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 Exponential-and-logarithmic-functions/274779: Can someone please explain why logbx/ logby does not equal logbx - logby? 1 solutions Answer 200502 by jim_thompson5910(28536)   on 2010-02-27 00:20:19 (Show Source): You can put this solution on YOUR website!To prove something false, we usually resort to a counterexample. A counterexample is an explicit example in which proves a theorem or equation false (usually by a contradiction of some sort). So let's say that and . This would then mean that Now plug and into to get So in short, and when and . Clearly which means that when and . --------------------------------------------------------------- Here's another way of looking at it. By the change of base formula, By another identity, So let's assume that . If this is the case, then Equate the bases and arguments to get and . The second equation simplifies to . Solve for b to get . Now if , this means that which is impossible (you can't divide by zero).
 Points-lines-and-rays/274776: I am having a hard time with my Geomarty homework I would really like if i were to get some kind of help. The question on the homework says... 1.Draw the following triangle A(-3,2) B(9,2) C(3,-6)1 solutions Answer 200500 by jim_thompson5910(28536)   on 2010-02-27 00:01:21 (Show Source): You can put this solution on YOUR website!Plot the 1st point A(-3,2) Plot the 2nd point B(9,2) Plot the 3rd point C(3,-6) Now connect the points to plot the triangle:
 Points-lines-and-rays/274778: Whats the distance between (4,4) and (7,8)1 solutions Answer 200499 by jim_thompson5910(28536)   on 2010-02-26 23:51:12 (Show Source): You can put this solution on YOUR website!Note: is the first point . So this means that and . Also, is the second point . So this means that and . 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 5 units.
 Graphs/274760: Evaluate f(x)= 4x^2 - 2 for f(1/2) 1 solutions Answer 200498 by jim_thompson5910(28536)   on 2010-02-26 23:49:56 (Show Source): You can put this solution on YOUR website! Start with the given equation. Plug in . Square to get . Multiply and to get . Combine like terms.
 Linear-equations/274739: Find the equation of the line through the points (-3,-1) and (-9,-6)1 solutions Answer 200469 by jim_thompson5910(28536)   on 2010-02-26 20:14:43 (Show Source): You can put this solution on YOUR website!First let's find the slope of the line through the points and 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 ------------------------------------------------------------------------------------------------ Now remember that the general slope intercept equation is where 'm' is the slope of the line and 'b' is the y-intercept. We can use this general equation to find the equation of the line. Since the line goes through the point (-3,-1), this means that x=-3 and y=-1. In addition, we know that the slope is . So we can use these values to solve for 'b'. Start with the general slope-intercept equation. Plug in , and Multiply. Reduce. Add to both sides to isolate 'b' Combine like terms. So the value of 'b' is Since and , we can plug these values into to get ====================================================================== Answer: So the equation of the line in slope-intercept form through the points (-3,-1) and (-9,-6) is
 Polynomials-and-rational-expressions/274710: Find the quotient of the polynomials. (3x^4-5x^3+2x-7) / (x-5)1 solutions Answer 200449 by jim_thompson5910(28536)   on 2010-02-26 19:24:07 (Show Source): You can put this solution on YOUR website!Let's use polynomial long division to get So this means that
 Functions/274719: Determine whether the correspondence is a function. Celebrity~~~~~~~~~~~~~~~~ Birthday Sigourney Weaver~~~~~~~~~~~ October 8 Jesse Jackson~~~~~~~~~~~~ October 8 Chevy Chase ~~~~~~~~~~~~~~ October 8 Muhammad Ali~~~~~~~~~~~~January 17 Jim Carrey~~~~~~~~~~~~~~January 17 Thank you for your help!1 solutions Answer 200447 by jim_thompson5910(28536)   on 2010-02-26 19:19:26 (Show Source): You can put this solution on YOUR website!Since each celebrity has ONLY ONE birthday, this means that each input (celebrity) has ONLY ONE output (birthday). So this relation is a function.
 Linear-equations/274720: I need to find an equation of the line containing the given pair of points; (-3,-1) and (-9,-6). The equation of the line in the slope-intercept is y=? m=y2-y1= -6-(-1)= -5 x2-x1 -9-(-3) -6 y-(-1)=m(x-x1) y-(-1)=-5(x-(-3)) -6 y-(-1)=-5x-5 -6 2 y=-5 -3 6x 2 Apparently I did this wrong somewhere and I am not sure what I did wrong. Can someone help me please.1 solutions Answer 200446 by jim_thompson5910(28536)   on 2010-02-26 19:17:54 (Show Source): You can put this solution on YOUR website! First let's find the slope of the line through the points and 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 Now let's use the point slope formula: Start with the point slope formula Plug in , , and Rewrite as Rewrite as Distribute Multiply Subtract 1 from both sides. Combine like terms. note: If you need help with fractions, check out this solver. So the equation that goes through the points and is
 Radicals/274702: how to write an equivalent expression using radical notation for (16a^6)^3/41 solutions Answer 200432 by jim_thompson5910(28536)   on 2010-02-26 18:09:34 (Show Source): You can put this solution on YOUR website!Recall that This means that We could optionally cube to get which would mean that
 expressions/274703: Evaluate the factorial expression (n+5)!/n+51 solutions Answer 200429 by jim_thompson5910(28536)   on 2010-02-26 18:06:44 (Show Source): You can put this solution on YOUR website!Since , this means that In other words,
Graphs/274663: what is the euclidean distance from the point (3,5) to the line y=2x
1 solutions

