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y=3x
or
y=3x+0
Compared to the equation of a line formula:
y=mx+b
y=any y value
m=the slope of the line
x=any corresponding x-value
b=the point where the line intersects the y-axis
For y=3x+0,
the line will cross or intersect the y=axis at (0, 0), starting from that point, use the slope of 3 or 3/1. That is, from (0, 0,) go up +3 and over +1 and plot that point (1,3)
From (1, 3) go up +3 and over +1 and plot that point (2, 6).
.
Now for the bottom of the graph, start again at (0, 0), and using the slope
-3/-1 (same as +3), go down -3 and to the left (-1); plot this point (-1, -3).
Repeat this pattern by going down -3 and over to the left -1; plot this point (-2, -6).
.
Connect the lines.
.
y=x+3
or y=1x+3
The slope = 1 or 1/1
The line crosses or intersects the y-axis at (0, 3)
So, repeat the instructions from above, only start at point (0, 3) and go up 1 and over 1, etc.
Graphing:

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3(x-5)+3x=4x+5
3x-15+3x=4x+5
6x-15=4x+5
6x-4x-15=4x-4x+5
2x-15=5
2x-15+15=5+15
2x=20
2x/2=20/2
x=10
.
check by plugging (x=10) back into the original equation and solve:
3(10-5)+3x=4x+5
3(5)+3(10)=4(10)+5
15+30=40+5
45=45 [checks out]

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10-16
-6

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Perimeter of a rectangle:
P=2(length+width)
So, Lenght=3x-4
and Width= 7x+10
.
P=2[(3x-4)+(7x+10)]
P=2(10x+6)

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Use the distance formula:
d=
.
d= [plug-in the values(1,4) and (-3,1)and solve]
.
d=
.
d=
.
d=
d=5

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If you are given the line in the slope-intercept format:
y=mx+b, the m-term represents the slope
.
If you are given two points (x1, y1) and x2, y2), use the slope formula:
Slope=m=

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Lines that are perpendicular have slopes that are the negative reciprocal of each other:
For example:
if slope #1 = 2, then the perpendicular slope = -1/2
.
Line #1
4x-5y=7 [re-write in the y=mx+b format]
4x-4x-5y=-4x+7
-5y=-4x+7
-5y/-5=-4/-5x+7/-5
y=4/5x-7/5
Slope # 1 = 4/5
.
So, slope of Line #2 =-5/4 [because I flipped 4/5 and multiplied it by (-1)]
y=mx+b [plug-in the values (2, 6) and (m=-5/4) and solve for the b-term]
6=(-5/4)(2)+b
6=-5/2+b
5/2+6=b
17/2=b
Plug-in just the values of the slope and the b-term:
y=-5/4x+17/2

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Use the formula y=mx+b where the m-term represents the value of the slope
.
1.
a. y = 7x
Slope = 7
.
b.y= -3x
Slope = m=-3
.
c. y= x
Slope=m=1
.
d. y= 1/2x
Slope=m=1/2

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3.
a. [cancel out the hour units; the dollar units are left]
.

$288/1 [multiply straight across
$288
.
b. [convert days to hours]
[cancel out the units wherever possible (hours and days cancel out]
.
[you are left with miles/hour]
16.0416 mi/hr [multiply straight across]
.

c. 35 people/1 party * 2.5 pizzas/10 people [cancel units wherever possible (people units)]
.
35 /1 party * 2.5 pizzas/10
35 people/1 party * 2.5 pizzas/10 people
8.75 pizzas/party [multiply straight across]

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8x-11y=5 [solve for the y-term]
8x-8x-11y=-8x+5
-11y=-8x+5
-11y/-11=-8x/-11+5/-11
y=(8/11)x-5/11

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[factor out the least common multiply amongst each factor and divide each term by it]
.
[this re-grouping doesn't work becasue each group has a different factor]
.
.
Try to re-arrange the terms, re-group and factor again:




[re-factor again; this factoring works because the groups now have a common factor (2x-3)]

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Lines that are parallel have the same slope. In the y=mx+b formula, the "m" is the slope. So look at each equation and find matching slopes:
(a)and (d) have matching slopes, so these two lines are parallel.

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–6x + 3y = 12 [re=write]
-6x+6x+3y=6x+12
3y=6x+12
3y/3=6x/3+12/3
y=2x+4
.
y=2x+4
y=2x+3
______
.
.
(-1)y=2x+4
y=2x+3
______
.
.
-y=-2=-4
y=2x+3
________
0y=0x+-1
o=-1 [untrue statement, so there is no solution or no points in common]
.
Also, note that the lines have the same slope (2), therefore the lines are parallel and will never intersect.

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I agree. Good job!!




0.83333

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2x+2y=0 [substitute (y=4x+0) for the y-term and solve for the x-term]
.
2x+2(4x+0)=0
2x+2(4x)=0
2x+8x=0
10x=0
10x/10=0/10
x=0
.
solve for the y-term by plugging x=0 back into one of the original equations:
2x+2y=0
2(0)+2y=0
2y=0
2y/2=0/2
y=0
.
check by plugging the values for x and y back into either eiquation and solve:
2x+2y=0
2(0)+2(0)=0
0=0 [checks out].
and
y=4x+0
0=4(0)+0
0=0 [also checks out]

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g(x) = –2x + 1
g(2a)=-2(2a)+1
g(2a)=-4a+1
.
checking:
-4a+1=-2x+1
-4a=-2x
-4a=-2(2a)
-4a=-4a [checks out]

