Linear Equations in Two Variables

Infinitely many solutions

Step 1: A linear equation in two variables always has infinitely many solutions. Step 2: For any value we choose for x, we can find a corresponding value of y that satisfies the equation. Step 3: For example, if x = 0, then 3y = 12, so y = 4, giving us (0, 4). Step 4: If x = 3, then 6 + 3y = 12, so 3y = 6, giving y = 2 and the solution (3, 2). Step 5: Since we can choose any real number for x and find a corresponding y, there are infinitely many solutions.

k = 2

Step 1: Since (2, k) is a solution of 3x + 2y = 10, it must satisfy the equation. Step 2: Substitute x = 2 and y = k into the equation: 3(2) + 2(k) = 10. Step 3: Simplify the left side: 6 + 2k = 10. Step 4: Subtract 6 from both sides: 2k = 10 - 6 = 4. Step 5: Divide both sides by 2: k = 4/2 = 2.

(3, 4)

Step 1: To find which point lies on the line x + y = 7, substitute each point's coordinates. Step 2: For (2, 4): 2 + 4 = 6 ≠ 7, so this point doesn't lie on the line. Step 3: For (3, 4): 3 + 4 = 7 ✓, this equals 7, so this point lies on the line. Step 4: For (1, 5): 1 + 5 = 6 ≠ 7, so this point doesn't lie on the line. Step 5: For (4, 2): 4 + 2 = 6 ≠ 7, so this point doesn't lie on the line.

3x + 2y = 25

Step 1: Identify the variables: cost of a pen = ₹x, cost of a pencil = ₹y. Step 2: We have 3 pens, so total cost of pens = 3 × x = 3x. Step 3: We have 2 pencils, so total cost of pencils = 2 × y = 2y. Step 4: Total cost = cost of pens + cost of pencils = 3x + 2y. Step 5: Since the total cost is ₹25, the equation is 3x + 2y = 25.