Pair of Linear Equations in Two Variables

The equations represent coincident lines, The system has infinitely many solutions, The equations are dependent, a₁/a₂ = b₁/b₂ = c₁/c₂

Rewriting in standard form: 2x + 3y - 12 = 0 and 4x + 6y - 24 = 0. Here a₁/a₂ = 2/4 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = -12/-24 = 1/2. Since all ratios are equal, the lines are coincident, giving infinitely many solutions.

6

For infinitely many solutions, a₁/a₂ = b₁/b₂ = c₁/c₂. Here a₁/a₂ = 2/k, b₁/b₂ = 3/9 = 1/3, c₁/c₂ = 7/21 = 1/3. For infinitely many solutions: 2/k = 1/3, so k = 6.

x + 2y = 5 and 2x + 4y = 15, 3x - y = 8 and 6x - 2y = 10, x + y = 3 and x + y = 7, x - y = 1 and 2x - 2y = 3

For parallel lines, a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Checking each: (1) 1/2 = 2/4 = 1/2, but -5/-15 = 1/3 ≠ 1/2 ✓ (2) 3/6 = 1/2, -1/-2 = 1/2, but -8/-10 = 4/5 ≠ 1/2 ✓ (3) 1/1 = 1, 1/1 = 1, but -3/-7 = 3/7 ≠ 1 ✓ (4) 2/4 = 1/2, 3/6 = 1/2, -6/-12 = 1/2 (coincident, not parallel) (5) 1/2 = 1/2, -1/-2 = 1/2, but -1/-3 = 1/3 ≠ 1/2 ✓

3

Multiply first equation by 3 and second by 2: 9x + 6y = 33 and 4x + 6y = 8. Subtracting: 5x = 25, so x = 5. Substituting in first equation: 3(5) + 2y = 11, so 15 + 2y = 11, 2y = -4, y = -2. Therefore x + y = 5 + (-2) = 3.