Expert: Paul Klarreich - 7/2/2007

Question
1. (x-1)squared +(y+2)squared= 10
                         x+y=  1

2. 25y sq - 16x sq = 400
   9y sq -  4x sq =  36

I need to know how to use substitution to answer these systems. The answer, according to my College Algebra book, for #1 is (0,1) and (4,-3) but, I couldn't figure out how they got those numbers. Sorry, I don't know how to write "squared" in numerical form on my computer, so I just spelled it out or abbreviated it (sq).

Answer
Questioner:   mel
Category:  Advanced Math
Private:  No
 
Subject:  systems of equations
Question:  1. (x-1)squared +(y+2)squared= 10
                        x+y=  1

2. 25y sq - 16x sq = 400
  9y sq -  4x sq =  36

I need to know how to use substitution to answer these systems. The answer, according to my College Algebra book, for #1 is (0,1) and (4,-3) but, I couldn't figure out how they got those numbers. Sorry, I don't know how to write "squared" in numerical form on my computer, so I just spelled it out or abbreviated it.
.................
Hi, Mel,

For your first example:
1. (x-1)^2 +(y+2)^2 = 10
         x+y=  1

To solve by substitution we start with:
x = 1 - y

((1 - y) - 1)^2 +(y+2)^2 = 10

(1 - y - 1)^2 + (y + 2)^2 = 10

(- y)^2 + (y + 2)^2 = 10

y^2 + y^2 + 4y + 4 = 10

2y^2 + 4y - 6 = 0

y^2 + 2y - 3 = 0

(y + 3)(y - 1) = 0

y = -3  -->  x = 1 - (-3) = 1 + 3 = 4

y = 1   -->  x = 1 - 1 = 0

Them's your solutions.

.............................
2. These are a pair of hyperbolas. (hyperbolae?)

25y^2 - 16x^2 = 400   (A)
9y^2 -  4x^2 =  36   (B)

Now these are in a nice enough form that we can use elimination:

25y^2 - 16x^2 = 400   (A)
36y^2 - 16x^2 = 144   (4B)
-----------------------
11y^2         = - 256  (4B - A)

Uh, oh.  Since y^2 is nonnegative, so is 11y^2, and it can't be equal to -256, so there are only imaginary solutions to these equations, which means the graphs won't intersect.  Still, we can do:

y^2 = - 256/11

y = +- 16i/sqrt(11)

(where i = sqrt(-1), the imaginary unit)

25(-256/11) - 16x^2 = 400

16x^2 = - 6400/11 - 400

16x^2 = - 6400/11 - 4400/11

16x^2 = - 10800/11

8x^2 = - 5400/11

4x^2 = - 2700/11

2x^2 = - 1350/11

x^2 = - 675/11

x = +- 15i sqrt(3)/11

Now if you have to, you can use substitution, like this:

25y^2 - 16x^2 = 400   (A)
9y^2 -  4x^2 =  36   (B)

Use A:

25y^2 = 400 + 16x^2
y^2 =  16 + 16x^2/25

Now substitute that into B:

9(16 + 16x^2/25) - 4x^2 = 36

144 + 144x^2/25 - 4x^2 = 36

144 + 144x^2/25 - 100x^2/25 = 36

44x^2/25 = - 108

11x^2/25 = - 27

etc.