Answer 200418 by jim_thompson5910(28536)   on 2010-02-26 16:16:50 (Show Source):
You can put this solution on YOUR website!
 Solved by pluggable solver: Finding a distance between a point given by coordinates (x, y) and a line given by equation y=ax+b We want to find the perpendicular distance between a point given by coordinates (,) and a line given by equation First, let's draw a diagram of general situation with point P (xo, yo) and line L: y= a.x + b. The required distance is PC. (in the diagram below) Methodology We will first find the vertices of the triangle in order to get the side lengths and then by applying Sine Rule on right angle triangle PAB and PBC we will calculate the desired distance PC. Step1 Calculation of the vertices of triangle PAB: Draw a vertical line passing through the point 'P'. This line will cut the given line 'L' at point 'A'. The X coordinate of A(x1) will be same as . To find the Y-coordinate of 'A' we will use the fact that point 'A' lies on the given line 'L' and satisfies the equation of the line 'L' . Now, plug this in to the equation of line: y=2*x+0 Hence, Point (A)(,) Similarly, Draw a horizontal line passing through the point 'P'. This line will cut the given line 'L' at point 'B'. The Y coordinate of B(y2) will be same as . To find the X-coordinate of B we will use the fact that point 'B' lies on the given line 'L' and satisfies the equation of the line 'L' . Now, plug this in to the equation of line: y=2*x+0 Hence, Point (B)(,) Now, we have all the vertices of the triangle PAB Step2 Calculation of the side lengths using distance formula: Hence, The side lengths PA, PB and AB are Step3 Apply Sine rule on common angle B in triangle PAB and triangle PBC. Both triangle PAB and triangle PBC are right angle triangle and points 'A', 'B' and 'C' lay on the given line L. PC is the required perpendicular distance of the point P (3, 5) from line given lineL1: y=2*x+0. For better understanding of this concept, look at the Lesson based on the above concept. Lesson

So the distance is approximately units.

Matrices-and-determiminant/274496: Use Cramer's rule to solve the systems:
1. 2x-9y-z=-72
x+3y+4z=51
-6x+y+z=-4

2. 4x+6y=14
2x+y=-3
1 solutions

Answer 200314 by jim_thompson5910(28536)   on 2010-02-26 01:40:09 (Show Source):
You can put this solution on YOUR website!
# 1

 Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables First let . This is the matrix formed by the coefficients of the given system of equations. Take note that the right hand values of the system are , , and and they are highlighted here: These values are important as they will be used to replace the columns of the matrix A. Now let's calculate the the determinant of the matrix A to get . To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver. Notation note: denotes the determinant of the matrix A. --------------------------------------------------------- Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'x' column so to speak). Now compute the determinant of to get . Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant. To find the first solution, simply divide the determinant of by the determinant of to get: So the first solution is --------------------------------------------------------- We'll follow the same basic idea to find the other two solutions. Let's reset by letting again (this is the coefficient matrix). Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'y' column in a way). Now compute the determinant of to get . To find the second solution, divide the determinant of by the determinant of to get: So the second solution is --------------------------------------------------------- Let's reset again by letting which is the coefficient matrix. Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix Now compute the determinant of to get . To find the third solution, divide the determinant of by the determinant of to get: So the third solution is ==================================================================================== Final Answer: So the three solutions are , , and giving the ordered triple (3, 8, 6) Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.

