Question

Find all pairs (x,y) of real numbers such that:

x + y = 10
x² + y² = 56

A) (2, 8)

B) (4, 6)

C) (-3, 13)

D) (0, 10)

Answer 1

To find the pairs (x, y) that satisfy the given equations, first express y in terms of x, substitute it into the second equation, simplify and solve the quadratic equation, find x and y, and verify by plugging them back into the original equations.

Step-by-step explanation:

The student's question is asking to find all pairs of real numbers (x, y) such that x + y = 10 and x² + y² = 56. To solve for these pairs, we will employ a step-by-step approach to simplify and solve the system of equations.

1. Use the first equation x + y = 10 to express y in terms of x: y = 10 - x.
2. Substitute this expression for y into the second equation to get x² + (10 - x)² = 56.
3. Expand and simplify the equation to solve for x: 2x² - 20x + 44 = 0. This is a quadratic equation that can be factored or solved using the quadratic formula.
4. Find the values of x that solve the equation, and then use these to find the corresponding values of y.
5. Once you have the pairs (x, y), verify each one by plugging them back into the original equations to make sure they satisfy both.

After solving the quadratic equation and checking the solutions, we would find the correct pairs (x, y) that meet the criteria given in the problem statement. Note that the pairs listed in options A) to D) are to be checked against the equations.