I am about to fail this math class. I would like to verify that my answers are right. Please help.
Find the distance from the given point to the vertical axis.
(7, −6)

Determine the value of k such that the points whose coordinates are given lie on the same line.
(k, 0), (0, −1), (10, −11)

Find the slope of the line containing the given points.
P1(5, −9), P2(1, 7)

Find the equation of the line that passes through the midpoint of the line segment between
P1(4, 1) and P2(−2, 3) and has slope of 3.
Let y be the dependent variable and let x be the independent variable.

Find the equation of the line that contains the given point and has the given slope.
P(0, 6), m = 1

Suppose a ball is being twirled at the end of a string and the center of rotation is the origin of a coordinate system. If the string breaks, the initial path of the ball is on a line that is perpendicular to the radius of the circle. Suppose the string breaks when the ball is at the point whose coordinates are P(3, 9). Find the equation of the line on which the initial path lies

During one month, a homeowner used 500 units of electricity and 100 units of gas for a total cost of $331. The next month, 400 units of electricity and 250 units of gas were used for a total cost of $326. Find the cost per unit of gas.

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\Rightarrow m=\frac{-11+1}{10-0}\Rightarrow m=-1\\-11=-1*(10)+b\therefore b=-1\Rightarrow f(x)=-x-1\\f(k)=-k-1\\0=-k-1\therefore k=-1[/tex]

So (k,0)=(-1,0)

3) To find the slope of the line, we must apply that formula used above.

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\Rightarrow m=\frac{7+9}{1-5}\Rightarrow m=\frac{16}{4}=-4[/tex]

4) To find the equation of the line which the midpoint (2,2) since Midpoint is given by

[tex]Midpoint=(\frac{x_{1}+x_{2}}{2}+\frac{y_{1}+y_{2}}{2})\\[/tex]

And the slope is 3, then m=3. Notice the formula is the same to calculate the slope, but we will only pick one point. Since (2,2) ∈ to the function let's use this point, as initial value (x0,y,0)

[tex]m(x-x_{0})=y-y_{0}\\3(x-2)=y-2\\3x-6=y-2\Rightarrow 3x-y=6-2\Rightarrow -y=4-3x\Rightarrow y=3x-4[/tex]

5) Similarly to the previous one:

[tex]m(x-x_{0})=y-y_{0}\\(x-0)=y-6\\x=y-6\Rightarrow -y=-x-6\Rightarrow y=x+6[/tex]

6) A ball being twirled. The center of the rotation is the origin of Coordinate System (0,0) when the string breaks at point (3,9)

If the line was straight from the origin to point (3,9)[tex]m=\frac{9-0}{3-0}\Rightarrow m=3\therefore 9=3(3)+b\Rightarrow b=0\Rightarrow f(x)=3x[/tex]

But the point (3,9) ∈ to a tangent line to the circumference described by the twirling.

Since it is perpendicular instead of m=3 it is -1/m i.e. -1/3, also the circumference intercepts the y-axis in ≈10

g(x)=-1/3x+10

7) In this case, we must also find the function.

x=units of electricity, y=units of gas |

500x+100y=331

400x+250y=326

The cost per unit of gas is y. Finding out the unit value for y

[tex]100y=331-500x\rightarrow y=\frac{331-500x}{100}\\500x=331-100y\\x=\frac{331-100y}{500}\\400(\frac{331-100y}{500})+250y=326\\52.96-80y+250y=326\Rightarrow y=\frac{9}{25} \,and \,x=\frac{59}{100}[/tex]