Linear Equations

23

Step 1: Let the unknown number be x. The statement 'a number increased by 12 equals 35' translates to x + 12 = 35. Step 2: Subtract 12 from both sides: x = 35 - 12 = 23. Step 3: Verify: 23 + 12 = 35 ✓. Step 4: Check other options - if x = 47, then 47 + 12 = 59 ≠ 35. Therefore, the number is 23.

Infinitely many solutions

Step 1: A linear equation in two variables always has infinitely many solutions. Step 2: We can verify by finding different solutions: when x = 0, y = 2; when x = 3, y = 0; when x = 1.5, y = 1. Step 3: Each pair (x, y) that satisfies the equation is a solution. Step 4: Since we can choose any value for x and calculate corresponding y (or vice versa), there are infinitely many solutions. This is a fundamental property of linear equations in two variables.

x = 10

Step 1: Start with (x + 2)/3 = 4. Step 2: Multiply both sides by 3 to eliminate the fraction: x + 2 = 4 × 3 = 12. Step 3: Subtract 2 from both sides: x = 12 - 2 = 10. Step 4: Verify by substituting: (10 + 2)/3 = 12/3 = 4 ✓. Common mistake: Students might forget to multiply the entire right side by 3.

4.5 cm

Step 1: Let breadth = x cm, then length = (x + 3) cm. Step 2: Perimeter = 2(length + breadth) = 2(x + 3 + x) = 2(2x + 3) = 4x + 6. Step 3: Given perimeter = 24, so 4x + 6 = 24. Step 4: Solve: 4x = 18, so x = 4.5 cm. Step 5: Verify: breadth = 4.5 cm, length = 7.5 cm, perimeter = 2(4.5 + 7.5) = 24 cm ✓.