Linear Equations in Two Variables — Class 9 (Expanded)

Important formulas, 2 main solved questions (with full steps), plus 5 practice problems — "agodar" style.

What is a linear equation in two variables?

A linear equation in two variables x and y can be written as ax + by + c = 0, where a and b are not both zero. Its graph is a straight line.

General form: ax + by + c = 0
Slope-intercept form: y = mx + c (here m = −a/b)

Important formulas & quick facts

  • Slope m = (y₂ − y₁) / (x₂ − x₁)
  • Y-intercept = value of y when x = 0 (point (0, c)).
  • X-intercept = value of x when y = 0 (point (−c/a, 0)).
  • Two-point form: Equation of line passing through (x₁,y₁) and (x₂,y₂): (y − y₁) / (x − x₁) = (y₂ − y₁) / (x₂ − x₁).
  • Parallel lines: have equal slopes. Perpendicular lines: product of slopes = −1.

Main Solved Question 1 (detailed)

Q: Solve the equation 3x + 4y = 12 for two different integer solutions. Then find the slope and intercepts of the corresponding line.
Solution — Step 1: find solutions
Let x = 0 ⇒ 3(0) + 4y = 12 ⇒ y = 3 → point A(0,3).
Let x = 4 ⇒ 3(4) + 4y = 12 ⇒ 12 + 4y = 12 ⇒ y = 0 → point B(4,0). Step 2: slope
m = (y₂ − y₁)/(x₂ − x₁) = (0 − 3)/(4 − 0) = −3/4. Step 3: intercepts
y-intercept = 3 (point A). x-intercept = 4 (point B). Answer: Two integer solutions: (0,3) and (4,0). Slope = −3/4. Intercepts: x = 4, y = 3.

Main Solved Question 2 (detailed)

Q: A shop sells pencils and erasers. 2 pencils + 3 erasers cost ₹50, while 5 pencils + 2 erasers cost ₹85. Let x be the cost of one pencil and y the cost of one eraser. Find x and y.
Solution
Translate into equations: 
2x + 3y = 50
5x + 2y = 85
We'll solve by elimination. Multiply first equation by 5 and second by 2 to eliminate x: 
10x + 15y = 250
10x + 4y  = 170
------------
      11y = 80
So y = 80/11 = ₹7.27 (approx). Now substitute into one equation, say 2x + 3y = 50: 
2x + 3(80/11) = 50
2x + 240/11 = 50
2x = 50 - 240/11 = (550 - 240)/11 = 310/11
x = 155/11 = ₹14.09 (approx)
So pencil ≈ ₹14.09 and eraser ≈ ₹7.27. Check: 2(155/11)+3(80/11)= (310+240)/11 = 550/11 = 50 ✓

Graphical interpretation (quick)

Each linear equation ax + by + c = 0 represents a line. A system of two linear equations corresponds to two lines; their intersection (if any) gives the simultaneous solution.

Methods to solve two linear equations

  • Substitution: solve one equation for a variable and substitute into the other.
  • Elimination: add/subtract equations to remove one variable (used in example 2).
  • Graphical: draw both lines and read intersection.

Extra solved tips

  • To get integer solutions quickly, set x or y to small integers (0,1,2...).
  • When coefficients are large, use elimination after multiplying equations to match coefficients.
  • Always check solutions by substitution into original equations.