Solve by Substitution x+y=-10 , (x+3)^2+(y+9)^2=10

Problem

{[x+y=−10],[(x+3)2+(y+9)2=10])

Solution

1. Isolate a variable in the linear equation. Subtract x from both sides of the first equation to solve for y

y=−x−10

2. Substitute the expression for y into the second equation. Replace y with −x−10 in the quadratic equation.

(x+3)2+(−x−10+9)2=10

3. Simplify the terms inside the second set of parentheses.

(x+3)2+(−x−1)2=10

4. Expand both squared binomials using the identity (a+b)2=a2+2*a*b+b2

(x2+6*x+9)+(x2+2*x+1)=10

5. Combine like terms and set the equation to zero.

2*x2+8*x+10=10

2*x2+8*x=0

6. Factor the quadratic equation to find the possible values for x

2*x*(x+4)=0

x=0* or *x=−4

7. Substitute each x value back into the linear equation y=−x−10 to find the corresponding y values.

If *x=0⇒y=−0−10=−10

If *x=−4⇒y=−(−4)−10=−6

Final Answer

(x,y)=(0,−10),(−4,−6)

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