You can put this solution on YOUR website!
Start with the given expression


Multiply the coefficients 7 and 6 to get 42


When you multiply monomials, simply add the exponents


Add


So our answer is

You can put this solution on YOUR website!
Let x=verbal score and y=math score

"Megan’s verbal and math scores on the SAT was 1094" translates to

"Her math score was 248 points higher than her verbal score" translates to


Start with the given system





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


Combine like terms on the left side


Subtract 248 from both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide




Now that we know that , we can plug this into to find



Substitute for each


Simplify


So our answers are and



So the verbal score was 423 and the math score was 671.

You can put this solution on YOUR website!

Start with the given expression.


Factor to get .


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where

You can put this solution on YOUR website!
# 1



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 3. This means that to go from point to point, we can go down 1 and over 3



So starting at , go down 1 unit


and to the right 3 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

---

# 2

Start with the given equation



Add to both sides


Rearrange the equation


Divide both sides by


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 3 and the run is 1. This means that to go from point to point, we can go up 3 and over 1



So starting at , go up 3 units


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

You can put this solution on YOUR website!

Start with the given system





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


Distribute


Combine like terms on the left side


Subtract 24 from both sides


Combine like terms on the right side


Divide both sides by -7 to isolate y



Divide




Now that we know that , we can plug this into to find



Substitute for each


Simplify


So our answer is and which also looks like



Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer.


Graph of (red) and (green)

You can put this solution on YOUR website!

Start with the given expression.


Factor to get .


Factor to get .


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where or


So the answer is choice C

You can put this solution on YOUR website!
Start with the given expression.



Factor to get .


Factor to get .


Factor to get .


Combine the fractions.


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where ,, or



So the answer choice is C

You can put this solution on YOUR website!
Start with the given inequality



Add 9 to both sides


Subtract 2x from both sides


Combine like terms on the left side


Combine like terms on the right side


Divide both sides by 3 to isolate x



Reduce

--------------------------------------------------------------
Answer:
So our answer is (which is approximately in decimal form)




So the answer in interval notation is


Here's the graph of the solution set

---

Start with the given inequality


Distribute




Distribute


Combine like terms on the left side


Add 20 to both sides


Add 2m to both sides


Combine like terms on the left side


Combine like terms on the right side


Divide both sides by 12 to isolate m



Reduce

--------------------------------------------------------------
Answer:
So our answer is (which is approximately in decimal form)



So the answer in interval notation is




Here's the graph of the solution set

You can put this solution on YOUR website!
This solution was generated from the linear algebra toolkit






So , and

You can put this solution on YOUR website!

Start with the given function



Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is .


Horizontal Asymptote:
Since the degree of the numerator and the denominator are the same, we can find the horizontal asymptote using this procedure:

To find the horizontal asymptote, first we need to find the leading coefficients of the numerator and the denominator.

Looking at the numerator , the leading coefficient is

Looking at the denominator , the leading coefficient is

So the horizontal asymptote is the ratio of the leading coefficients. In other words, simply divide by to get


So the horizontal asymptote is





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



Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

Set the denominator equal to zero


Now let's use the quadratic formula to solve for x. If you need help with the quadratic formula, check out this solver.

After using the quadratic formula, we get the solutions
or

So this means the vertical asymptotes are or

Notice if we graph , we can visually verify our answers:

Graph of with the horizontal asymptote (blue line) and the vertical asymptotes and (green lines)

note: for some reason the graph didn't format correctly, but you get the idea. This should look better on a graphing calculator.

You can put this solution on YOUR website!
Any rational zero can be found through this equation

where p and q are the factors of the last and first coefficients


So let's list the factors of 10 (the last coefficient):



Now let's list the factors of 8 (the first coefficient):



Now let's divide each factor of the last coefficient by each factor of the first coefficient









Now simplify

These are all the distinct rational zeros of the function that could occur

You can put this solution on YOUR website!
Start with the given equation


Combine the logs using the identity



Foil


Rewrite the equation using the property: ====>


Square 10 to get 100


Subtract 100 from both sides


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 possible answers are or

However, since you cannot take the log of a negative number, this means that the only answer is





-------------------------------
Answer:


So our answer is which approximates to

You can put this solution on YOUR website!
Let's evaluate f(g(x))

Start with the first function


Plug in


Rewrite the right side using the identity


Take the log base 2 of 2 to get 1. Note:


Multiply



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



Let's evaluate g(f(x))

Start with the second function


Plug in



Let . So this means that by use of the property ====>


Replace with "y"


Now replace with "x"




So both f(g(x)) and g(f(x)) equal x.

You can put this solution on YOUR website!

Start with the given function



Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is .


Horizontal Asymptote:

Since the degree of the numerator (which is ) is less than the degree of the denominator (which is ), the horizontal asymptote is always

So the horizontal asymptote is



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



Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

Set the denominator equal to zero


Now let's use the quadratic formula to solve for x. If you need help with the quadratic formula, check out this solver.

After using the quadratic formula, we get the solutions
or

So this means the vertical asymptotes are or

Notice if we graph , we can visually verify our answers:

Graph of with the horizontal asymptote (blue line) and the vertical asymptotes and (green lines)

You can put this solution on YOUR website!
I'm assuming you want to solve for x. I'll solve the first one to get you started.


# 1




Start with the given equation.


