept” = “never meet”)
Coincident: All three ratios equal → Infinite solutions (think “same line” = “every point is a solution”)
Comparison Table — All Methods
🌍 5. Real-Life Applications
Linear equations in two variables appear in numerous real-world contexts. MP Board exams often include word problems based on these applications:
- Age Problems: Sum of ages of two people is given, and the difference in their ages at different times is known
- Money & Cost Problems: Cost of items, ticket prices, profit/loss calculations
- Speed & Distance: Two vehicles traveling at different speeds; upstream/downstream boat problems
- Number Problems: Two-digit numbers where digits are related; sum and difference of two numbers
- Geometry: Perimeter and area relationships; angles of triangles
- Supply & Demand: Price-demand equilibrium in economics
- Mixture Problems: Mixing two solutions of different concentrations
Solution: Let tens digit = x, units digit = y. Number = 10x + y, Reversed = 10y + x.
Eq 1: (10x + y) + (10y + x) = 121 → 11x + 11y = 121 → x + y = 11
Eq 2: x – y = 3 (or y – x = 3; since sum is 11, x > y, so x – y = 3)
Solving: x = 7, y = 4 → Number = 74 ← Check: 74 + 47 = 121 ✓
📋 6. Previous Year Questions (2017–2026)
❓ 7. Frequently Asked Questions (FAQ)
Q1: What is a linear equation in two variables?
A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, c are real numbers and a and b are not both zero. It represents a straight line when plotted on a coordinate plane.
Q2: How many methods are there to solve a pair of linear equations?
There are four main methods: (1) Graphical Method — plotting both lines and finding intersection; (2) Substitution Method — expressing one variable in terms of the other; (3) Elimination Method — eliminating one variable by adding/subtracting equations; (4) Cross-Multiplication Method — using the formula x/(b₁c₂-b₂c₁) = y/(c₁a₂-c₂a₁) = 1/(a₁b₂-a₂b₁).
Q3: When does a pair of linear equations have no solution?
A pair of linear equations has no solution when the lines are parallel, i.e., a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Such a system is called inconsistent.
Q4: What is the difference between consistent and inconsistent equations?
A consistent system has at least one solution (unique or infinite). An inconsistent system has no solution. Consistent systems represent intersecting or coincident lines; inconsistent systems represent parallel lines.
Q5: Which method is best for MP Board exams?
The elimination method is most frequently tested in MP Board exams. It is systematic and less error-prone. For word problems, use substitution. Use graphical method only for graph-based questions (3-4 marks). Cross-multiplication is good for verification but not recommended as the primary method due to calculation complexity.
Q6: How do I check if my solution is correct?
Substitute the values of x and y back into both original equations. If both equations are satisfied (LHS = RHS), the solution is correct. This is called verification and MP Board awards partial marks even if your final answer is wrong but you correctly verify.
Q7: What are word problems in linear equations?
Word problems are real-life scenarios that can be modeled using two variables and two equations. Common types include age problems, number problems, money/cost problems, speed and distance problems, and fraction problems. The key is to identify two variables and translate the conditions into equations.
Q8: How many marks come from this chapter in the MP Board exam?
Approximately 8-10 marks come from Chapter 3 (Pair of Linear Equations in Two Variables) in the MP Board Class 10 Mathematics exam. This includes 3-4 marks from graphical questions (where you need to plot lines on graph paper) and 4-6 marks from algebraic methods and word problems.
Q9: What is the condition for infinite solutions?
A pair of linear equations has infinite solutions when the three ratios are all equal: a₁/a₂ = b₁/b₂ = c₁/c₂. This means the two equations represent the same line (coincident lines). Every point on the line is a solution.
Q10: How do I solve age-based word problems?
Define variables for present ages (e.g., x = father’s age, y = son’s age). Form the first equation from the given present-age relationship. Form the second equation from the “n years ago” or “n years later” relationship (e.g., (x – 5) = 3(y – 5) means 5 years ago, father was 3 times older). Solve using substitution or elimination.
Q11: Can I use a calculator in the MP Board exam?
No, calculators are not allowed in MP Board Class 10 Mathematics exams. All calculations must be done manually. Practice mental arithmetic and quick multiplication/division to save time.
Q12: What common mistakes should I avoid?
Common mistakes: (1) Sign errors when moving terms across the equals sign; (2) Forgetting to multiply all terms when scaling equations in elimination; (3) Plotting graph points incorrectly — always check your points; (4) Not verifying the solution in both equations; (5) Mixing up the cross-multiplication formula — practice writing it correctly; (6) Inconsistent checking of ratios for infinite/no solution conditions.