Solve the following system of equations:
x^2+y^2=24
x-y=2
x=y+2
(y+2)^2+y^2=24
2y^2+4y+4=24
.5y^2+y=5
quadratic formula
y=-1+sqrt(11),-1-sqrt(11)
x=1+sqrt(11),1-sqrt(11)
Thank you so much I really appreciate it!
To solve this system of equations, we can start by solving one equation for one variable and substituting it into the other equation. Let's solve the second equation for x:
x - y = 2
Adding y to both sides, we get:
x = 2 + y
Now we can substitute this expression for x in the first equation:
(2 + y)^2 + y^2 = 24
Expanding and simplifying, we have:
4 + 4y + y^2 + y^2 = 24
2y^2 + 4y - 20 = 0
Dividing the entire equation by 2, we get:
y^2 + 2y - 10 = 0
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac))/(2a)
In this case, a = 1, b = 2, and c = -10. Substituting these values into the formula, we have:
y = (-2 ± √(2^2 - 4(1)(-10)))/(2(1))
y = (-2 ± √(4 + 40))/2
y = (-2 ± √(44))/2
y = (-2 ± 2√11)/2
y = -1 ± √11
So we have two possible values for y: -1 + √11 and -1 - √11.
Now we can substitute each value of y back into the second equation to find the corresponding values of x.
When y = -1 + √11:
x - (-1 + √11) = 2
x + 1 - √11 = 2
x - √11 = 1
Adding √11 to both sides:
x = 1 + √11
When y = -1 - √11:
x - (-1 - √11) = 2
x + 1 + √11 = 2
x + √11 = 1
Subtracting √11 from both sides:
x = 1 - √11
So the solutions to the system of equations are:
x = 1 + √11 and y = -1 + √11
x = 1 - √11 and y = -1 - √11