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Question

1. Find the slope of the line that passes through the points (4, -7) and (-2, -5).




2. Find the equation of the line with a slope of -5 and passes through the point (2, -4).




3. Write an equation of the line that passes through (0, -4) and is parallel y = (3/4)x + 2. Write the answer in slope-intercept form.




4. Solve the system of equations by substitution:

x + 2y = 9
3x - y = 13




5. Solve the system of equations by substitution:

4x - 3y = 1
12x - 9y = 3

Answer 1

1. slope is change in y (you find change by subtracting) divided by change in x, so for y, -7 → -5 is a change of 2, and for x, 4 → -2 is a change of -6, so slope is 2/-6 = -1/3

2. a line with slope -5 has equation y = -5x + b. To find b, plug in (2,-4):

-4 = -5(2) + b
-4 = -10 + b
6 = b

so y = -5x + 6

3. parallel lines have the same slope, so y = (3/4)x + b. find b by plugging in (0,-4). -4 = (3/4)(0) + b, -4 = b,

so y = (3/4)x - 4

4. solve 1st equation for x (or 2nd for y): x + 2y = 9, x = 9 - 2y, and use the expression in place of x in the 2nd:

3(9 - 2y) - y = 13
27 - 6y - y = 13
-7y = -14
y = 2

x = 9 - 2(2)
x = 5

5. the 2nd equation is 1st equation times 3. if you graphed them both, one line sits right on top of the other -- same line, same set of points. so there are infinitely many solutions, one for each point on the line.

if you tried it algebaically, you end up with 0 = 0:

4x - 3y = 1
4x = 1 + 3y
x = 1/4 + (3/4)y

12( 1/4 + (3/4)y) - 9y = 3
3 + 9y - 9y = 3
3 = 3
0 = 0

Answer 2

Ok I will try to help you

1) slope is a fraction that basically comes down to being, change in rise divded by change in run. To go from (4,-7) to (-2,-5), you see that you go down 6 units (a run of -6) and you go to the right 2 units ( a rise of +2), so the slope is +2/-6 or -1/3

2) I know there is a specific equation for this type of problem but I always solved this differently. I used the slope intercept form (y=mx+b, where m is the slope and b is the y intercept). You have a point (2,-4) and the slope is -5. If you plug these into the slope intercept formula, you get -4=-5(2)+b. You solve this for b and get b=6. Now you have the slope and y-intercept so you can put it into slope intercept form, y=-5x+6

3) parallel lines have the same slope, so since the equation is in slope intercept form you know that the slope is (3/4). Again you have a slope and a point so do the same thing.
-4=(3/4)(0)+b, so b=-4 and the equation is y=(3/4)x-4

4) rearranging the first equation, you get x=9-2y. You substitute this into the second equation like this...
3(9-2y) -y =13, this simplifies to 27 -7y=13 and y=2 and putting y=2 into either equation, you get x=5

5) You do the same thing as the last problem, 4x=1+3y
The other equation can be written like this... 3(4x) -9y=3
You substitute the first equation into the second equation and get 3(1+3y) -9y =3, simplify this to 3=3. This result means that there are inifnitely many solutions. On a side note, you can see this becuase if you multiply the first equation by 3, you get the second equation, so they are the same line

Answer 3

PROBLEM 1: Find the slope of the line that passes through the points (4, -7) and (-2, -5).

slope m = (y2 - y1)/(x2-x1)

so where point 1 (either of the two)... has x-y coordinates (x1, y1) and the other point which will be point 2 has x-y coordinates (x2, y2)....

so we'll say that point 1 is (4, -7) and point 2 is (-2, -5)

so....

point 1...........point 2
(4, -7).............(-2, -5)
(x1, y1)..........(x2, y2)

sooo.... slope m = (y2 - y1)/(x2-x1) = (-5 -[-7]) / (-2-4) = (-5+7)/(-6) = 2/-6 = -1/3

so the line that passes through the given two points has a slope m = -1/3


PROBLEM 2: Find the equation of the line with a slope of -5 and passes through the point (2, -4).

slope of -5 means you know that m= -5...

so substitute m= - 5 into your general line equation

y = mx + b becomes y = -5x + b

you know that the line needs to pass through given point (2, -4).... if the line needs to pass through the given point... that means that the point has to be on the line.... if the point is on the line.... and you know y = -5x + b..... you can 'solve' for b (the y-intercept) by substituting the x-y coordinates of the point (2, -4). That point (2, -4) has x-value of 2 (x = 2) and y-value of -4 (y = -4)

y = -5x + b .... after substituting x = 2 and y = -4 becomes...

-4 = -5(2) + b...

-4 = -10 + b...

add +10 to both sides of eqn like this...

-4+10 = -10 + b + 10

6 = b.... which is the same as b = 6....

substitute b = 6 in y = -5x + b and you get...

y = -5x + 6 which is a line equation with a slope of -5 which also passes through the point (2, -4).


PROBLEM 3: Write an equation of the line that passes through (0, -4) and is parallel y = (3/4)x + 2. Write the answer in slope-intercept form.

