SOLUTION: x^2 + y^2 = 9 x + y = 2 x = ? or ? y = ? or ? Give answer to 2 decimal places

Algebra ->  College  -> Linear Algebra -> SOLUTION: x^2 + y^2 = 9 x + y = 2 x = ? or ? y = ? or ? Give answer to 2 decimal places      Log On


   

Question 846641: x^2 + y^2 = 9
x + y = 2
x = ? or ?
y = ? or ?
Give answer to 2 decimal places
Found 2 solutions by Fombitz, josh_jordan:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Intersection of a circle and a straight line.
Substitute,
y=2-x
x%5E2%2B%282-x%29%5E2=9
x%5E2%2B%284-4x%2Bx%5E2%29=9
2x%5E2-4x%2B4=9
2x%5E2-4x-5=0
Use quadratic formula,
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-4%29+%2B-+sqrt%28+%28-4%29%5E2-4%2A2%2A%28-5%29+%29%29%2F%282%2A2%29+
x+=+%284%2B-+sqrt%28+16%2B40%29%29%2F%284%29
x+=+%284%2B-+sqrt%28+56%29%29%2F%284%29
x+=+%284%2B-+2%2Asqrt%28+14%29%29%2F%284%29
x+=+1%2B-+sqrt%28+14%29%2F2

Then,
y=2-x
y=2-%281%2B-sqrt%2814%29%2F2%29%29
y=1+%2B-+sqrt%2814%29%2F2 Remember the +/- sign should be -/+ but the symbol doesn't exist here.
(1-sqrt%2814%29%2F2,1%2Bsqrt%2814%29%2F2)
and
(1%2Bsqrt%2814%29%2F2,1-sqrt%2814%29%2F2)
.
.
.

Answer by josh_jordan(263) About Me  (Show Source):
You can put this solution on YOUR website!
To solve this system of equations, first, rewrite the second equation in terms of either x or y. I will rewrite it in terms of y. So, I will rewrite x + y = 2, as y = 2 - x. Next, replace y in equation 1 with 2 - x:

x%5E2%2B%282-x%29%5E2=9 ----->

x%5E2%2B%282-x%29%2A%282-x%29=9 ----->

x%5E2%2B4-4x%2Bx%5E2=9

Next, combine like terms:

2x%5E2-4x%2B4=9

Now, subtract 9 from both sides, giving us this equation in quadratic form:

2x%5E2-4x%2B4-9=0 ----->

2x%5E2-4x-5=0

You will notice this equation cannot be factored with integers, so we can use the quadratic equation to find the values of x:

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-4x%2B-5+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-4%29%5E2-4%2A2%2A-5=56.

Discriminant d=56 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+56+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+56+%29%29%2F2%5C2+=+2.87082869338697
x%5B2%5D+=+%28-%28-4%29-sqrt%28+56+%29%29%2F2%5C2+=+-0.870828693386971

Quadratic expression 2x%5E2%2B-4x%2B-5 can be factored:
2x%5E2%2B-4x%2B-5+=+2%28x-2.87082869338697%29%2A%28x--0.870828693386971%29
Again, the answer is: 2.87082869338697, -0.870828693386971. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-4%2Ax%2B-5+%29



Rounding to two decimal points gives us x: 2.87 or -0.87

To find y, replace each of these values for x in our second equation x + y = 2:

2.87 + y = 2 ----->

y = 2 - 2.87 ----->

y = -.87

AND

-0.87 + y = 2 ----->

y = 2 + 0.87 ----->

y = 2.87

SO, x = 2.87 or -0.87 and y = 2.87 or -0.87