Saturday, July 24th, 2010

### Solution:

First, we can write

Then we can substitute this into the second equation you get

We can simplify this to

If multiply both sides by 25 and expand this out we get

This can be simplified to: Or

Thus the solutions for a are 6 and -4

The solutions for b can be found by the equation (b=4 when a=6, and b=-2 then a=-4)

- Another way to solve this is by looking at the equations graphically. When you plot the two equations, you can see there they intersect.

As you can see, they intersect at two places:

You can plug the answers into both equations to confirm that they are both true using these two values of a and b.

Saturday, July 24th, 2010

### Solution:

1. 3x + 4y=25

2. 2x – 5y=9

We could rewrite these as follows by multiplying the first equation by 5 and the second by 4:

1. 15x + 20y = 125

2. 8x – 20y = 36

If we add the above two equations, we can eliminate y:

23x = 161

So x =3D 7, and plugging this into the first equation we can figure out

y: **21 + 4y =3D 25**,

so **y = 1**

Saturday, July 24th, 2010

### Solution:

- 2x – =y=61
- 2x + y= -7

If you subtract the second equation from the first, you can eliminate x and you are left with **-4y = 68.**

So **y =3D -17**, and plugging this into the first equation we can figure out

y: 2x = (-51)=61,

so **x = 5**

Friday, July 23rd, 2010

### Solution:

1. Out of 100 women, we expect 1.2 to actually have breast cancer (12 out of 1000)

2. However, only 88% (880 out of 1000) of these will be detected, so 1.056 (i.e. 88% of 1.2) will be detected

3. In addition, of the 98.8 women who don’t have breast cancer 9% (90 out of 1000) will trigger a false positive” (9% of 98.8 = 8.892)

4. So out of 100 women, 9.948 will have positive mammography readings (1.056+9.892)

5. Of these 9.948, only 1.056 will actually have breast cancer

6. 10.6% = (1.056/1.056+9.892) This ratio is essentially an application of Bayes Theorem

Please click on http://www.thegodofreason.com/bayesintro.pdf to download a PDF that explains how Bayes Theorem works. Page 3 gives an example that is very similar to your problem:

Friday, July 23rd, 2010

### Solution:

- 2x – =y=61
- 2x + y= =7

If you subtract the second equation from the first, you can eliminate x and you are left with -4y = 68.

So y = 3D -17, and plugging this into the first equation we can figure out

y: 2x = (-51)=61,

so **x = 5**

Friday, July 23rd, 2010

### Solution:

The expression “9/10 of 80” can be written as

This is equal to 72.

Friday, July 23rd, 2010

### Solution:

A six-sided polygon is known as a Hexagon:

Friday, July 23rd, 2010

### Solution:

First write down the two equations and then pick several values of x and use those values to calculate the corresponding values of y. You can use the x,y pairs to then plot each line. You will see that the two lines intersect at 2,8 =i.e. x=2 and y=8).