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Formula for the perimeeter of a rectangle:
P=2[(length + width)]
P=56
Length=2w+4
Width=w
.
so,
56=2[(2w+4)+w)]
56/2=2/2(2w+4+w)]
28=3w+4
28-4=3w
24=3w
24/3=3w/3
8=w
.
So,
P=56
Length=2w+4 = 2(8)+4=16+4=20
Width=w=8
.
Plugging everything back into the formula for a perimeter:
P=2[(length + width)]
56=2(20+8)
56=2(28)
56=56 [checks out]

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Good job!! I agree with your answer. You correctly initially squared both sides of the equation in order to get rid of the radical; used FOIL on the left side; and solved for the x-term.
.
If your instructor/book wanted you to solve for the x=term, just continue the process:
0=x^2-7x+10
(x-5)(x-2)=0
x-5=0
x=5
or x-2=0
x=2
.
Try plugging each x-value back into the original equation to make sure each solution is a true solution. One of the values of x could possibly be an erroneous solution:
.
=
Let x=5
=
=[the square root of 4 is either a plus or minus 2]
2=2 [so, x=5 checks out]
.
=
Let x=2
= [the square root of 1 is either a plus or minus (1). In this case, the (-1) works]
-1 = -1 [ture, so x=2 is also a solution]

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[factor out a perfect square (25)]

[multiply by in order to get rid of the radical in the denominator]
.



[cancel out wherever possible]
.
or, just cancel out radicals wherever possible:

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Starting at the origin of (0,0) on the graph, go three spaces to the left (-3) and than down three spaces (-3). Plot the point.
.
For coordinate points, (x, y) the first number tells you how many spaces to move on the x-axis (left or right). The second number tells you how many spaces to move on the y-axis (up or down)

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1.factor: y2-4y-5
(y-5)(y+1)
.
2. factor 3a2+9a+6
(3a+3)(a+2)
.
3.evaluate: x(x+1)(x-1)
set each factor equal to zero and solve for the x-term:
x=0
x+1=0
x=-1
and x-1=0
x=1
So, x= 0, -1 and 1
.
4.factor: 6y2-2yz
factor out (2y)

(2y)(3y-z)
.

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Standard for is:
Ax+By=C
y=2/3x+1 [get the variables to one side and the constant (1) to the other side]
-2/3x+y=-2/3x+2/3x+1
-2/3x+y=1

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1. factor 6f2+5f+1

(3f+1)(2f+1)=0 [use trial-and-error to find the factors that have a product (multiplied) of the first and last term; and a sum (add) of the middle term]
Example: factor =(3f)(2f)
factor the last term of (1) = (1)(1)
Try to find the combination of the factors that, when added togeter, will equal the middle term of 5f.
(3f+1)(2f+1)

.
factor 12w2+10w-8

(6w+8)(2w-1) [use trial-and-error to find the factors that have a product (multiplied) of the first and last term; and a sum (add) of the middle term]
Example: 12 = (6)(2) or (12)(1) or (3)(4)
and -8 = (-1)(8) or (1)(-8) or (-4)(2) or (-2)(4)
Try to find the combinations that, when added, will equal the middle term of (10w)
.
3. factor: 20y2-33y+10

(5y-2)(4y-5)[use trial-and-error to find the factors that have a product (multiplied) of the first and last term; and a sum (add) of the middle term]
.
factor: -x2-x-6

(-x-3)(x-2)[use trial-and-error to find the factors that have a product (multiplied) of the first and last term; and a sum (add) of the middle term]

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y<1
First, establish a boundary line by using an equal sign:
y=1 (see the graph:

.
In this case, the boundary line will be a dashed line because the inequality is "less than" "1". If the inequality had been just "less than or equal to", the boundary line would have been a solid line instead.
.
Now, shade every thing below or less than the line (which is everything < 1) since "y" is every point less than the line (see the graph below):

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3x=6x-15 [solve for the x-term=
3x-6x=6x-6x-15
-3x=-15
-3x/-3=-15/-3
x=5
[checking:
3(5)=6(5)-15
15=30-15
15=15 [checks out]
.
So, x=5 and
x+8= 5+8=13

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Using the formula: y=mx+b;
y=any y-value on the line
m=the slope of the line
x=any corresponding x-value
b=the y-intercept
.
f(x)= f(x)=3x+2
y=mx+b
The slope = 3
The y-intercept = 2
Graphing:

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Using the formula: y=mx+b;
y=any y-value on the line
m=the slope of the line
x=any corresponding x-value
b=the y-intercept
.
f(x)=-2x+2
y=mx+b
The slope = -2
The y-intercept = 2
Graphing:

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Use the formula for the perimeter of a rectangle:
P=2(length+width)
P=36
Length = 3in more than twice its width = 2 x Width +3 = 2w+3
Width=w
.
P=2(l+w) [plug-in the values and solve for the width(w)]
36=2[(2w+3)+w]
36/2=2/2(2x+3+w)
18=2w+3+w
18=3w+3
18-3=3w
15=3w
15/3=3w/3
5=w
.
checking:
Length = 2w+3=2(5)+3=13
Width=5
.
36=2(13+5)
36=2(18)
36=36 [checks out]

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[multipliply the entire fraction by the denominator in order to get rid of the radical in the denominator]
.

[simplify wherever possible]


5

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f(x)=-x-4 [substitute (a-3) for the x-term and solve as much as possible
f(a-3)=-x-4
f(a-3)=-(a-3)-4
f(a-3)=-a+3-4[multiply through by (-1)
f(a-3)=-a-1
f(a-3)=a+1 [multiply through by (-1)

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y=2x-4 [substitute for the y-term in the first equation]
8x-4y=16
8x-4(2x-4)=16 [simplify]
8x-8x+16=16
0+16=16
16=16 [Infinate number of solutions because the equations represent the same line]