How do I find two solutions for both x and y when I have two sample equations?

Question

_{Solve for x and y: x2 +y2 =17 There are two solutions: (x , y ) and (x , y ).
                       y −1= x}

 

_{Evaluate: x1 + y1 + x2 + y2 =}

 

AND

 

Solve the system: x2 +y2 =37 There are two solutions: (x , y ) and (x , y ).

                           3x - 9 = y

 

Evaluate: x1 + y1 + x2 + y2 =

Answer 1

Let's (x₁, x₂) and (x₂, y₂) are solutions to the system 

x² + y² = 17
y-1=x

This means that

x₁²+y₁² = 17 (1a)
y₁-x₁=1          (1b)

and 

x₂²+y₂² = 17  (2a)
y₂-x₂=1          (2b)

 

Substructing (2a) from (1a) and from (2b) from (1b) results

(x₁-x₂)(x₁+x₂)+(y₁-y₂)(y₁+y₂)=0 and y₁-y₂=x₁-x₂. 

Since x₁≠x₂, substituting  the second equation into the first one gives

(x₁-x₂)[(x₁+x₂)+(y₁+y₂)]=0, from which follows that x₁+x₂+y₁+y₂=0.

 

Answer 2

I am going to answer your question and then let you do the problem, i.e. I will get you started

 

Each problem is a circle intersected by a straight line (and it will help you to use your graphing calculator or to make a sketch).

 

In each case you should solve the linear equation for one of the variables in terms of the other and make the substitution, like this:

 

x^2 + y^2 = 17

x = y-1

 

then (y-1)^2 + y^2 = 17

expand and collect terms: 2y^2 - 2y -16 = 0

 

Use the quadratic formula to solve for y, then use the equation x = y-1 to get x

 

The second problem goes the same way.