Given that x is an integer, kelvin conjectured that x^2> x+1 . which value is a counterexample to kelvin's conjecture? −5 0 12 7

We calculate the expressions  and 

for each of -5, 0, 12 and 7 to see whether the inequality holds:

for x=-5,

for x=-5

the inequality holds, 

for x=0,

for x=0

1 is not larger that 0, so x=0 is an counterexample that  
is not larger than for all integers.

0

Part a)

Part b) The equation in point slope form is and the equation in slope intercept form is

Part c) The system is consistent independent

Part d) The coordinates of the point of intersection are x=1 and y=5

Step-by-step explanation:

Part a) Write the equation for Sidewalk 1 in slope-intercept form

we know that

The equation of the line in slope intercept form is equal to

Find the slope m

we have the points(2,7) and (0,3)

The y-intercept is the point (0,3)

so

substitute

Part b) Write the equation for Sidewalk 2 in point-slope form and then in slope-intercept form

step 1

The equation of the line in point slope form is equal to

Find the slope m

we have the points

(1,5) and (3,3)

take the point (1,5)

substitute

step 2

Convert the equation in slope intercept form

Isolate the variable y

Part c) Is the system of equations consistent independent, coincident, or inconsistent?

we have

> equation A

>equation B

we know that the lines are not parallel (the slopes are different), therefore will intersect at a single point, and the system will have only one solution

Remember that

If a system has at least one solution, it is said to be consistent .

If a consistent system has exactly one solution, it is independent .

If a consistent system has an infinite number of solutions, it is dependent

If a system has no solution, it is said to be inconsistent

therefore

In this problem the system is consistent independent

Part d) Use the substitution method to solve your system. If the two sidewalks intersect, what are the coordinates of  the point of intersection?we have

> equation A

> equation B

Substitute equation A in equation B

Solve for x

Find the value of y

substitute the value of x

The solution is the ordered pair (1,5)

therefore

The coordinates of the point of intersection are x=1 and y=5