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The slope-intercept form is: y = mx+b, where m is the slope and b is the y-intercept (the point were x=0).
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The slope is defined as the rise over the run, which is the change in 'y' divided by the change in 'x'.
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m = (y₂-y₁)/x₂-x₁)
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(x₁,y₁) = (5,4)
(x₂,y₂) = (-4,-3)
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m = (-3 -4)/(-4-5)
m = -7/-9 = 7/9
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y = 7/9x + b
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Substitute what we know already to find 'b'.
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For example, we know the point (5,4) is on the line.
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4 = 7/9(5) + b
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Multiply through by 9 to remove the fraction
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36 = 7(5) + 9b
36 = 35 + 9b
1 = 9b
9b = 1
b = 1/9
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Now we can find the x-intercept, which is the point where y=0.
0 = 7/9x + 1/9
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multiply through by 9
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9(0) = 7x + 1
0 = 7x + 1
x = -1/7
(-1/7,0)
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So, to summarize:
Equation is: y = 7/9x + 1/9
y-intercept is: (0, 1/9)
x-intercept is: (-1/7, 0)
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Graphing it helps confirm your work.

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Done

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When solving solution problems, you need to keep track of how much 'pure' stuff you need. In this case the problem states you need 40 gal. of 8% nitrogen fertilizer: 40*.08 = 3.2 gal. 'pure' nitrogen.
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The materials on hand include 2% and 10% nitrogen in liquid form, so we assume we have as much of these two as we may need.
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Since the final amount needed is 40 gal., we can define the two mixtures in terms of the variable 'x'.
x = amount of 2% nitrogen fertilizer
40 - x = amount of 10% fertilizer.
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The objective equation we need to solve thus becomes:
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.02x + .10(40-x) = .08*40 = 3.20 gal.
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multiplying through by 100 will remove the decimals
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2x + 10(40-x) = 320
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expanding
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2x + 400 - 10x = 320
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collecting
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-8x = 320 - 400 = -80
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divide both sides by -8
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x = -80/-8 = 80/8 = 10
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Since we need 10 gal. of 2% solution, we need 40-10 = 30 gal of 10%.
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40-x = 30
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Therefore, we think that we need 10 gal. of 2% and 30 gal. of 10%.
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But we always need to check our answer...
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How much 'pure' stuff do we have in the 40 gallons we propose?
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10 gal. @ 2% = .02*10 = 0.2 gal.
30 gal. @ 10% = .10*30 = 3 gal.
0.2+3 = 3.2 gal.
Correct
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Answer:
Mix 10 gallons of 2% and 30 gallons of 10% nitrogen fertilizer to produce 40 gallons of 8% solution.
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Done.

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We have two numbers: x & y.
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The difference is 34.
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x - y = 34
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The larger number is 4 more than 3 times the smaller.
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Well, since I decided that x-y is a positive number, then x > y.
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x = 3y + 4
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Since x - y = 34, then
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x = y + 34
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Substitute x = y+34 to find the value of 'y'
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y+34 = 3y + 4
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subtract y from both sides
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y -y +34 = 3y-y + 4
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34 = 2y + 4
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subtract 4 from both sides
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34-4 = 2y +4-4
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30 = 2y
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so
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2y = 30
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divide both sides by 2
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y = 15
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substitute y=15 into the first equation to find 'x'
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x - 15 = 34
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x = 49
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check this answer by substituting these values for x and y into the second equation
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x = 3y + 4
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49 = 3(15) + 5
Correct
.
Answer:
x = 49
y = 15
.
Done.

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Assuming, the expression is:

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.

.

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Done

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Volume of a cylinder= pi*r^2*h
where
pi = the constant pi
r = radius
r^2 = r*r
h = height
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diameter = 12, so radius =6
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height = 15
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V = pi * 6 ft * 6 ft * 15 ft
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V = pi * 36 ft^2 * 15 ft
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V = pi * 540 ft^3
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V = 1696.46 ft^3
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Done.