What do you know about parallel lines? What do they have in common? Parallel lines have the same slope m.... but different y-intercepts since they cross the vertical y-axis at different points... so same "m" value as the other line, but different "b" values in y = mx + b...

So you are given line with equation y = (3/4)x + 2....

that line has slope m = 3/4 and y-intercept b = 2

The parallel line will also have slope m = 3/4

so... the equation will look something like this....

y = mx + b.... becomes y = (3/4)x + b

You also want this line to pass through given point (0,-4)....

where is point (0,-4).... it is on the vertical y-axis..... If it is on the line and it is also on the y-axis.... it is a coincidence that the given point just so happens to also be the y-intercept b... so b = 4

sooo.... y = (3/4)x + 4 is an equation of the line that passes through (0, -4), that is also parallel to the line with equation y = (3/4)x + 2, and is written in slope-intercept form.


PROBLEM 4: Solve the system of equations by substitution:

x + 2y = 9
3x - y = 13

x + 2y = 9 can also be written x = 9 -2y

now plug that expression 9 -2y for "x" in the 2nd equation given... like this....

3x - y = 13 becomes...

3(9-2y) - y = 13

doing the math... you get...

3*9 + 3(-2y) - y = 13.... then....

27 - 6y - y = 13....

combine like terms to get...

27 -7y = 13

add +7y to both sides... like this...

(27 - 7y) +7y = 13 + 7y....

combine like terms ... to get...

27 = 13 + 7y.... which is the same as 13 + 7y = 27

add -13 to both sides....

(13 + 7y) -13 = 27 -13

combine like terms to get...

7y = 14....

divide both sides by 7

7y / 7 = 14/7

7's on left side cancel out leaving you with '1y'... and 14/7 = 2...

sooo... y = 2...

put y into first equation x + 2y =9 ... and solve for x.

x + 2(2) = 9....

x + 4 = 9...

x = 5 ... so now you know x = 5 and y = 2

now check your work by substituting x = 5 and y = 2 into the 2nd given equation to see if the statement is true.... if it is true... then you did everything right...

3x - y = 13.... becomes...

3(5) - 2 = 13 Is that true???

15 - 2 = 13 ?

Yes.... 3 = 3..... so your solution for this problem is x = 5 and y = 2


PROBLEM 5: Solve the system of equations by substitution:

4x - 3y = 1
12x - 9y = 3

these two equations are one and the same.... because the 2nd equation is just "3 TIMES" the 1st equation given...

3(4x - 3y) = 1 * 3

3*4x -9y = 3...

12x - 9y = 3 See?...

so 4x - 3y = 1 is the equation of a line.... it's not written in y = mx + b format... it's the same line... just with the terms rearranged... but no matter how you write the line... it's still the SAME line...

If you were to write the line in the general line equation y=mx + b format.... this is how the line would look like rewritten...

4x - 3y = 1....

multiply the whole equation by ' -1 '... so you get this.... (you just have to change the signs of the terms)

-4x + 3y = -1...

add "+ 4x" to both sides...

(-4x + 3y) + 4x = 4x - 1

combine like terms...

3y = 4x - 1

now divide both sides by 3....

3y / 3 = (4x -1) / 3

3's on left cancel out leaving '1y' and on right... divide each of the terms in the ' ( ) ' by 3...

now you have... y = (4/3)x - 1/3

this is the given equations in y = mx + b format

It is a line... so the "solution" that will make these equations true will be any point (x, y) that is on that given line... and only those points... There are an infinite number of sets of (x, y) that will make this statement true.... the only constraint is that that these sets have to be on the line 4x - 3y = 1

Answer 4

Slope formula

m = y₂- y₁/ x₂- x₁

Ordered Pair

(4, - 7)(- 2, - 5)

m = - 5 - (- 7) / - 2 - 4

m = - 5 + 7 / - 6

m = 2 / - 6

m = - 1/3

- - - - - - --

Sloope INtercept form

y = mx + b

- 7 = - 1/3(4) + b

- 7 = - 4/3 + b

- 7 + 4/3 = - 4/3 + b + 4/3

- 7 + 4/3 = b

- 213 + 4/3 = b

- 17/3 = b

- - - - -

The equation

y = - 1/3x - 17/3

- - - - - - -s-

Answer 5

1. Don't remember how... sorry

2. Equation of the line formula:

y1 - y2 = m ( x1 - x2 )

So, all you have to do is plug in and solve.

m= -5
y1= -4
x1= 2

-4 - y = -5 ( 2 - x)

Solve for y:

-4 - y = -10 + 5x

-y = -14 + 5x

y = 14 - 5x <---answer

3. Follow same instructions.

m = (3/4)
y1 = -4
x1 = 0

4. Solve equation 1 (x+2y=9) for x:

x + 2y = 9

x = 9 - 2y

Then Plug in for x in equation 2 (3x-y=13):

3x - y = 13

3 (9 - 2y) - y = 13

27 - 6y - y = 13

27 - 7y = 13

-7y = -14

y = 2

Now plug in y back into the original equation to get x:

x + 2y = 9

x + 2 (2) = 9

x + 4 = 9

x = 5

5. Follow same instructions as problem #4.

Answer 6

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