The equation of the line is defined in the following forms:

Point-slope form:  \(\displaystyle y-y_1 = m(x-x_1)\)

Standard form:  \(\displaystyle Ax+By=C\)

Slope intercept form:  \(\displaystyle y=mx+b\)

Find the slope of the two points using the slope formula.

\(\displaystyle m = \frac{y_2-y_1}{x_2-x_1} = \frac{9-(-1)}{3-9} = \frac{10}{-6} = -\frac{5}{3}\)

Using the slope and any point  \(\displaystyle (9,-1)\) or \(\displaystyle (3,9)\), we can substitute either into the point-slope form.

\(\displaystyle y-(-1) = -\frac{5}{3}(x-9)\)

The answer is:  \(\displaystyle y+1 = -\frac{5}{3}(x-9)\)

Write the slope formula.

\(\displaystyle m = \frac{y_2-y_1}{x_2-x_1}\)

Substitute the points.

\(\displaystyle m = \frac{-9-(-3)}{6-2} = \frac{-6}{4} = -\frac{3}{2}\)

The answer is:  \(\displaystyle -\frac{3}{2}\)

In order for a point to be on the line, the point must satisfy the equation given. Thus, plug in the \(\displaystyle x\) and \(\displaystyle y\) coordinates to see if they will give you a true equation.

If you plug in \(\displaystyle (3, 0)\) into the equation, you will get the following:

\(\displaystyle 4(3)-8(0)=12\)

\(\displaystyle 12=12\)

Thus, \(\displaystyle (3, 0)\) satisfies the equation and must be on the line.

Which of the following points is on the following line? 

\(\displaystyle y=-12x+6\)

\(\displaystyle (0,6)\) \(\displaystyle (4,4)\) \(\displaystyle (0,-4)\) \(\displaystyle (-2,-18)\)

So, to test this, we can plug in each choice and solve to see if they make sense. 

To save time, let's test the easier ones first. Recall that anything times 0 is 0, so we should try out the options with 0's first.

Recall that ordered pairs represent an x and a y value with the x coming first: (x,y)

\(\displaystyle y=-12x+6\)

\(\displaystyle (0,-4)\)

\(\displaystyle -4=-12(0)+6\)

\(\displaystyle -4=0+6\)

\(\displaystyle -4\neq6\)

So, this is not our answer. However, it does give us a hint as to the correct answer.

When we plugged in 0 for x, we got 6 on the right hand side. This means that if we plug in 0 for x, then we should get 6 for y. So, let's try out our next point.

\(\displaystyle y=-12x+6\)

\(\displaystyle (0,6)\)

\(\displaystyle 6=-12(0)+6\)

\(\displaystyle 6=6\)

So, our answer must be (0,6)

In order for a point to be on a line, the point must satisfy the equation. Plug in the values of \(\displaystyle x\) and \(\displaystyle y\) from the answer choices to see which one satisfies the equation.

Plugging in \(\displaystyle (-2, -3)\) will give the following:

\(\displaystyle 7(-2)-8(-3)=10\)

\(\displaystyle -14+24=10\)

\(\displaystyle 10=10\)

Since \(\displaystyle (-2, -3)\) satisfies the given equation, it must be on the line.

Find the y-coordinate which would would make the slope between the following points equal to 5.

\(\displaystyle (13,65)\) \(\displaystyle (43,y_2)\) 

To find the slope of a line, use the following formula:

\(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\)

Now, we know all of these values except one. Let's find it.

\(\displaystyle 5=\frac{y_2-65}{43-13}\rightarrow5=\frac{y_2-65}{30}\)

\(\displaystyle 5=\frac{y_2-65}{30} \rightarrow 150=y_2-65\)

\(\displaystyle 150=y_2-65\rightarrow 150+65=y_2\)

\(\displaystyle y_2=215\)

So, our answer is

\(\displaystyle y_2=215\)