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.

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Add 3x to both sides...


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Add 3 to both sides


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Divide both sides by 5


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Check...
.




Correct
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Answer: x=1
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Done

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In this case, speed is measured in miles per hr, which is shown as 'mph' but really is miles/hours.
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The average speed = distance / time.
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miles/hr
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Done

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The key to this problem is to keep track of your units of measurement.
Oftentimes the units of measurement will help you figure out what to divide by what or what to multiply by what.
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The shrimp is priced at $ per lb, which is shown as $/lb.
That tells you you need to divide the $ by the lb.
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He paid $x for 100 lb.
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So it would be $x/100 lb = x/100 $/lb
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Done.

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Here's a start...

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Start with 1950, which which we are told should be year t=0.
y = 151 million in 1950.
y = mx + b is the slope-intercept form.
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We're using the year to predict the population, so t=x-axis.
The population is plotted on y-axis as the dependent variable.
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So we would define it as:
y = mt + 151, which define defines the point (0,151)
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1990 would be t=40
1990 population is 249 million
So it would be plotted at (40,249)
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The slope = rise/run
rise = change in y = 249-151 = 98
run = change in x = change in t = 40-0 = 40
m = slope
m = 98/40
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That means our equation is:
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y = 98/40*t + 151
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Graph it to check
.

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L = length
W = width
P = 2L + 2W = perimeter
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P = 40 :: given
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L = 4W -5 :: length is 5 less than 4*Width
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Substitute and solve for W.
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2L + 2W = P
2(4W-5) + 2W= 40
8W -10 + 2W = 40
10W -10 = 40
10W = 50
W = 5
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L = 4W-5
L = 4(5) -5
L = 20 - 5
L = 15
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Checking the perimeter will tell if this is the right answer
2L + 2W = 40??
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2(15) = 30
2(5) = 10
30+10 = 40
Correct.
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Answer:
The dimensions of the rectangle are length = 15 and width = 5.
.
Done.

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When solving solution problems, you need to focus on how much 'pure' stuff you need.
In this case, you need to produce 4 gal. of a desired 45% solution.
That means you will need .45*4 = 1.8 gal. of the 'pure' stuff.
The simplest way would be to take 1.8 of 'pure' stuff and add 2.2 gal. of water, but you do not have that option.
Instead you have to use 90% and 30% solutions.
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Defining the amounts of is critical.
x = amount of 90% solution
y = amount of 30% solution
4 = total amount of 45% solution
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So y = 4-x, which will make the job of solving easier.
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.9x + .3(4-x) = .45*4
.9x + 1.2 -.3x = 1.8
.6x + 1.2 = 1.8
.6x = .6
x = 1
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That means you would propose to use 1 gal. of 90% solution + 3 gal. of 30% solution.
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Check by determining if that is enough 'pure' stuff.
1*.9 = .9
3*.3 = .9
.9 + .9 = 1.8, which is what you needed.
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Answer:
Mix 1 gal. of 90% solution with 3 gal. of 30% solution to obtain 4 gal. of 45% solution.
.
Done.

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7x^2 + xy + 2y + 14x
.
7x^2 + 14x + xy + 2y
.
7x(x+2) + y(x+2)
.
(7x+y)(x+2)
.
Done

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y = mx+b is slope-intercept form, where
m = slope
b = y-intercept (defined by x=0)
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Two points can be used to draw a line.
(-5,2) and (4,1)
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You also can determine the slope by computing the rise over the run.
rise = change in y = y₂ - y₁
run = change in x = x₂ - x₁
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We can define either point what we want, but we have to keep consistent.
(x₁, y₁) = (-5, 2)
(x₂, y₂) = (4, 1)
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rise = 1-2 = -1
run = 4 -(-5) = 4+5 = 9
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So,
m = -1/9
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That means y= -1/9x + b
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Now we have to compute 'b' to ensure it goes through the points given.
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We know that when x=4, y=1
Substituting, we find 'b'
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1 = -1/9*4 + b
1 + 4/9 = b
9/9 + 4/9 = b
13/9 = b
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So, we now have the equation: y = -1/9x + 13/9
Slope = -1/9
y-intercept = (0, 13/9)
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The x-intercept occurs where y=0, so we can solve by substitution again.
0 = -1/9x + 13/9
The obvious answer is x=13, so the point is (13,0).
.
Graphing is a good way to check your work.

