Worked example 1
Question: Solve 2x+3y=13 and 3x−2y=0.
Solution: Multiply the first equation by 2: 4x+6y=26. Multiply the second by 3: 9x−6y=0. Add: 13x=26, so x=2. Substitute into 3x−2y=0: 6−2y=0, so y=3.
What this chapter covers
Represent two conditions with linear equations, interpret their graphs, and solve consistent or inconsistent systems by substitution and elimination.
Graphical meaning
Each linear equation is a straight line. Intersecting lines give one solution; parallel distinct lines give no solution; coincident lines give infinitely many solutions. For a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0, compare coefficient ratios to identify these cases.
Algebraic methods
Substitution isolates one variable and replaces it in the other equation. Elimination adds or subtracts suitable multiples of equations to remove one variable. Define variables before translating a word problem.
Key formulae: Unique solution if a₁/a₂ ≠ b₁/b₂; no solution if a₁/a₂ = b₁/b₂ ≠ c₁/c₂; infinitely many if all three ratios are equal.
Worked examples: CBSE question patterns
These are original practice questions modeled on common board-exam patterns; they are not reproduced past-paper questions.
Worked example 2
Question: Adult tickets cost ₹50 and child tickets ₹30. A group buys 12 tickets for ₹480. How many of each?
Solution: Let adults=a and children=c. a+c=12; 50a+30c=480. Divide the second by 10: 5a+3c=48. Subtract 3(a+c)=36: 2a=12, so a=6 and c=6.
Practical activity / investigation
Choose two equations such as x+y=5 and 2x−y=1. Make a value table, plot both lines on graph paper, read the intersection, then verify it by substitution in both equations.
Record: Write the aim, materials, steps, observations and conclusion in your notebook. Follow teacher supervision for any classroom activity.
Quick practice with answers
Try each question before opening its solution.
Practice 1: Solve x+y=9 and x−y=3.
Add the equations: 2x=12, x=6; then y=3.
Practice 2: Classify 2x+4y=8 and x+2y=5.
The left coefficients are proportional but constants are not (2/1=4/2 ≠ 8/5), so there is no solution.
Practice 3: A pen and notebook together cost ₹85. Three pens and two notebooks cost ₹195. Find the price of each.
Let p and n be prices: p+n=85 and 3p+2n=195. Subtract 2(p+n)=170 from the second equation: p=25; then n=60.
Revision checklist
- Can I explain the key idea without memorising a worked solution?
- Have I selected the right formula and written units where needed?
- Can I show each algebraic or geometric step clearly?
- Can I check my answer by substitution, estimation or a diagram?
Aligned to the Class X Mathematics curriculum for 2026–27 and the NCERT Mathematics textbook. Always confirm any school-specific internal assessment with your teacher.