There was 52 people at a football game. Three times as many children as adults. Find how many children and adults went to the game by using a system.

c=3a
a+c=52 

so 4a=52
a=13 
c=3a=39

39 children

find a rectangular form of the equation r=5cos(t)

The length of a football field is 180 feet more than its width. If the perimeter of the field is 1,060 feet, find the length of the field.

P=2(l+w)
l=180+w 

1060=2(180+w+w) 
2w+180=1060/2
2w=530-180
w=175

Without making a drawing, how can you tell that the graph of y = x - 30 passes through three (3) quadrants?

When  both x and y are negative (quad III).


When  x is positive and y is negative (quad IV). 

When  they are both positive (quad I))).

A math test has 25 problems. Some problems are worth 2 points and some are worth 3 points and the total for the test is 66. How many 2 points and 3 points promblems are there.

x-->2pt questions
y-->3pt questions

1)  x+y=25  <--- total questions
2)  2x+3y=66 <--- total points

Using substitution:
2x+3(25-x)=66
2x+75-3x=66
75-66=x
x=9 
y=25-9=16 
there are 9 - 2pt problems and 16 - 3pt problems

The larger of two positive integers is 1 less than twice the smaller. If twice the square of the smaller is decreased by the larger, the result is 39. Find the larger.

Not possible...
choose
y > x
then
y=2x-1
and so
2x^2-y=39
2x^2-2x+1=39, but this equation has no integer solutions

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What are the rational routes of 8x^3+2x^2-5x+1=0 ?

By using synthetic division we can write:
8x^3+2x^2-5x+1=(x+1)(8x^2-6x+1)
then by the quadratic equation, factoring, or some other method we get
8x^3+2x^2-5x+1=(x+1)(2x-1)(4x-1)
The roots are: {-1, 1/2 1/4}