Question 1: Solve for x and y: x + y = 7 x^2 - y^2 = 20

Answer: {"x":5,"y":2}

Question 2: Find the values of x and y satisfying: 2x + y = 10 y = x^2 - 4

Answer: {"x":3,"y":2}

Question 3: Evaluate and solve: x^2 + 3x = 4x + 5

Answer: [1,-5]

Question 4: Construct a quadratic equation with roots at x=2 and x=4, and a linear equation that intersects it at one point. Write their equations.

Answer: {"linear":"y = 3x - 2","quadratic":"y = (x - 2)(x - 4) = x^2 - 6x + 8"}

Question 5: Solve the system: x^2 - y = 3 y = 2x + 1

Answer: {"x":2,"y":5}

Question 6: A rectangle's length is given by y = 2x + 3, and its width by y = x^2 - 4. Find the points where the rectangle's length and width are equal.

Answer: {"x":1,"y":5}

Question 7: Solve for x and y: x + y = 12 x^2 + y^2 = 85

Answer: {"x":7,"y":5}

Question 8: A quadratic equation is y = x^2 + 2x + 1. Find the point of intersection with the line y = -x + 4.

Answer: {"x":-1,"y":3}

Question 9: An algebraic problem involves solving: x^2 + 5x = 14 x + y = 7

Answer: {"x":2,"y":5}

Question 10: Identify and correct the mistake in solving the system: x + y = 6 x^2 - y^2 = 12 Common mistake: Solution: x=4, y=2

Answer: The mistake is substituting or solving incorrectly for the second equation; correct approach involves factoring or substitution.

Question 11: Challenge: Find all solutions to the system: x^2 + y^2 = 25 2x + y = 10

Answer: [{"x":3,"y":4},{"x":4,"y":2}]