Homework Solution

After expanding the three equations and finding the linear equations that pass throught the points of intersection, students should be prepaired to solve this system of equations using substitution and/or elimination methods.

 

Given the following system of equations:

BA = (x + 1)^2 + (y - 2)^2 = 5^2

CA = (x + 6)^2 + (y - 6)^2 = 4^2

DA = (x + 9)^2 + (y - 2)^2 = 3^2

 

 

We can solve using the standard algebraic methods of elemination and substitution:

There is a small short cut. Notice that BA and DA have a common term (y - 2)^2. If you subtract one from the other, the "y" cancels out completely leaving "x" easy to solve for. You should encourage students to look for these shortcuts.

First, expand the three equations:

BA = (x + 1) (x + 1) + (y - 2) (y - 2) = 25

= x^2 + 2x + 1 + y^2 - 4y + 4 = 25

CA = (x + 6) (x + 6) + (y - 6) (y - 6) = 16

= x^2 +12x + 36 + y^2 - 12y + 36 = 16

DA = (x + 9) (x + 9) + (y - 2) (y - 2) = 9

= x^2 +18x +81 + y^2 - 4y + 4 = 9

 

Next, subtract BA from DA :

x^2 +18x +81 + y^2 - 4y + 4 = 9

-(x^2 + 2x + 1 + y^2 - 4y + 4 = 25)

0 +16x +80 + 0 + 0 + 0 = - 16

Now, solve for "x":

x = -6

Next, subtract BA from CA :

x^2 +12x + 36 + y^2 - 12y + 36 = 16

- (x^2 + 2x + 1 + y^2 - 4y + 4 = 25)

10x +35 - 8y +32 = -9

 

Substitute x = -6 into equation (CA - BA) and solve for "y":

10x + 35 - 8y + 32 = -9

= 10 (-6) + 35 - 8y + 32 = -9

= -60 + 35 - 8y + 32 = -9

7 - 8y = -9

- 8y = -16

y = 2

So, the solution is x = -6, y = 2.

If you look at the GSP presentation with the axes showing, you will see that point A (our location on Earth is located at (-6, 2) which confirms our solution for this system of equations.

Although the equations get a little larger than students are used to working with, the algebra is still the same. They just need to roll up their sleeves and give it a try.

Who knows, they might impress themselves!