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# 2

 Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 2 variables First let . This is the matrix formed by the coefficients of the given system of equations. Take note that the right hand values of the system are and which are highlighted here: These values are important as they will be used to replace the columns of the matrix A. Now let's calculate the the determinant of the matrix A to get . Remember that the determinant of the 2x2 matrix is . If you need help with calculating the determinant of any two by two matrices, then check out this solver. Notation note: denotes the determinant of the matrix A. --------------------------------------------------------- Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'x' column so to speak). Now compute the determinant of to get . Once again, remember that the determinant of the 2x2 matrix is To find the first solution, simply divide the determinant of by the determinant of to get: So the first solution is --------------------------------------------------------- We'll follow the same basic idea to find the other solution. Let's reset by letting again (this is the coefficient matrix). Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'y' column in a way). Now compute the determinant of to get . To find the second solution, divide the determinant of by the determinant of to get: So the second solution is ==================================================================================== Final Answer: So the solutions are and giving the ordered pair (-4, 5) Once again, Cramer's Rule is dependent on determinants. Take a look at this 2x2 Determinant Solver if you need more practice with determinants.

 real-numbers/274513: Hello, I am having a question on 1 problem for my online Algebra class that just are not clicking in my head. I have tried to do them several times. I would really appreciate if you could help with these two :) Thank You it read a-2/5 + a-3/4 = 6/5 1 solutions Answer 200312 by jim_thompson5910(28536)   on 2010-02-26 01:37:15 (Show Source): You can put this solution on YOUR website!I'm assuming that the problem is Start with the given equation. Multiply both sides by the LCD to clear any fractions. Distribute and multiply. 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 solution is which approximates to .
 Expressions-with-variables/274504: After you show me how to solve this question I can help my son complete the rest. Luke has $5 more than sam. Together they have$73. How much does each have? 1 solutions Answer 200306 by jim_thompson5910(28536)   on 2010-02-26 00:25:45 (Show Source): You can put this solution on YOUR website!Let x = amount that luke has and y = amount that sam has So "Luke has $5 more than sam" means that and "Together they have$73" tells us that Start with the given equation. Plug in 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. So this means that sam has $34 Go back to the first equation Plug in Add So this means that Luke has$39
Polynomials-and-rational-expressions/274509: I can't find two numbers that multiplied give me negative 270 and added give me negative 49
1 solutions

Answer 200304 by jim_thompson5910(28536)   on 2010-02-26 00:20:11 (Show Source):
You can put this solution on YOUR website!
Let's list all of the factors of (the product needed).

Factors of :
1,2,3,5,6,9,10,15,18,27,30,45,54,90,135,270
-1,-2,-3,-5,-6,-9,-10,-15,-18,-27,-30,-45,-54,-90,-135,-270

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

These factors pair up and multiply to .
1*(-270) = -270
2*(-135) = -270
3*(-90) = -270
5*(-54) = -270
6*(-45) = -270
9*(-30) = -270
10*(-27) = -270
15*(-18) = -270
(-1)*(270) = -270
(-2)*(135) = -270
(-3)*(90) = -270
(-5)*(54) = -270
(-6)*(45) = -270
(-9)*(30) = -270
(-10)*(27) = -270
(-15)*(18) = -270

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

First NumberSecond NumberSum
1-2701+(-270)=-269
2-1352+(-135)=-133
3-903+(-90)=-87
5-545+(-54)=-49
6-456+(-45)=-39
9-309+(-30)=-21
10-2710+(-27)=-17
15-1815+(-18)=-3
-1270-1+270=269
-2135-2+135=133
-390-3+90=87
-554-5+54=49
-645-6+45=39
-930-9+30=21
-1027-10+27=17
-1518-15+18=3

From the table, we can see that the two numbers and add to . Since each pair multiplies to -270, this means that 5 and -54 also multiply to -270.

So the two numbers 5 and -54 both add to -49 and multiply to -270.

 Inequalities/274500: How do I graph this inequality and what is the solution? y < 4x - 3 x - y _>_ -12 Note: _>_ Means that the sign is underline, meaning "Or equal to." How would I graph this inequality and what is the estimated point of solution? I already know that one line produces as true while the other produces a false answer when I substitute it with (0, 0). This leaves me stumped on where to shade, since it is an inequality. I also need to include a point of solution, and this inequality doesn't appear to have any... Thank you for any help. 1 solutions Answer 200302 by jim_thompson5910(28536)   on 2010-02-26 00:16:26 (Show Source):
Matrices-and-determiminant/274498: Evaluate the determinant:
|3 3 4|
|6 1 2|
|3 2 2|
1 solutions

