Hi,

2^5/2 - 2^3/2 = 

 

Hi,

2x - y = 4   



3x + y - z = 10

y + z = 3

   3x + 2y = 13

   3x + 2(2x-4) = 13

        7x = 21,   x = 3  and y= 2  and z = 1



Using Cramer's rule



 
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, 2, 1)

  

  

  

  

  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.

  

  

  

 

Hi,

f(x)=8x+9, g(x)=x^2+7x 

(fg)(x) = (8x+9)(x^2+7x) 

(fg)(2) =  

 

Hi,

P(x) = x4 + 9x3 - 9x + 2

P(-2) = 16 -72 +18 + 2 = -36

 

Hi,

B

Might recommend downloading the FREE graph software at:

http://www.padowan.dk.com

 

Hi,

y = -x3 + 5x2 - 11x + 55   ||x = 5 one root  -125 + 125 - 55 + 55 = 0

using Synthetic Divsion

5	-1	5	-11	55

 	 	-5	0	-55

 	-1	0	-11	0

results in:

 -x^2 - 11 = 0

 x^2 + 11 = 0

   x^2 = -11

   x = ± i√11

roots are:  i√11,-i√11,5

 

Hi,

y = x^2 + 2x + 2   Pt(0,2) on the curve

y =  x^2 + 2x +1 -1 + 2  

y = (x + 1)^2 + 1  parabola, V(-1,1) line of symmetry x = -1

Graph, NO real solutions

 

Hi,

 college math club consists of 16 men and 20 women.  total

a. in how many different ways can a committee of 5 members be chosen?  

 b. in how many different ways can a committee of 5 be chosen 

and have it consist of 2 men and 3 women? 

 c. it the 5 committee members are randomly selected, 

what is the probability that the committee will consist of 2 men and 3 women

  

 

Hi,

2 3/4   this is a mixed fraction, 11/4 as an improper fraction

2(3/4)  the parenthesis indicates multiplication

 

Hi,

(3 + i) - (2 - 2i) = 3-2 + i + 2i = 1 + 3i

 

Hi,

if your traveling 6 mph how long will it take you to go 197 feet?

  ||| Cross Multiplying to solve

    x = 197/31680 = .006 hr

 

Hi,

-2x2 - 5x + 5 = 0   ||Standard form:  ax^2 + bx + c = 0

 

Hi,  This was an example today

y = x^2 +2x - 8  ||factoring (x+4)(x-2) = 0 , x = -4  0r  x = 2 are the solutions

,  x = -1 ± 3

y = x^2 +2x +1 - 1 - 8  Completing the Square

y = (x+1)^2 - 9  V(-1,-9) and line of symmetry is x = -1

and when y  = 0 ⇒   (x+1)^2 = 9, x = -1 ± 3, Pt(2,0) & (-4,0) are the x-intercepts

 

Hi,  Previously Posted

y = x^2 +2x - 8 

y = x^2 +2x +1 - 1 - 8 

y = (x+1)^2 - 9  V(-1,-9) and line of symmetry is x = -1

and when y  = 0 ⇒   (x+1)^2 = 9, x = -1 ± 3, Pt(2,0) & (-4,0) are the x-intercepts

 

Hi,

In a geometric proof, statements following the given are a means to the end

 Ask yourself, starting with the givens, how do I reach the final conclusion..

(final conclusion being... what is being 'proved' to be true)



What steps must be taken (what statements used...)

 

Hi,

what are four consecutive even whole numbers whose sum is less than 35.

 x + (x+2) + (x+4) + (x+6) < 35

       4x < 23

        x < 5.75

4, 6, 8, 10

and..

 4 + 6 + 8 + 10 = 28 < 35

 

Hi,  Previously Posted

y = x^2 +2x - 8 

y = x^2 +2x +1 - 1 - 8 

y = (x+1)^2 - 9  V(-1,-9) and line of symmetry is x = -1

and when y  = 0 ⇒   (x+1)^2 = 9, x = -1 ± 3, Pt(2,0) & (-4,0) are the x-intercepts

 

Hi,

y = x^2 +2x - 8 

y = x^2 +2x +1 - 1 - 8 

y = (x+1)^2 - 9  V(-1,-9) and line of symmetry is x = -1

and when y  = 0 ⇒   (x+1)^2 = 9, x = -1 ± 3, Pt(2,0) & (-4,0) are the x-intercepts

 

Hi,

Number of Dollars: D

Number of quarters:  3D + 5 

CENTS makes sense

 100D + 25(3D + 5) = 7650¢

        175D = 7650 - 125

        175D = 7525

           D = 43, the number of dollar coins.  There are 134 quarters. 

and..

 

 

Hi,

 half life of radioactive substance is 10,000 years



        ln(.5)/10,000 = r = -.000069 , the rate of decay

 

Hi,

factoring the numerator and denominator...

 

Hi,

two angles  and are complementary. 

one of the angles is 8 degrees more than half the other.

Translate into an equation: 

    |||Multiplying thru by 2 so as all denominators = 1

 2x = (90°-x) + 16

  3x = 106

   x = 35 1/3°   and it complement is 54 2/3 °

and...

  

 

Hi,

A store decreased its price on a computer by $112    to $478. 

What was the original price?  $478 + $112 =  $590

 

Hi,

Two of the boxes have something special in them, while the third one is empty.

select two boxes ...

P(selecting boxes/prizes) = 

How do you maximize the probability of winning...leave that up to you :)

 

Hi,

a vehicle traveling at the posted speed of 25 mph  0r 

will take no less than 20 seconds to move from point a to point b,

  = .14mi

 a nd b are .14mi  apart.  (rounded to the nearest tenth)

 

Hi,

Annual profit in thousands of dollars is given by the function,

 P(x) = -.1x2 + 50x - 300, where x is the number of items sold, x ≥ 0.

 the -300 would be the adjustments to the annual profits due to 'fixed cost'



that is to say if NO items were sold..there would be a loss of $300,000

and...solving -1x^2 + 50x - 300 = 0 would tell us how many need to be sold

to break even.

 

Hi,

 -3y ≥ 6x - 3

   y ≤ -3x + 1

1st graph y = -3x + 1   

(0,1) and (1,-2) on this line, plot pts, connect with a line 

2nd 

Shaded area 'BELOW' (and including the line) is the graph of y ≤ -3x + 1 

will let you do that shading

 

Hi,

 y = 2|x|   (0,0), (1,2) and (-1,2) on this graph

 

Hi,

  s + w = 2280mi/4hr = 570mph

  s - w = 2280mi/4.75hr = 480mph

    2s = 1050mph

     s = 525mph, speed of the plane in still air.  w = 45mph

 

Hi,

 

Hi,

Line:

y=5/7x-12   |||Multiplying thru by 7 so as all denominators = 1

7y = 5x - 84

 5x - 7y = 84  ||The Line in standard form