Algebra

Subtracting the first equation from the second: (sqrt(3) - sqrt(2))x = sqrt(2) - sqrt(3). This implies x = -1. Substituting x = -1 into the first equation: -sqrt(2) + 3y = sqrt(3), so 3y = sqrt(3) + sqrt(2), which gives y = (sqrt(3) + sqrt(2)) / 3.

Adding the equations: 94x + 94y = 282, so x + y = 3. Subtracting: -12x + 12y = -12, so -x + y = -1. Adding these two results: 2y = 2, y = 1. Then x = 2.

Solving the system x+3y=6 and 3x-5y=18: multiply the first by 3 to get 3x+9y=18. Subtracting the second from this gives 14y=0, so y=0 and x=6. Substituting into 7x+3y=K gives 7(6)+3(0)=42.

For two linear equations to be coincident, their coefficients must be proportional: k/3 = m/1 = 2/1. This implies k/3 = 2, so k = 6.

Equation 1: 2x - 3y = -5. Equation 2: -4x + 6y = 10, which simplifies to 2x - 3y = -5. Since both equations are identical, there are infinite solutions.

Equations: 6ax + 6by = 3a + 2b and 6bx - 6ay = 3b - 2a. Multiply first by a and second by b: 6a^2x + 6aby = 3a^2 + 2ab; 6b^2x - 6aby = 3b^2 - 2ab. Adding: 6x(a^2 + b^2) = 3(a^2 + b^2) => x = 1/2. Substituting x = 1/2: 3a + 6by = 3a + 2b => 6by = 2b => y = 1/3.

From 4x - 6y = -4, we get 2x - 3y = -2, so 2x = 3y - 2. Substitute into 8x - 2y = 48: 4(3y - 2) - 2y = 48 -> 12y - 8 - 2y = 48 -> 10y = 56 -> y = 5.6. Then 2x = 3(5.6) - 2 = 16.8 - 2 = 14.8, so x = 7.4.

Subtracting the first equation (a - 2b = 2) from the second (3a - 2b = 6) gives 2a = 4, so a = 2. Substituting a = 2 into the first equation: 2 - 2b = 2, so -2b = 0, b = 0. The solution is (2, 0).

The second equation 10x + 12y = 0.5 can be rewritten as 5x + 6y = 0.25 by dividing by 2. Since 5x + 6y = -1 and 5x + 6y = 0.25 are contradictory, the system is inconsistent.

For a system a1x + b1y = c1 and a2x + b2y = c2, if a1/a2 != b1/b2, the system is consistent and independent. Here, 5/2 != -2/5, so it is consistent.

The second equation is -1/6x + 6y = -48. Multiply by -6 to get x - 36y = 288. The first equation is x - 36y = 8. Since 8 is not equal to 288, the lines are parallel and the system is inconsistent.

From x - y = 5, x = y + 5. Substitute into x^2 + y^2 = 25: (y + 5)^2 + y^2 = 25. y^2 + 10y + 25 + y^2 = 25. 2y^2 + 10y = 0. 2y(y + 5) = 0. So y = 0 or y = -5. If y = 0, x = 5. If y = -5, x = 0. Solutions are (5, 0) and (0, -5).

From the second equation, x=2. Substituting into the first equation: 3(2) - y = 0, so y=6.

The equations are x - 2y = 6 and x - 2y = -6. Since the left sides are identical but the constants are different, the lines are parallel and never intersect, meaning there is no solution.

Adding the equations: (x+y) + (x-y) = 5 + 1 => 2x = 6 => x = 3. Substituting x=3 into x+y=5 gives y = 2. The solution is (3,2).