.

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3y - 3w + yt - tw = 3(y-w) + t(y-w) = (3+t)(y-w)
.
Done.

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L = length
W = width
Area= L*W, for a rectangle
A = 50 ft^2 :: given
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L = W + 5 :: Length is 5 ft longer than the width, so you have to add 5 to W to make them equal
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Substitute L=W+5
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(W+5) * W = 50
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Collect and simplify
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W^2 + 5W = 50
W^2 + 5W - 50 = 0
(W + 10)(W - 5) = 0
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So our candidate solutions are W= -10 and W=5.
But a negative width is nonsensical, so our only practical option is W=5.
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Substituting W=5, we can find L.
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L = W+5
L = 5+5 = 10
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Checking the answer, we see
5*10 = 50, which matches the setup
10 = 5+5, which also matches the setup
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Answer:
The larger dimension is the length, which is 10.
The width is 5.
.
Done.

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Solve using a system of linear equations.
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4x +2y = 6
4x -5y = -1
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subtract them
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7y = 7
y = 1
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substitute y=1 into the first equation
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4x +2(1) = 6
4x = 4
x = 1
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So it appears the solution is (1,1).
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Graph the two equations it to check, but first you need to get them into y = mx+b form.
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4x+2y = 6
2y = -4x+6
y = -2x + 3
.
4x -5y = -1
-5y = -4x -1
y = 4/5x + 1/5
.

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Two consecutive page numbers would be, for example, 1 & 2 or 323 and 324 or x and x+1.
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x(x+1) = 600
x^2 + x - 600 = 0
(x-24)(x+25) = 0
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So there are two solutions: x=24 and x=-25.
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But a negative page number does not make any sense. So the answer is:
.
x = 24
x+1= 25
.
Done

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x^2 +4x -12 = (x +6)(x -2)
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check using FOIL (first, outer, inner, last)
.
x*x + x*(-2) + 6*x + 6*(-2) = x^2 +4x -12
.
Done

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Assuming you problem is to solve:
.
x^2 -x -30 = 0
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Factoring this is the best approach.
We notice that we need a + and a - term to result in -30.
And the two factors need to 1 apart.
There are a number of ways we can multiply two numbers to result in 30.
3*10 = 30, but they are 7 apart.
5*6 = 30, and they are 1 apart. So that's it.
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(x -6)(x +5) = 0
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checking this using FOIL (first, outer, inner, last):
.
x*x + x*5 + (-6)x + (-6)*5 = x^2 -x -30.
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So, the solutions that make the equation true are: x=6 and x= -5.
.
We also can graph it.

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y=3x
x+y = 8
.
substitute y=3x into the second equation
x+y = 8
x + 3x= 8
4x = 8
x = 2
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substitute x=2 into the first equation
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y = 3(2) = 6
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check by substituting x=2 and y=6 into the second equation
.
x+y = 8??
2 + 6 = 8
correct
.
Answer:
x = 2
y = 6
.
Done.

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First, get it into slope-intercept form: y = mx + b
where
y = value of y that depends on the value of x
m = slope of the line
b = y-intercept (the point where x=0)
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2y = 6x+4
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Divide both sides of the equation by 2.
.
y = 3x + 2
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When x=0
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y = 3*0 + 2 = 2
which defines the point (0,2).
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The x-intercept occurs where y=0.
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0 = 3x + 2
or
3x +2 = 0
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Subtract 2 from both sides.
.
3x = -2
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Divide both sides by 3.
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x = -2/3, which defines the point (-2/3,0).
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Given you have two points, you can graph a straight line.
.

