Linear Equations in Two Variables — Karnataka (SSLC) Class 9 Mathematics Solutions (Free)

Equation: 2x + 3y = 12 Method: Choose values for x and solve for y (or vice versa). Solution 1: Let x = 0 2(0) + 3y = 12 3y = 12 y = 4 Solution: (0, 4) Solution 2: Let x = 3 2(3) + 3y = 12 6 + 3y = 12 3y = 6 y = 2 Solution: (3, 2) Solution 3: Let x = 6 2(6) + 3y = 12 12 + 3y = 12 3y = 0 y = 0 Solution: (6, 0) Solution 4: Let x = -3 2(-3) + 3y = 12 -6 + 3y = 12 3y = 18 y = 6 Solution: (-3, 6) Answer: Four solutions are (0, 4), (3, 2), (6, 0), and (-3, 6)

Equations: 2x + y = 7 ... (1) x - y = 2 ... (2) Method: Elimination Step 1: Add equations (1) and (2) to eliminate y (2x + y) + (x - y) = 7 + 2 3x = 9 Step 2: Solve for x x = 9/3 = 3 Step 3: Substitute x = 3 in equation (2) 3 - y = 2 y = 3 - 2 = 1 Step 4: Verify solution in both equations Equation (1): 2(3) + 1 = 6 + 1 = 7 ✓ Equation (2): 3 - 1 = 2 ✓ Answer: x = 3, y = 1

Equation: 2x + 3y - 6 = 0 Step 1: Solve for y 3y = -2x + 6 y = -2x/3 + 6/3 y = (-2/3)x + 2 Step 2: Compare with y = mx + c This is in the form y = mx + c where: m = -2/3 (slope) c = 2 (y-intercept) Step 3: Interpret the slope Slope m = -2/3 < 0, so the line is falling (decreasing from left to right). For every 3 units moved right, the line drops 2 units. Step 4: Interpret the y-intercept The line crosses the y-axis at (0, 2). Answer: y = (-2/3)x + 2 Slope (m) = -2/3 y-intercept (c) = 2

Point: (2, 1) means x = 2, y = 1 Equations: x + 2y = 4 ... (1) 2x - y = 3 ... (2) Step 1: Check equation (1) Substitute x = 2, y = 1: 2 + 2(1) = 2 + 2 = 4 ✓ The point satisfies equation (1). Step 2: Check equation (2) Substitute x = 2, y = 1: 2(2) - 1 = 4 - 1 = 3 ✓ The point satisfies equation (2). Step 3: Conclusion Since (2, 1) satisfies both equations, it is the solution of the system. Answer: Yes, (2, 1) is a solution of the given system.

Equations: x + y = 5 ... (1) x - y = 1 ... (2) Step 1: Find solutions for equation (1): x + y = 5 Let x = 0: y = 5 → (0, 5) Let x = 5: y = 0 → (5, 0) Let x = 2: y = 3 → (2, 3) Step 2: Find solutions for equation (2): x - y = 1 Let x = 0: y = -1 → (0, -1) Let x = 1: y = 0 → (1, 0) Let x = 3: y = 2 → (3, 2) Step 3: Plot both lines Line 1 (x + y = 5): passes through (0, 5) and (5, 0) Line 2 (x - y = 1): passes through (0, -1) and (1, 0) Step 4: Find intersection point Both lines intersect at a single point. From the graphical analysis, the intersection is at (3, 2). Step 5: Verify algebraically Add equations: (x + y) + (x - y) = 5 + 1 2x = 6 → x = 3 Substitute in equation (1): 3 + y = 5 → y = 2 Answer: Solution is (3, 2)

Point: (1, 2) means x = 1, y = 2 Equations: ax + by = 8 ... (1) 2ax - by = 5 ... (2) Step 1: Substitute (1, 2) into equation (1) a(1) + b(2) = 8 a + 2b = 8 ... (3) Step 2: Substitute (1, 2) into equation (2) 2a(1) - b(2) = 5 2a - 2b = 5 ... (4) Step 3: Solve the system of equations (3) and (4) From (3): a + 2b = 8 From (4): 2a - 2b = 5 Add equations (3) and (4): (a + 2b) + (2a - 2b) = 8 + 5 3a = 13 a = 13/3 Step 4: Find b Substitute a = 13/3 in equation (3): 13/3 + 2b = 8 2b = 8 - 13/3 2b = 24/3 - 13/3 2b = 11/3 b = 11/6 Step 5: Verify Equation (1): (13/3)(1) + (11/6)(2) = 13/3 + 22/6 = 13/3 + 11/3 = 24/3 = 8 ✓ Equation (2): 2(13/3)(1) - (11/6)(2) = 26/3 - 22/6 = 26/3 - 11/3 = 15/3 = 5 ✓ Answer: a = 13/3, b = 11/6

Given: Point (1, 1), Slope m = 2 Step 1: Use point-slope form y - y₁ = m(x - x₁) y - 1 = 2(x - 1) Step 2: Expand y - 1 = 2x - 2 y = 2x - 2 + 1 y = 2x - 1 Step 3: Convert to standard form ax + by + c = 0 2x - y - 1 = 0 Step 4: Verify that (1, 1) lies on the line 2(1) - 1 - 1 = 2 - 1 - 1 = 0 ✓ Step 5: Verify slope From 2x - y - 1 = 0, solving for y: y = 2x - 1 Slope = coefficient of x = 2 ✓ Answer: 2x - y - 1 = 0

Equations: 3x + 2y = 12 ... (1) 2x - 3y = 5 ... (2) Step 1: Express x from equation (2) 2x = 5 + 3y x = (5 + 3y)/2 ... (3) Step 2: Substitute (3) into equation (1) 3[(5 + 3y)/2] + 2y = 12 Step 3: Multiply by 2 to clear fraction 3(5 + 3y) + 4y = 24 15 + 9y + 4y = 24 13y = 24 - 15 13y = 9 y = 9/13 Step 4: Find x using equation (3) x = (5 + 3(9/13))/2 x = (5 + 27/13)/2 x = (65/13 + 27/13)/2 x = (92/13)/2 x = 92/26 x = 46/13 Step 5: Verify in equation (1) 3(46/13) + 2(9/13) = 138/13 + 18/13 = 156/13 = 12 ✓ Step 6: Verify in equation (2) 2(46/13) - 3(9/13) = 92/13 - 27/13 = 65/13 = 5 ✓ Answer: x = 46/13, y = 9/13