+152 Find all ordered pairs x, y of real numbers such that x+y=10 and x^2+y^2=64.
For example, to enter the solutions (2, 4) and (-3, 9), you would enter "(2,4),(-3,9)" (without the quotation marks).
+86 We can solve for x and y in this system of equations by using substitution or the method of completing the square. Here, we'll use substitution:
Solve one equation for one variable:
From the first equation, we can express one variable in terms of the other:
x = 10 - y (Equation 1)
Substitute into the second equation:
Substitute this expression for x in the second equation:
(10 - y)^2 + y^2 = 64
Expand and solve for y:
Expand the squared term:
100 - 20y + y^2 + y^2 = 64 Combine like terms: 2y^2 - 20y + 36 = 0 Factor the equation: 2(y^2 - 10y + 18) = 0
This factors further as: 2(y - 6)(y - 3) = 0
Therefore, the possible values for y are:
y = 6 or y = 3
Substitute y back to find x:
Substitute each value of y back into Equation (1) to find the corresponding value of x:
If y = 6:
x = 10 - 6 = 4
If y = 3:
x = 10 - 3 = 7
Solutions:
Therefore, the ordered pairs (x, y) that satisfy the system of equations are:
(x, y) = (4, 6)
(x, y) = (7, 3)
