I thing the easiest would be to pick point (3,3) and point (4,4)

x1 = 3

y1 = 3

x2 = 4

y2 = 4

m = (y1-y2)/(x1-x2) = (3-4)/(3-4) = (-1)/(-1) = +1

m = (y2-y1)/(x2-x1) = (4-3)/(4-3) = (1)/(1) = +1

PROBLEM 2A:

Point (2,0) and (5,0) are two points on a horizontal line (because the x-axis is a horizontal line)

x1 = 2

y1 = 0

x2 = 5

y2 = 0

Now let’s find the slope…

m = (y1-y2) / (x1-x2) OR m = (y2-y1)/(x2-x1) …. choose 1 of them

I’ll choose the first…

m = (0-0) / (2-5) = 0 / -3 = 0

OR if you chose the 2nd…

m = (0-0) / (5-2) = 0/3 = 0

That is how come a horizontal line has slope m = 0

PROBLEM 2B:

Point (0,2) and (0,5) are two points on a vertical line (because the y-axis is a vertical line)

x1 = 0

y1 = 2

x2 = 0

y2 = 5

Now let’s find the slope…

m = (y1-y2) / (x1-x2) OR m = (y2-y1)/(x2-x1) …. choose 1 of them

I’ll choose the first…

m = (2-5) / (0-0) = -3 / 0 = UNDEFINED (because a slope that has 0 in the denominator is UNDEFINED)

OR if you chose the 2nd…

m = (5-2) / (0-0) = 3/0 = UNDEFINED

That is how come a vertical line has slope m = UNDEFINED

PROBLEM 3A:

Let’s look at 2x +3y = 5 and 2x + 3y = 6….

To find out why they are parallel, if you put them into the general line equation format y = mx + b… you will see that the slope m of equation is the same…

Let’s put the 1st equation into y = mx + b format… This is how you do that…

2x + 3y = 5

add “-2x” to both sides of the equation like this…

(2x + 3y) -2x = -2x + 5

now combine “like” terms… to get…

3y = -2x + 5

Now divide the whole equation (each term) by 3… so that you have a “lone y” on the left side…

like this…

3y/3 = (-2x + 5)/3

y = -2x/3 + 5/3

y = (-2/3)x + 5/3

so m = -2/3 (This is the slope of this line….)

NOW…. let’s put the 2nd equation into y = mx + b…. If this equation also has a slope m = -2/3…. then this proves that the two lines are parallel…

2x + 3y = 6

remember the steps that we did for the 1st equation?

(2x + 3y) -2x = -2x + 6

3y = -2x + 6

3y/3 = (-2x + 6)/3

y = -2x/3 + 6/3

y = (-2/3)x + 2

YES…. slope me of this line ALSO equals -2/3….

So the two lines are parallel for reasons of having the same slope…. the difference between the two lines is that they have different y-intercepts (b-values… remember y = mx + b). The different “b-values” or y-intercepts mean that the two line have the same slop and are parallel… but cross the y-axis at “different” points…

PROBLEM 3B:

Now what about 2x+3y = 5 and 4x +6y = 10…. why are they the same line?

because 4x + 6y= 10 is the same as multiplying the 1st equation by “2”

2 * (2x + 3y = 5) is equal to 4x + 6y = 10….. you see, each of the terms in the 1st equation are multiplied by 2… Do you see that?

You can also prove this to be true by putting both of the equations into y=mx+b format…. if you get the same end equation… it is the same line…

2x + 3y = 5

adding “-2x” to both sides, you get…

3y = -2x + 5

divide the whole equation by “3”… and you get…

y = -2x/3 + 5/3

y = (-2/3)x + 5/3….. 1st line in y=mx+b format…

Now put the 4x+6y=10 into y=mx+b format… like this…

4x + 6y = 10

adding -4x to both sides, you get…

6y = -4x + 10

divide the whole equation by “6”… and you get…

y = -4x/6 + 10/6

which is the same as

y = (-2/3)x + 5/3….. This is the 2nd line in y=mx+b format…

Guess what? Do you notice that this is the same as the 2x+3y=5 equation in y=mx+b formation that you figured out before?

Because both of them resulted in y = (-2/3)x + 5/3…

2x + 3y = 5 and 4x + 6y = 10 are one and the same line…

Hope this helped!!!

Your slope functions are both true– just remember you must take them in the same order– top and bottom. they just describe the slope of the line going from the first to the second or the second to the first…

if the value of 2 points in the “Y” (vertical) direction is the same– but the “X” (horizontal) are different . the line between is always flat– with the horizon….therefore y1=y2 therefore slope= 0

but if the “X” numbers are the same your fraction is divided by 0 “zero” which is an undefined number..

the last question is very simple if you graph it out and pick a few points to prove it!

Have fun– it gets easier once you have these basic concepts down pat!