Vardaan
Class 9 Maths Ӣ Chapter 4 Ӣ NCERT + RS Aggarwal + RD Sharma

Linear Equations in Two Variables

Vardaan Learning Institute  |  School-Exam Focused Notes

📚 1. Key Definitions

Linear Equation in Two Variables: An equation of the form ax + by + c = 0
where a, b, c are real numbers and a, b are not both zero.

Ӣ Solution: An ordered pair (x, y) = (p, q) that satisfies the equation when substituted
Ӣ Infinite solutions: A linear equation in 2 variables has infinitely many solutions
Ӣ Graph: The graph of ax + by + c = 0 is always a straight line

Standard Forms

General Form
ax + by + c = 0
a, b ≠ both zero
Slope-Intercept
y = mx + c
m = slope, c = y-intercept
Equation of X-axis
y = 0
Equation of Y-axis
x = 0

🔢 2. Finding Solutions (Solution Table Method)

Example: Find 4 solutions of 2x + y = 6
Rearrange: y = 6 ∠’ 2x

x y = 6 ∠’ 2x Solution (x, y)
0 6 (0, 6)
1 4 (1, 4)
2 2 (2, 2)
3 0 (3, 0)
All these points lie on the line 2x + y = 6. Connecting them gives the straight line graph.

📊 3. Types of Lines and Their Equations

Type of Line Equation Example Characteristic
Horizontal line y = k (constant) y = 3, y = ∠’2 Parallel to X-axis; no x term
Vertical line x = k (constant) x = 4, x = ∠’1 Parallel to Y-axis; no y term
Passes through origin ax + by = 0 → y = (∠’a/b)x 2x ∠’ 3y = 0 Always has (0,0) as solution
X-axis y = 0 — b=1, a=0, c=0
Y-axis x = 0 — a=1, b=0, c=0

📈 4. Graphing a Linear Equation — Step-by-Step

Graph 3x + 2y = 12:
Step 1: Find x-intercept (put y=0): 3x = 12 → x = 4 → point: (4, 0)
Step 2: Find y-intercept (put x=0): 2y = 12 → y = 6 → point: (0, 6)
Step 3: Find one more point (put x=2): 6 + 2y = 12 → y = 3 → point: (2, 3)
Step 4: Plot all 3 points on graph paper and join with a straight line.

Minimum 2 points needed to draw a line; 3rd point is used for verification.
X Y O 1 2 3 4 1 2 3 4 6 (0,6) (4,0) (2,3) 3x + 2y = 12

🌍 5. Equations of Practical/Word Problems

Word Problem → Equation Examples (RS Aggarwal style):

Ex 1: The cost of 2 apples and 3 bananas is ₹16. Express as linear equation.
Let apples cost x, bananas cost y → 2x + 3y = 16

Ex 2: Age of Rani is 5 years more than twice age of her son.
Let son's age = x, Rani's age = y → y = 2x + 5 or 2x ∠’ y + 5 = 0

Ex 3: A boat goes 30 km upstream and 44 km downstream in 10 hours. The speed of stream is 3 km/h. Let speed of boat = x. Then form the equation.
📝 Important Special Cases ”¢ x = a (e.g., x = 3) — vertical line parallel to Y-axis (does NOT depend on y)
”¢ y = b (e.g., y = ∠’2) — horizontal line parallel to X-axis (does NOT depend on x)
”¢ y = mx (c=0) — line through origin. E.g., y = 2x passes through (0,0), (1,2), (2,4).

🔢 6. Number of Solutions (Geometric Interpretation)

Case Graph Number of Solutions
One linear equation in 2 variables One straight line (infinite points) Infinite
One variable only: 2x = 6 (i.e., x = 3) Vertical line x = 3 on number line; single point on 1D 1 in 1D; infinite in 2D
A point (xâ‚€, yâ‚€) Single point Checks if point satisfies equation
✏️ Practice Questions — Linear Equations in Two Variables (30 Questions)

Section A — 1 Mark Easy

Q1. Express x = 5 as a linear equation in two variables.
Q2. How many solutions does 2x + 3y = 10 have?
Q3. Is (2, 3) a solution of x + 2y = 8? Verify.
Q4. Write the equation of X-axis in standard form.
Q5. Find the x-intercept of the line 3x ∠’ 4y = 12.
Q6. Find the y-intercept of 2x + 5y = 20.
Q7. What type of line does ax + by + c = 0 always represent?
Q8. Write a linear equation in two variables passing through (0, 0) and (2, 3).

