Given that the length of each side of a quadrilateral is a distinct integer and that the longest side is not greater than 7, how many different possible combinations of side lengths are there? (a) 21(b) 24(c) 32(d) 34(e) 35

(E) 35

Step-by-step explanation:

The possible lengths for a side are: {1, 2, 3, 4, 5, 6, 7}

Therefore 7 possible lengths.

The quadrilateral has 4 sides.

It is clearly stated that the sides lengths should be distinct, so it indicates we are dealing with either permutations or combinations.

The order here does not matter. For example the quadrilateral with sides {1, 2, 3, 4} is the same quadrilateral as if it was {4, 3, 2, 1}

Therefore we are dealing with combinations.

The formula is:

Here n is 7 and r is 4, since we have a group of 7 items to pick (lengths) and we will form groups of 4 items (quadrilaterals)

Therefore the formula becomes:

We re-write 7! as 7(6)(5)4! to be able to cancel the 4!:

And notice 3! is 3(2)(1)=6 so we finally get:

Therefore there are 35 possible combinations of side lengths

a.   we have two

lines:   y = 4-x   and  

y = 8-x^-1

given two simultaneous equations that are both to be

true, then the solution is the points where the lines cross. the intersection is

where the two equations are equal. therefore the solution that works for both

equations is when

4-x = 8-x^-1

this is where the two lines will cross and that is the

common point that satisfies both equations.

b.   4-x = 8-x^-1

  x           4-x       8-x^-1

-3           7        

8.33

-2           6         8.5

-1           5         9

  0           4        

-

  1           3        

7

  2           2        

7.5

  3           1         7.67

the table shows that none of the x values from -3 to 3 is

the solution because in no case does

4-x = 8-x^-1

to find the solution we need to rearrange the equation to

find for x:

4-x = 8-x^-1

multiply both sides with x:

4x-x^2 = 8x-1

x^2+4x-1=0

x= -4.236, 0.236

therefore there are two points that satisfies the

equation.

find y:  

x=-4.236

y = 4-x   = 4 – (-4.236)

= 8.236

y = 8-x^-1 =   .236)^-1

= 8.236

x=0.236

  y = 4-x   = 4 – (0.236) = 3.764

y = 8-x^-1 =   8-(0.236)^-1

= 3.764

thus the two lines cross at 2 points:

(-4.236, 8.236) & (0.236, 3.764)

c.   to solve

graphically the equation 4-x = 8-x^-1

we would graph both lines: y = 4-x   and   y

= 8-x^-1

the point on the graph where the lines cross is the

solution to the system of equations.

just graph the points on part b on a cartesian coordinate

system and extend the two lines.   the

solution is, as stated, the point where the two lines cross on the graph.

step-by-step explanation:

mark brainliest!

what are the statments

step-by-step explanation: