geno3141

Check the second equation:  4x-6y+16; should it be 4x - 6y = 16?

If it is:   3x + 6y = -9                                                      4x - 6y  =  16

                 6y  =  -3x - 9                                                      -6y  =  -4x + 16

                    y  =  (-3/6)x - 9/6                                                   y  =  (-4/-6)x + 16/-6

                slope = -1/2                                                       slope = 2/3

        y-intercept =  -3/2                                                   y-intercept = -8/3

Start at point (0,-3/2) on the y-axis                               Start at point (0,-8/3) on the y-axis   

go over 2 and down 3 (place a dot)                                 go over 3 and up 2 (place a dot)

repeat; draw a line.                                                      repeat; draw a line.

If the lines are parallel (slopes are equal and y-intercepts are not equal), they have no solutions.

If the lines are coincident (one in exactly the same place as the other) (slopes are equal and y-interceps are also equal), they have an infinite number of solutions, all the points that they have in common.

If the lines intersect (slopes are not equal), they have one solution, the point where they intersect.

The number of solutions is the number of points in which the two lines cross.

If the problem is 32 ^ (2/5) then

-- the 2 means square the number

-- the 5 means find the fifth root

You can do these in either order:

  32 squared is 1024; the fifth root of 1024 is 4

  the fifth root of 32 is 2; 2 squared is 4

To find the square root on a calculator, enter:  ^ ( 1 / 5)   <-- you must use parentheses

Find 25% of $72.65; round your answer to the nearest penny.

Subtract this answer from the original price. This is the sale price. 

Check the first equation:  3x=5y=-5; should it be 3x - 5y = -5?

If it is:   3x - 5y = -5                                                      6x - 10 y  =  -10

                 -5y  =  -3x - 5                                                     -10y  =  -6x - 10

                    y  =  (3/5)x + 1                                                   y  =  (3/5)x + 1

                slope = 3/5                                                       slope = 3/5

        y-intercept =  1                                                   y-intercept = 1

Start at point (0,1) on the y-axis                                   Start at point (0,1) on the y-axis   

go over 5 and up 3 (place a dot)                                   go over 5 and up 3 (place a dot)

repeat; draw a line.                                                      repeat; draw a line.

If the lines are parallel (slopes are equal and y-intercepts are not equal), they have no solutions.

If the lines are coincident (one in exactly the same place as the other) (slopes are equal and y-interceps are also equal), they have an infinite number of solutions, all the points that they have in common.

If the lines intersect (slopes are not equal), they have one solution, the point where they intersect.

The number of solutions is the number of points in which the two lines cross.

30 hours x $25.00 per hour  =  $750.00

The markup of 30% on $4500.00 is  .30 x 4500  =  $1350.00

The total made by the designer is  $750.00 +  $1350.00  =  $2100.00

Total cost to the employer: $2100.00 + $4500.00  =  $6600.00

In 90 ml that is 75% acid, the amount of acid is 90 ml x .75  =  67.5 ml of acid.

A)  67.5 ml + 35 ml  =  102.5 ml.

B) The total amount in the mixture is 90 ml + 35 ml  =  125 ml.

     Of this, 102.5 ml is acid. To find the percentage:  102.5 / 125  =  82%.

1+ 1/3  = 4/3

4(2/3 - 4/3)  =  -(-1/9 + x)

4(-2/3)  =  1/9 - x

-8/3  =  1/9 - x

-8/3 - 1/9  =  -x

-24/9 - 1/9  =  -x

-25/9  =  -x

x  =  25/9  

Written as unreduced radicals and arranged in order, the problem is:

1/(√16 + √14) + 1/(√14 + √12)+1/(√12 + √10)+ 1/((√10 + √8) * 1/(√8 + √6) + 1/(√6 + √14)

Now multiply each fraction by (its conjugate divided by its conjugate):

1/(√16 + √14)·(√16 - √14)/(√16 - √14) =  (√16 - √14)/(16 - 14) = (√16 - √14)/2

1/(√14 + √12)·(√14 - √12)/(√14 - √12) =  (√14 - √12)/(14 - 12) = (√14 - √12)/2

1/(√12 + √10)·(√12 - √10)/(√12 - √10) =  (√12 - √10)/(12 - 10) = (√12 - √10)/2

...

1/(√6 + √4)·(√6 - √4)/(√6 - √4) =  (√6 - √4)/(6 - 4) = (√6 - √4)/2

Note that all the denominators are 2, so the numerators can be added.

Now note that when you add all the numerators together, all the intermediate terms cancel and the only terms left in the numerator is the first and the last:  √16 - √4  = 4 - 2  = 2

So this sumes to  2/2  = 1  <-----  A quite surprising result! And, a super neat problem!

Oh, Phil is so right; I forgot a few at the start!

Sorry!

If you have access to a calculator such as TI-84+, use it.

You can also you a spreadsheet, such as Excel.

You can also do it by hand, using a formula such as found at