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That's about all you need to know about this equation.
.
y = 3x + 2
The slope = 3, which means it goes up to the right.
The y-intercept = (0,2).
The x-intercept = (-2/3,0).
The graph is shown above.
You could test another value of x to make sure it is on the line, but I leave that to you.
.
Done.

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Show the multiplication to help keep yourself straight...
6(2b+7)3 = 6 * (2b + 7) * 3
.
Since it doesn't matter what you multiply first, you can multiply what is outside the parentheses...
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6 * (2b + 7) * 3 = 6*3 * (2b+7) = 18 * (2b + 7)
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Now multiply that times what is inside the parentheses...
.
18 * (2b+7) = 18*2b + 18*7
.
Simplify..
.
18*2b + 18*7 = 36b + 126
.
Done

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The ratio of 5:4 means that 4/9 are vanilla.
Given there are 45 in total...
.
4/9 = x/45
180 = 9x
20 = x
x=20
.
There are 20 vanilla.

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The setup ignores the perfect square statement and instead focuses on the relationship.
x+6=2x-4
.
Solving for 'x'
.
-x=-10
x = 10
.
Now we can check for the perfect square...
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x+6 = 16 :: a perfect square
.
Continue checking for the other condition...
2x-4= 20-4 = 16, which =x+6
Correct!
.
Answer: The number is 10.
.
Done.

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1 ft = 12 inches
8 ft = 8 ft * 12 inches/ft = 96 inches
.
Divide 96 by 9 = 10 2/3.
.
So, you can make 10 banners at 9 inches each.
That leaves you with 6 inches of material (96 -90 = 6).
.
Note that the remainder of '2/3' when you divided by 9 is '2/3 of 9' = 2/3*9 = 2*3 = 6.
.
Done

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3x^3+4x^2-32x=0
.
separate the common 'x'
.
x*(3x^2+4x-32)=0
.
factor
.
(3x^2+4x-32)=(3x - 8 )(x + 4)
.
so...
.
x(3x-8)(x+4)=0
.
That means there are three roots:
x = 0
x = 8/3
x = -4
.

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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=0 is zero! That means that there is only one solution: . Expression can be factored: Again, the answer is: 0, 0. Here's your graph:

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The perimeter of an equilateral triangle = 3s, where s=length of the each side.
3s = 45
s = 15
.
A rectangle has perimeter = 2(l+w)
We're told
w = l-3 and perimeter = 48
48 = 2(l + l-3)
2(2l -3) = 48
2l -3 = 24
2l = 27
l = 13.5
.
w = l-3 = 13.5-3 = 10.5
.
checking perimeter
2(13.5 + 10.5) = 48??
2(24) = 48
correct!
so
l = 13.5
w = 10.5
.
The question asks which figure has the longest side.
Answer:
Equilateral triangle has the longest sides at 15 inches.
.
Done

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Equations have a equal sign.
Expressions do not.

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First, get the main equation into slope-intercept form: y = mx + b
.
3x -y = 8
-y = -3x +8
y = 3x - 8
.
Second, you recall that any line with slope = 3 will be parallel to this line.
Your problem is define such a line that goes through the point (7,2).
Well, plug x=7 into the equation and then adjust 'b' to make y=2.
.
y = 3x -b
2 = 3(7) -b
2 = 21 -b
-19 = -b
b = 19
.
so,
y = 3x -19 will be parallel and will go through the point.
.
Check by graphing the two equations. y=3x-8 is red; y=3x-19 is green.

.
Rechecking the question, this equation needs to be shown in slope-intercept and in standard form.
y = 3x-19 :: slope-intercept form
.
Standard form is: Ax + By = C
.
subtract 3x from both sides
-3x + y = -19
.
multiply both sides by -1
.
3x -y = 19 :: standard form
.
Done