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here are the problems:

1. The graphs of these two equations intersect in how many points? 

x^2 + y^2 =25

x^2-y^2=3

 

2. For every value of a, the graph of each of these three equations is a line:

ax+y=1

x+ay=1

x+y=6

For what value of a do all three of these lines intersect at the same point?

 

3. A and B are constants such that the graphs of xy=B and y=Ax^2 intersect at the point (2,6).
Write the ordered pair (A,B).

 

thanks!

 Apr 17, 2021
 #1
avatar+118586 
+1

1. The graphs of these two equations intersect in how many points? 

x^2 + y^2 =25

x^2- y^2 =   3

 

Add these equations and we  get  that

 

2x^2  = 28           divide  through  by 2

 

x^2   = 14            take both roots

 

x = sqrt (14)     or  x  = -sqrt (14)

 

When  x  =  sqrt (14)  we  have

 

(sqrt 14)^2  + y^2  = 25

 

14 + y^2  =  25

 

y^2  =  25 - 14

 

y^2  =  11

 

y = sqrt (11)     or y = -sqrt (11)

 

So.......two points of intersection  are  (sqrt 14, sqrt 11)   and (  sqrt 14, - sqrt 11)

 

Due  to symmetry......the  graphs will  also intersect  at (-sqrt 14, sqrt 11)  and (-sqrt 14 , - sqrt 11)

 

So.....4 points of  intersection

 

See  the  graph here  :  https://www.desmos.com/calculator/ie0t00e5nl

 

 

cool cool cool

 Apr 17, 2021
edited by CPhill  Apr 17, 2021
 #2
avatar+118586 
+1

2. For every value of a, the graph of each of these three equations is a line:

ax+y=1

x+ay=1

x+y=6

For what value of a do all three of these lines intersect at the same point?

 

Note  that  the last equations  can  be writtern   as   y   =  6 - x

 

Subbing  this  into  the  other two equations we  get

 

ax  + (6 - x)  =  1       ⇒   x(a - 1)  =  -5

 

x + a(6 - x)  = 1      ⇒   x((1 - a) = 1  -6a  ⇒  x ( a - 1)  =  6a  - 1

 

This implies that   6a  - 1  =  -5  ⇒   6a  =  -4  ⇒  a = -4/6  = -2/3

 

Subbing  into  the first  equation  for  a  and  y we  have  that

 

(-2/3)x  +   6 - x   =  1

 

-(5/3)x  +  6  =  1

 

-(5/3)x  =  -5

 

x = 3

 

And  y  =6- x =  6 - 3  = 3

 

The intersection  pt is (3,3)

 

See the grpah here :  https://www.desmos.com/calculator/nmla6yjdjt

 

 

cool cool cool

 Apr 17, 2021
 #3
avatar+118586 
+1

3. A and B are constants such that the graphs of xy=B and y=Ax^2 intersect at the point (2,6).
Write the ordered pair (A,B).

 

2*6  =  B  =  12

 

6  = A(2)^2

6  = 4A

A  = 6/4  = 3/2

 

 

 

See the  graph here :  https://www.desmos.com/calculator/nvkmus8bpa

 

 

cool cool cool

 Apr 17, 2021

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