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Start with the given system of equations:



Multiply the both sides of the second equation by -1.


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.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Subtract from both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So the solutions are 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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Start with the given system of equations:



Multiply the both sides of the first equation by 2.


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.


Simplify.


Divide both sides by to isolate .


Reduce.


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


Now go back to the first equation.


Plug in .


Multiply.


Add to both sides.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


So the solutions are 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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Start with the given equation.


Distribute.


Combine like terms on the left side.


Combine like terms on the right side.


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 solution is which approximates to .

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Start with the given inequality.


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 . note: Remember, the inequality sign flips when we divide both sides by a negative number.


Reduce.


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

Answer:

So the solution is

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Start with the given inequality.


Distribute.


Subtract from both sides.


Add to both sides.


Combine like terms on the left side.


Combine like terms on the right side.


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

Answer:

So the solution is

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Start with the given system of equations:



Multiply the both sides of the second equation by 6.


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.


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 the solutions are 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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Start with the given equation.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "r":


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Subtract from to get


Multiply and to get .


Take the square root of to get .


or Break up the expression.


or Combine like terms.


or Simplify.


So the solutions are or

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Remember the sum of two numbers is simply the result of adding them. So the sum of x and y is . In this particular problem, the first addend is the unknown number 'x' and the second addend is 36 times its reciprocal. In other words, the second number is


So add these two expressions to get: and set that equal to -13 to get


I'll let you take it from here. All you need to do now is solve for 'x'.

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Start with the given equation.


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 solution is

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Start with the given system of equations:



Multiply the both sides of the first equation by 2.


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.


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 the solutions are 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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Start with the given expression.


Multiply the fraction by .


Combine the fractions.


FOIL the numerator.


FOIL the denominator.


Multiply.


Replace with -1.


Multiply.


Combine like terms.


Break up the fraction.


Reduce.


So .


So the expression is now in standard form where and

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The variable is simply the unknown number. In the case of 6y-3=6y+8, the variable is the letter y. The letter y simply stands for some unknown number.

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Hint: Since AND , this means that we can set the right sides equal to one another to get . Solve this equation for 'x' to find its value. Once you have 'x', you can plug it into either given equation to find 'y'.

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Hint: the slope formula is . So plug in the given coordinates and the slope to get and simplify to get . Now solve for 'r' to get -->


So the second y coordinate is -5. It turns out that to have a slope of zero, the y coordinates must be the same.

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Hint: Remember that . So this means that is equivalent to


Square both sides to get . From here, you basically expand and then use the quadratic formula to solve for 'x'. Don't forget to check your answers.

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Since there are 47 men and 40 women, this means that there are 47+40=87 people total.

Probability prize goes to woman = # of women/# total = 40/87 = 0.4597

which is about a 45.97% chance

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Start with the given expression.


Break up the log using the identity


Break up the first log using the identity


Convert to rational exponent notation.


Pull down the exponents using the identity


So

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The unfortunate part of sequences and series is that there is usually more than one answer. However, if you know what to look for, then it's not so bad.

Try breaking up the sequence to get:

12,10,16,17,8,6,18


Notice how the red terms are decreasing by 2 while the blue terms are increasing by 1. The red and blue terms alternate in sets of 2. In other words, there are two subsequences (a smaller sequence) alternating every two terms. So up next is a blue term.

To get the next term, add 1 to the last blue term to get 18+1=19. After that, subtract 2 from the last red term to get 6-2=4.


So the next two terms are 19 and 4. The obvious problem with this solution is that you have to know the answer to have an idea of what to do here. However, it is possible to just stumble onto it.

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Hint: if we let and apply the appropriate substitutions, we go from to . Now factor . Once you have factored out, plug back in.

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Hint: Since we're given an "equilateral octagon", this means that all of the sides are of equal length. So . Simply solve this equation for 'x' to find the side lengths.

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Hint: Since the angles are supplementary, this means that the angles add to 180 degrees. Algebraically, this means that . So solve this equation for 'x' to find 'x' and the two angles.

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Let . So the expression then becomes


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .


Now multiply the first coefficient by the last term to get .


Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?


To find these two numbers, we need to list all of the factors of (the previous product).


Factors of :
1
-1


Note: list the negative of each factor. This will allow us to find all possible combinations.


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

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

From the table, we can see that the two numbers and add to (the middle coefficient).


So the two numbers and both multiply to and add to


Now replace the middle term with . Remember, and add to . So this shows us that .


Replace the second term with .


Group the terms into two pairs.


Factor out the GCF from the first group.


Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


Combine like terms. Or factor out the common term


Condense the terms.



So factors to .


In other words, .


Now plug in to go from to . Now simplify to get


===============================================================

Answer:


So factors to .


In other words, .

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Start with the given polynomial.


Plug in .


Square to get .


Multiply and to get .


Multiply and to get .


Combine like terms.


So when

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First you have to convert into polar form .

To do that, use the formulas and


In this case, we're given which means that and . So and

So this means that the rectangular expression is equivalent to the polar form .

From here, we can now use De Moivre's Theorem. De Moivre's Theorem states that if , then

Since , using De Moivre's Theorem gets us






... Evaluate to the 12th power to get


... Evaluate 2 to the 6th power to get 64


... Evaluate the trig functions.


... Simplify.


... Multiply


Because we let , this means that

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Start with the given equation.


Notice that the quadratic is in the form of where , , and


Let's use the quadratic formula to solve for "x":


Start with the quadratic formula


Plug in , , and


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


or Break up the expression.


So the solutions are or


which approximate to or

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Note: is the first point . So this means that and .
Also, is the second point . So this means that and .


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


Reduce


So the slope of the line that goes through the points and 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.


So the equation is now in slope intercept form where the slope is and the y-intercept is note: the y-intercept is the point

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Start with the given expression.


Expand.


Distribute.


Multiply.


Now combine like terms.


So expands to .


In other words, .

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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.


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


Rewrite as


Distribute


Multiply


Add 6 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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I'll get the equations into slope intercept form and let you do the graphing.



Start with the first equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


So the equation is now in slope intercept form where the slope is and the y-intercept is note: the y-intercept is the point

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



Now move onto the second equation.


Subtract from both sides.


Rearrange the terms.


So the equation is now in slope intercept form where the slope is and the y-intercept is note: the y-intercept is the point

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The expression isn't defined when the denominator is equal to zero. So solve to get . So if , then the expression is undefined.