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You can simply count the spaces between the two points vertically to get 6 units, but here's the formal method to find the distance:




Start with the distance formula.


Plug in , , , and .


Subtract from to get .


Subtract from to get .


Square to get .


Square to get .


Add to to get .


Take the square root of to get .


So our answer is


So the distance between the two points is 6 units.

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Start with the distance formula.


Plug in , , , and .


Subtract from to get .


Subtract from to get .


Square to get .


Square to get .


Add to to get .


So our answer is


Which approximates to


So the distance between the two points is approximately 1.414 units.

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Note: and

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Start with the given equation


Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 81 and it's opposite -81




Now lets focus on the first equation


Subtract 142 from both sides


Combine like terms on the right side

---

Now lets focus on the second equation



Subtract 142 from both sides


Combine like terms on the right side





So the solutions are:

or

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Start with the given equation


Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 81 and it's opposite -81




Now lets focus on the first equation


Add 6 to both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Reduce

---

Now lets focus on the second equation



Add 6 to both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Reduce





So the solutions are:

and

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Start with the given equation.


Subtract from both sides.


Combine like terms on the left side.


Divide both sides by to isolate .


Reduce.


-------------------------------------


Move onto the given expression


Plug in


Multiply


Add


So when

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Start with the given equation.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


We can see that the equation has a slope and a y-intercept .


Now to find the slope of the perpendicular line, simply flip the slope to get . Now change the sign to get . So the perpendicular slope is .


Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope and the coordinates of the given point .


Start with the point slope formula


Plug in , , and


Distribute


Multiply


Add 3 to both sides.


Combine like terms.


So the equation of the line perpendicular to that goes through the point is .


Here's a graph to visually verify our answer:
Graph of the original equation (red) and the perpendicular line (green) through the point .

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Start with the given equation.


Multiply both sides by


Multiply.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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# 1



Start with the given equation.


Subtract from both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

---

# 2



Start with the given equation.


Distribute.


Combine like terms on the left side.


Combine like terms on the right side.


Add to both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is which in decimal form is .

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# 1

Cannot solve since is NOT an equation (it's missing an equals sign)

---

# 2




Start with the given equation.


Multiply EVERY term by the LCD to clear the fractions.


Multiply.


Add to both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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So you want to graph ?





Table of Contents:
Step 1: Finding the Vertex
Step 2: Finding two points to left of axis of symmetry
Step 3: Reflecting two points to get points right of axis of symmetry
Step 4: Plotting the Points (with table)
Step 5: Graphing the Parabola

In order to graph , we can follow the steps:


Step 1) Find the vertex (the vertex is the either the highest or lowest point on the graph). Also, the vertex is at the axis of symmetry of the parabola (ie it divides it in two).


Step 2) Once you have the vertex, find two points on the left side of the axis of symmetry (the line that vertically runs through the vertex).


Step 3) Reflect those two points over the axis of symmetry to get two more points on the right side of the axis of symmetry.


Step 4) Plot all of the points found (including the vertex).


Step 5) Draw a curve through all of the points to graph the parabola.


Let's go through these steps in detail


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Step 1)

Finding the vertex:

In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: .


Start with the given formula.


From , we can see that , , and .


Plug in and .


Multiply 2 and to get .


Divide.


So the x-coordinate of the vertex is . Note: this means that the axis of symmetry is also .


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the y-coordinate of the vertex is .


So the vertex is .


---------------------------------------------------------------------


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Step 2)

Find two points to the left of the axis of symmetry:

Let's find the y value when


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the first point to the left of the axis of symmetry is (-3,4)


---------------------


Let's find the y value when


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So the second point to the left of the axis of symmetry is (-2,1)


---------------------------------------------------------------------


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Step 3)

Reflecting the two points over the axis of symmetry:

Now remember, the parabola is symmetrical about the axis of symmetry (which is )


This means the y-value for (which is one unit from the axis of symmetry) is equal to the y-value of (which is also one unit from the axis of symmetry). So when , which gives us the point (0,1). So we essentially reflected the point (-2,1) over to (0,1).


Also, the y-value for (which is two units from the axis of symmetry) is equal to the y-value of (which is also two units from the axis of symmetry). So when , which gives us the point (1,4). So we essentially reflected the point (-3,4) over to (1,4).


---------------------------------------------------------------------


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Step 4)

Plotting the points:

Now lets make a table of the values we have calculated:

x	y
-3	4
-2	1
-1	0
0	1
1	4

Now let's plot the points:



---------------------------------------------------------------------


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Step 5)

Drawing a curve through all of the points:

Now draw a curve through all of the points to graph :


Graph of

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Start with the given equation


Plug in


Subtract 32 from both sides.