Section B — 2/3 Mark Medium

Q9. Find 3 solutions of x + 2y = 6. Make a table.
Q10. Find 3 solutions of 2x ∠’ y = 4.
Q11. Draw the graph of y = 3x + 1 (use 3 points).
Q12. Draw the graph of x + y = 5 and find x-intercept and y-intercept.
Q13. The cost of a pencil is ₹1.50 more than the cost of an eraser. Write this as a linear equation.
Q14. Express the statement: "perimeter of a rectangle with length l and breadth b is 56 cm" as a linear equation.
Q15. If (4, k) lies on x ∠’ 2y + 6 = 0, find k.
Q16. Find the value of a if (a, 3) is a solution of 2x + ay = 16.

Section C — 3/4 Mark Medium

  1. Q17. Draw the graph of 2x + 3y = 12. Shade the region where 2x + 3y ≤ 12.
  2. Q18. The age of a father is 3 times the age of his son. 10 years later, the father will be twice the age. Write both equations and solve graphically.
  3. Q19. Rupesh has some ₹5 and ₹10 coins totalling ₹200. Write a linear equation and find all possible positive integer solutions where number of ₹10 coins is ≤ 15.
  4. Q20. A fraction becomes 9/11 when 2 is added to both. Write the linear equation and find 2 other pairs.
  5. Q21. Draw the graph of x = ∠’3 and y = 2 on the same axes. What is the intersection point?
  6. Q22. If the graph of ax + 5y = 20 passes through (2, 2), find a. Then draw the graph.

Section D — 5 Mark Challenge Hard

  1. Q23. Draw the graph of 4x + 3y = 24, 2x + y = 6 on the same axes. Where do they intersect?
  2. Q24. The sum of two numbers is 18, and their difference is 6. Form linear equations and solve graphically. Also solve algebraically.
  3. Q25. A linear equation passes through (3, 2) and (∠’1, 4). Find the equation.
  4. Q26. Monu earns ₹600 per day working as a carpenter and ₹400 per day as a painter. He worked for 20 days and earned ₹9000. Write linear equations and find how many days he worked at each job.
  5. Q27. In a class, number of girls is 8 more than half the number of boys. Total students = 40. Form and solve graphically.
  6. Q28. Prove geometrically that any linear equation ax + by + c = 0 always represents a straight line by showing that any three distinct solution points are collinear.
  7. Q29. The equation kx + 3y = 12 passes through (∠’1, 5). Find k. Is this line parallel to the x-axis or y-axis? Plot and determine.
  8. Q30. Two supplementary angles are in the ratio 2:3. Let them be 2x and 3x. Form linear equation and also express in the form ax + by + c = 0.
✅ Key Answers: Q3:Yes(2+6=8✓) | Q5:x=4 | Q6:y=4 | Q15:k=5 | Q16:a=2 | Q21:(-3,2) | Q22:a=5 | Q23:(3,4) approx. | Q24:(12,6) | Q25:x+2y=7 | Q26:5 days carpenter, 15 days painter | Q29:k=-3, line is neither parallel to x nor y

📝 Quick Revision

  1. Standard form: ax + by + c = 0 where a, b are not both zero.
  2. Every linear equation in 2 variables has infinitely many solutions.
  3. Graph of a linear equation = always a straight line.
  4. To draw a graph: find at least 2 solutions (intercepts are easiest), plot, join.
  5. x = k → vertical line (parallel to Y-axis). y = k → horizontal line (parallel to X-axis).
  6. y = mx → passes through origin. Every equation of form ay = bx passes through (0,0).