Answer 200301 by jim_thompson5910(28536)   on 2010-02-26 00:06:00 (Show Source):
You can put this solution on YOUR website!
 Solved by pluggable solver: Finding the Determinant of a 3x3 Matrix If you have the general 3x3 matrix:the determinant is: Which further breaks down to:Note: , and are determinants themselves. If you need help finding the determinant of 2x2 matrices (which is required to find the determinant of 3x3 matrices), check out this solver--------------------------------------------------------------From the matrix , we can see that , , , , , , , , and Start with the general 3x3 determinant. Plug in the given values (see above) Multiply Subtract Multiply Combine like terms.======================================================================Answer:So , which means that the determinant of the matrix is

 Polynomials-and-rational-expressions/274372: Thanks in advance for your help... Here's my problem- The product of two consecutive positive odd integers is 38 less than the square of the greater integer. Find the integers. Now I know that product means mulitiplication so I thought I set the problem up as such: x(x+2)= (x+2)^2-38. Would this be correct?1 solutions Answer 200250 by jim_thompson5910(28536)   on 2010-02-25 19:37:47 (Show Source): You can put this solution on YOUR website!You have the right equation. Now let's solve for 'x'. Start with the given equation. Distribute FOIL Subtract from both sides. Combine like terms on the right side. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the solution is So the two numbers are 17 and 19.
Polynomials-and-rational-expressions/274329: x^2+10x-128. I've had this problem a few times now (I'm a tutor myself) and it asks to solve for x. I just cannot factor this. Please help me refresh my algebra skills and help me find a way to solve for x on this one. Thank you.
1 solutions

Answer 200222 by jim_thompson5910(28536)   on 2010-02-25 18:19:03 (Show Source):
You can put this solution on YOUR website!
For more factoring help, check out this quadratic formula solver.

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)
In order to factor , first multiply the leading coefficient 1 and the last term -128 to get -128. Now we need to ask ourselves: What two numbers multiply to -128 and add to 10? Lets find out by listing all of the possible factors of -128

Factors:

1,2,4,8,16,32,64,128,

-1,-2,-4,-8,-16,-32,-64,-128, List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to -128.

(-1)*(128)=-128

(-2)*(64)=-128

(-4)*(32)=-128

(-8)*(16)=-128

Now which of these pairs add to 10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 10

||||||||
 First Number | Second Number | Sum 1 | -128 | 1+(-128)=-127 2 | -64 | 2+(-64)=-62 4 | -32 | 4+(-32)=-28 8 | -16 | 8+(-16)=-8 -1 | 128 | (-1)+128=127 -2 | 64 | (-2)+64=62 -4 | 32 | (-4)+32=28 -8 | 16 | (-8)+16=8

None of these factors add to 10. So the quadratic cannot be factored.

So you must use the quadratic formula

So let's use the quadratic formula to solve

For more help with the quadratic formula, check out this quadratic formula solver.

 Solved by pluggable solver: Quadratic Formula Let's use the quadratic formula to solve for x: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=1, b=10, and c=-128 Square 10 to get 100 Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 1 to get 2 So now the expression breaks down into two parts or Now break up the fraction or Simplify or So the solutions are: or

Quadratic_Equations/274322: The roots of the equation 2x squared-10x+8=0 represent the dimension of the rectangle. what is the area of the rectangle
1 solutions

Answer 200221 by jim_thompson5910(28536)   on 2010-02-25 18:15:46 (Show Source):
You can put this solution on YOUR website!
For more help, check out this quadratic formula solver.

 Solved by pluggable solver: Quadratic Formula Let's use the quadratic formula to solve for x: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=2, b=-10, and c=8 Negate -10 to get 10 Square -10 to get 100 (note: remember when you square -10, you must square the negative as well. This is because .) Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 2 to get 4 So now the expression breaks down into two parts or Lets look at the first part: Add the terms in the numerator Divide So one answer is Now lets look at the second part: Subtract the terms in the numerator Divide So another answer is So our solutions are: or

Since the roots are or , this means that the length is 4 units and the width is 1 unit. So the area is simply A=4*1=4 square units.