Combine like terms.


Divide both sides by 1.8 to isolate C


Divide (and round to the nearest tenth)


So the answer is approximately


So this means that -50 F is approximately -45.6 C

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Let x=# of adult tickets and y=# of senior tickets


"a theater sold 478 tickets for a play" translates to

and


"thickets cost $20 for adults and $13 for senior citizens.,if the total receipts were $6970" translates to






So we have the system of equations:





Let's solve the system by substitution


Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation

Start with the first equation


Subtract from both sides


Rearrange the equation


---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Combine like terms on the left side


Subtract 6214 from both sides


Combine like terms on the right side


Divide both sides by 7 to isolate x



Divide





-----------------First Answer------------------------------


So the first part of our answer is:









Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is:









-----------------Summary------------------------------

So our answers are:

and


So this means that 108 adult tickets and 370 senior tickets were sold.

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Let x=side length of square

Since "A house had two rooms of equal area. One room is square and the other room is a rectangle 4ft narrower and 5ft longer than the square one.", this means that


Start with the given equation.


FOIL


Subtract from both sides.


Combine like terms.


Add 20 to both sides.



So the answer is which means that the side length of the square is 20 ft. So the area of the square is square feet.


So the length of the rectangle is ft and the width is ft. So the area of the rectangle is square feet.

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First solve for "x":


Start with the given equation.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


Now plug the given value of "x" into the expression


Start with the given expression


Plug in


Multiply


Add


So if

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Start with the given inequality.


Multiply both sides by


Multiply.


Divide both sides by to isolate . Note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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Start with the given equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is -5 and the run is 2. This means that to go from point to point, we can go down 5 and over 2



So starting at , go down 5 units


and to the right 2 units to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and

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Start with the given system of equations:



Multiply the both sides of the first equation by -4.


Distribute and multiply.


So we have the new system of equations:



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:





Group like terms.


Combine like terms. Notice how the x terms cancel out.


Simplify.


Divide both sides by to isolate .


Reduce.


------------------------------------------------------------------


Now go back to the first equation.


Plug in .


Multiply.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So our answer is and .


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of (red) and (green)

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I'll do the first three to get you started

# 1




Start with the given equation.


Distribute.


Combine like terms on the left side.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


----------------------------------------------------------------------

Answer:

So the answer is which approximates to .

---

# 2



Start with the given equation.


Distribute.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is which approximates to .

---

# 3




Start with the given equation.


Combine like terms on the left side.


Subtract from both sides.


Add to both sides.


Combine like terms on the left side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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Note: factors to and factors to





So where or

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Start with the given equation.


Combine like terms on the left side.


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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Looking at we can see that the first term is and the last term is where the coefficients are 1 and 1 respectively.

Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the middle coefficient 1? Let's list all of the factors of 1:



Factors of 1:
1

-1 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 1
1*1
(-1)*(-1)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1

First Number	Second Number	Sum
1	1	1+1=2
-1	-1	-1+(-1)=-2

None of these pairs of factors add to 1. So the expression cannot be factored

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So where

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Note: factors to


So where

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Take note that factors to





So


Note:

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Note: factors to


So where

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Start with the given equation


Cross Multiply


Distribute.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the answer is

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Do you want to graph? Please post full instructions.




Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is


Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis

So we have one point




Now since the slope is comprised of the "rise" over the "run" this means


Also, because the slope is , this means:




which shows us that the rise is 1 and the run is 1. This means that to go from point to point, we can go up 1 and over 1



So starting at , go up 1 unit


and to the right 1 unit to get to the next point



Now draw a line through these points to graph

So this is the graph of through the points and

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Note: you must surround the expression by three curly braces (not parenthesis) to make the expression show up the way you want it.


Also, use parenthesis to show grouping. So write the expression as (3a+b)/(6a+2b)






So where

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Table of Contents:


Jump to Table
Jump to Graph


In order to graph , we need to plot some points.


Let's find y when
(Note: you can start with any x-value):


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


----------------------------------------------------


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


----------------------------------------------------


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


----------------------------------------------------


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


----------------------------------------------------


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


----------------------------------------------------


Let's find y when :


Start with the given equation.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when , then .


So we have the point .


----------------------------------------------------





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Now let's make a table of the values we just found.

Table of Values:

x
y
0
4
1
0
2
-2
3
-2
4
0
5
4

Now let's plot the points:







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Graph:

Now draw a curve through all of the points to graph :


Graph of

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So where