x and y are real numbers that satisfy the equations x2+y2=125 and x+y=13. What is the sum of all (distinct) possible values of x?
To find the sum of all distinct possible values of x, we first need to solve the given system of equations.
Let's start by rearranging the second equation, x + y = 13, to solve for y: y = 13 - x.
Substitute this expression for y into the first equation, x^2 + y^2 = 125:
x^2 + (13 - x)^2 = 125
Expanding and simplifying this equation, we get:
x^2 + 169 - 26x + x^2 = 125
2x^2 - 26x + 44 = 0
Divide through the equation by 2 to simplify:
x^2 - 13x + 22 = 0
Now, we can solve this quadratic equation for x by factoring or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula states that for an equation ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying this formula to our equation x^2 - 13x + 22 = 0, we have:
x = (-(-13) ± √((-13)^2 - 4(1)(22))) / (2(1))
x = (13 ± √(169 - 88)) / 2
x = (13 ± √81) / 2
x = (13 ± 9) / 2
So we have two possible values for x:
x1 = (13 + 9) / 2 = 11
x2 = (13 - 9) / 2 = 2
Now, we need to determine the corresponding values of y for each value of x:
For x1 = 11, substitute this value into the equation y = 13 - x:
y1 = 13 - 11 = 2
For x2 = 2, substitute this value into the equation y = 13 - x:
y2 = 13 - 2 = 11
Therefore, the distinct values of x are 11 and 2. The sum of all distinct values of x is 11 + 2 = 13. So the sum of all (distinct) possible values of x is 13.