 expressions/274325: I need help on try to figure out how to evaluate an expression 48 over 2x + 41 solutions Answer 200219 by jim_thompson5910(28536)   on 2010-02-25 18:13:39 (Show Source): You can put this solution on YOUR website!Unless you've been given a value of x, you can only simplify: . So where .
 Polynomials-and-rational-expressions/274297: Hi. I'm trying to solve 2xy+5x-2y-5 over 3xy+4x-3y-4. Thank you!1 solutions Answer 200197 by jim_thompson5910(28536)   on 2010-02-25 17:13:49 (Show Source): You can put this solution on YOUR website!The key to this problem is factoring Let's factor Start with the given expression. Group the terms. Factor out the GCF 'x' from the first group. Factor out the GCF -1 from the second group. Factor out the GCF from the entire expression. So factors to -------------------------------- Now let's factor Start with the given expression. Group the terms. Factor out the GCF 'x' from the first group. Factor out the GCF -1 from the second group. Factor out the GCF from the entire expression. So factors to ------------------------------------------------ Now that we have the factorizations, we can now simplify: Start with the given expression. Factor the numerator (see the first factorization above) Factor the denominator (see the second factorization above) Highlight the common terms. Cancel out the common terms. Simplify. So simplifies to In other words, Note: One thing to point out is that the value of 'x' does not affect the final outcome of the expression since the 'x' terms cancel out. This isn't so obvious when you look at the original expression, but it becomes clear when we reach the last step.
 Polynomials-and-rational-expressions/274285: please help solve this problem 11/2x + 4/7x I know first you find the LCD which is 14x but i don't know what you do after this 11/2x *14 + 4/7x *141 solutions Answer 200188 by jim_thompson5910(28536)   on 2010-02-25 16:50:40 (Show Source): You can put this solution on YOUR website! Start with the given expression. Multiply both the numerator and denominator of the first fraction by (to get the denominator equal to the LCD) Multiply both the numerator and denominator of the second fraction by (to get the denominator equal to the LCD) Multiply. Combine the fractions (this is now possible since the denominators are now equal). Add So where
 Linear-equations/273991: If a line with equation 3x -ky= 4 has a slope 5/6, find the value of k? Help?1 solutions Answer 200033 by jim_thompson5910(28536)   on 2010-02-24 21:50:45 (Show Source): You can put this solution on YOUR website!Solve for 'y' to get . Notice that the slope is . Because the given slope is , this means that . Cross multiply to get . Finally, solve for k to get
 Linear-equations/273995: Put the equation 2x - 3y = 15 into slope-intercept form. Think you get two answers...right1 solutions Answer 200032 by jim_thompson5910(28536)   on 2010-02-24 21:48:07 (Show Source): You can put this solution on YOUR website! 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
 Functions/273994: Evaluate f(x)= 4x^2 - 2 for f(1/2) 1 solutions Answer 200031 by jim_thompson5910(28536)   on 2010-02-24 21:47:27 (Show Source): You can put this solution on YOUR website! Start with the given equation. Plug in . Square to get . Multiply and to get . Combine like terms.
 Exponents/273983: My daughter is very confused by this problem: Is there any difference between -(3)(squared) and -3 (squared)?1 solutions Answer 200028 by jim_thompson5910(28536)   on 2010-02-24 21:40:33 (Show Source): You can put this solution on YOUR website!If you're talking about and , then the answer is no. There is no difference between and since and However, if you're talking about and , then there is a difference since and
 Functions/273851: I am completely stumped on this question. Given (f+g)(x)=10-3x and (f-g)(x)=5x-14, find f(x) and g(x). Thanks so much.1 solutions Answer 200001 by jim_thompson5910(28536)   on 2010-02-24 17:52:33 (Show Source): You can put this solution on YOUR website!Since , this means that . Solve for 'f(x)' to get Also, because , we know that Now plug into to get Combine like terms to get From here, simply solve for 'g(x)' (think of g(x) as any other variable). Once you have g(x), use that to find f(x).
1 solutions

Answer 199992 by jim_thompson5910(28536)   on 2010-02-24 17:12:14 (Show Source):
You can put this solution on YOUR website!
For more help, check out this quadratic formula solver.

 Solved by pluggable solver: Quadratic Formula Let's use the quadratic formula to solve for x: Starting with the general quadratic the general solution using the quadratic equation is: So lets solve ( notice , , and ) Plug in a=4, b=-15, and c=9 Negate -15 to get 15 Square -15 to get 225 (note: remember when you square -15, you must square the negative as well. This is because .) Multiply to get Combine like terms in the radicand (everything under the square root) Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Multiply 2 and 4 to get 8 So now the expression breaks down into two parts or Lets look at the first part: Add the terms in the numerator Divide So one answer is Now lets look at the second part: Subtract the terms in the numerator Divide So another answer is