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


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 our answers are or

You can put this solution on YOUR website!
# 1


Start with the given function


Plug in


Raise 2 to the 2nd power to get 4


Multiply 2 and 4 to get 8


Multiply 7 and 2 to get 14


Subtract 14 from 8 to get -6


Add -6 and 3 to get -3




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


# 2

Start with the given function


Plug in


Raise -3 to the 2nd power to get 9


Multiply 2 and 9 to get 18


Multiply 7 and -3 to get -21


Subtract -21 from 18 to get 39


Add 39 and 3 to get 42

You can put this solution on YOUR website!
Start with the given equation


Subtract k from both sides


Multiply both sides by



So our answer is

You can put this solution on YOUR website!
Start with the given expression


Factor to get . So the LCD is


Multiply the second fraction by


Combine the fractions



Foil to get


Foil to get


Distribute


Since the two fractions have the same denominator, this means we can combine them.


Combine like terms


So simplifies to .


In other words, where or

You can put this solution on YOUR website!
# 1
D: all real numbers
R: all real numbers



# 2
D: all real numbers
R:


# 3
D: all real numbers
R:

You can put this solution on YOUR website!
I'll do the first two to get you started

# 1



Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is note: really looks like


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 6 and the run is 1. This means that to go from point to point, we can go up 6 and over 1



So starting at , go up 6 units


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

---

# 2



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 3 and the run is 1. This means that to go from point to point, we can go up 3 and over 1



So starting at , go up 3 units


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

You can put this solution on YOUR website!

Start with the 1st inequality


Replace the inequality sign with an equals sign


So in order to plot , we need to plot the equation first


Graph of


Now plug in a test point (0,0) into the inequality


Plug in and


Evaluate and simplify


Since the inequality is true, this means that we shade the entire region that contains the point (0,0). So this means that we shade everything below the line . Note: Since the inequality is a "less than" sign, this means that the line should be a dotted line.


Graph of with the shaded region in green

You can put this solution on YOUR website!
Start with the given expression


Multiply both the numerator and the denominator by the LCD


Distribute. Notice how the inner denominators cancel


Simplify

Factor to get .


Highlight the common terms.


Cancel out the common terms.


Simplify.


So simplifies to .


In other words, where , , or

You can put this solution on YOUR website!
Start with the given inequality



Divide both sides by 4 to isolate x



Divide

--------------------------------------------------------------
Answer:
So our answer is



Here's the graph of the solution set


Note: at there is a open circle (which means this point is excluded)

You can put this solution on YOUR website!
Graph of



Note: at there is a closed circle (which means this point is included) and at there is a open circle (which means this point is excluded)

You can put this solution on YOUR website!


Start with the given inequality





Add 4 to all sides






Divide every side by -1 to isolate z. Remember, dividing every side of an inequality by a negative number flips the inequality signs




Rearrange the inequality.


So the solution in interval notation is: (-15,-8)


The solution in set notation is


Now let's graph the solution set



Note: at there is a open circle (which means this point is excluded) and at there is a open circle (which means this point is excluded)

You can put this solution on YOUR website!
Start with the given expression


Factor out the GCF


Subtract


Rearrange the terms

You can put this solution on YOUR website!

Start with the given system of equations:





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


Add to both sides


Rearrange the equation


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

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



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



Distribute the negative


Combine like terms on the left side


Subtract 4 from both sides


Combine like terms on the right side


Divide both sides by -2 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

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

You can put this solution on YOUR website!
Start with the given expression


Factor out the GCF


Add


Rearrange the terms

You can put this solution on YOUR website!

Start with the given equation.


Get all terms to the left side.


Notice we have a quadratic equation in the form of where , , and


Let's use the quadratic formula to solve for x


Start with the quadratic formula


Plug in , , and


Negate to get .


Square to get .


Multiply to get


Rewrite as


Add to to get


Multiply and to get .


or Break up the expression.


So our answers are or


which approximate to or

You can put this solution on YOUR website!
Start with the first equation


Plug in


Multiply both sides by the LCM of 2. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



Distribute and multiply the LCM to each side



Add 4 to both sides


Subtract x from both sides


Combine like terms on the left side


Combine like terms on the right side


Divide both sides by -5 to isolate x



Reduce. This is the first part of our answer.


Go back to the first equation


Plug in


Multiply


Combine like terms



So our answer is and

You can put this solution on YOUR website!

# 1
Jump to problem #2





Start with the given system of equations:





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


Add to both sides


Rearrange the equation


Divide both sides by


Break up the fraction


Reduce



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

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



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



Distribute the negative


Combine like terms on the left side


Subtract 4 from both sides


Combine like terms on the right side


Divide both sides by -2 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

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

---

# 2
Jump to problem #1






Start with the given system of equations:





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





In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for , we would have to eliminate (or vice versa).


So lets eliminate . In order to do that, we need to have both coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.



So to make the coefficients equal in magnitude but opposite in sign, we need to multiply both coefficients by some number to get them to an common number. So if we wanted to get and to some equal number, we could try to get them to the LCM.



Since the LCM of and is , we need to multiply both sides of the top equation by and multiply both sides of the bottom equation by like this:




Multiply the top equation (both sides) by
Multiply the bottom equation (both sides) by




Distribute and multiply





Now add the equations together. In order to add 2 equations, group like terms and combine them



Combine like terms and simplify



Notice how the x terms cancel out




Simplify



Now plug this answer into the top equation to solve for x

Start with the first equation



Plug in




Multiply



Subtract 2 from both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide




So our answer is
and



which also looks like




Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).