1a) To solve this problem, firstly we have to know the equation of the line and the equation of its slope:
Line Equation:
[tex]Y=mX+b[/tex] (1)
Where m is the slope and b the Y-intersection of the line
Slope Equation:
[tex]m=\frac{Y_{2}-Y_{1}}{ X_{2}-X_{1}}[/tex] (2)
Where the Ys and the Xs, are the points represented in the Cartesian plane.
In this case we will use equation (2) with the data given, points (4,10) and (2,5):
[tex]m=\frac{10-5}{4-2}[/tex]
[tex]m=\frac{5}{2}[/tex] This is the slope
1b) The slope is positive, this means the amount of rain increases at a rate of [tex]\frac{5}{2}[/tex] with the time.
2) The equation of the graphed line in the second picture can be expressed using equation (2) to find the slope with the given points (0,7) and (8,-2):
[tex]m=\frac{Y_{2}-Y_{1}}{X_{2}-X_{1}}[/tex]
Which also can be expressed as follows:
[tex]Y_{2}-Y_{1}=m({X_{2}-X_{1}})[/tex] (3)
[tex]m=\frac{7-(-2)}{0 -8}[/tex]
[tex]m=-\frac{9}{8}[/tex] This is the slope
Now, we use equation (3) with one of the points given (with any of them the resulting equation will be the same). Let’s prove it:
With (0,7):
[tex]Y-7=-\frac{9}{8}({X-0)[/tex]
[tex]Y=-\frac{9}{8}X+7[/tex]
With (8,-2):
[tex]Y-(-2)=-\frac{9}{8}({X-8)[/tex]
[tex]Y+2=-\frac{9}{8}({X-8)[/tex]
[tex]Y=-\frac{9}{8}X+7[/tex]
3) Find the slope intercept of the linear equation [tex]2X-8Y=32[/tex]
To find the slope, we only have to convert this equation in the [tex]Y=mX+b[/tex] form:
[tex]-8Y=32-2X[/tex]
[tex]Y=\frac{32-2X}{-8}[/tex]
[tex]Y=-\frac{32}{8}+\frac{2}{8}X[/tex]
[tex]Y=-4+\frac{1}{4}X[/tex]
[tex]Y=\frac{1}{4}X-4[/tex]
Remember that b is the Y-intersection of the line, in this case b=-4. Then, the slope interecept is -4
4) In the figure we have three given points: (-8,9), (-2,0) and (0,-2).
These will be enough to find the slope of the line. We can use two of any of these three points to solve the equation. It doesn’t matter which two points you choose, the result will be the same for this case.
Using (-2,0) and (0,-2):
[tex]m=\frac{-2-0}{0-(-2)}[/tex]
[tex]m=-\frac{2}{2}=-1[/tex]
Using (-2,0) and (-8,6):
[tex]m=\frac{0-6}{-2-(-8)}[/tex]
[tex]m=-\frac{6}{6}=-1[/tex]
Finally, the slope of this equation is -1
