geno3141

The following works for all bases, including base 8:

log(x² - 16) - log(x - 4)

Since logs are subtracted when the problem is a division problem:

=  log( (x² - 16) / (x - 4) )

=  log( (x - 4)(x + 4) / (x - 4) )

=  log( x + 4)

The series 2 + 4 + 6 + 8 (etc) is an arithmetic series and you want its sum.

The formula for an arithmetic series is:  Sum  = N(F + L)/2

where     N = number of terms     F = first term     L = last term

Each sum starts with 2, so F = 2.

The value for N is x.

Also, the last term is x, so L = x

The sum then is:  y  =  x(2 + x)/2

Let's see if this works:

x = 2:  y  =  2(2 + 2)/2  = 4                            <---   Each y value is exactly 2 times too big.

x = 4:  y  =  4(2 + 4)/2  =  12

x = 6:  y  =  6(2 + 6)/2  =  24

...

x = 16:  y  =  16(2 + 16)/2  =  144

---> If the formula were corect it would be:  y  =  x(2 + x)/2

        But the formula gives answers which are 2 times too big, so  we have to divide that formula by 2:

--->  y  = x(2 + x)/4     <---

x + y  =  27    --->       (x)     + y  =  27

x  =  2y + 3    --->   (2y + 3) + y  =  27

                                      3y + 3  =  27

                                            3y  =  24

                                              y  =  8

x + y  =  27   --->   x + 8  =  27     --->  x  =  19

-2x - 3y - 3z  =  -3

 3x + y + 2z  =  -2

 3x - 3y +  z  =  -2

Usually, you look for a single term (a variable that has a coefficient of 1); there are two, y in the second equation and z in the third equation.

I'm going to work with the y in the second equation and combine the first eq. and the second eq.:

Leave the first equation alone:  -2x - 3y - 3z  =  -3

Multiply the second eq. by 3:    9x + 3y + 6z  =  -6

Add down the columns:                   7x + 3z  =  -9

Now, work with the second eq. and the third eq.:

Multiply the second eq. by 3:     9x + 3y + 6z  =  -6

Leave the third equation alone:   3x - 3y +  z  =  -2

Add down the columns:                  12x + 7z  =  -8

Combine these two equations together:  7x + 3z  =  -9   --->  multiply by 7  --->   49x + 21y  =  -63

                                                        12x + 7z  =  -8   --->  multiply by -3 --->  -36x - 21y  =  24

                                                                                   Add down the columns:          13x  =  -39

                                                                                                                                  x  =  -3

Substitute x = -3 into the equation 7x + 3z  =  -9   --->   7(-3) + 3z  =  -9

                                                                                     -21 + 3z  =  -9

                                                                                              3z  =  12

                                                                                                z  =  4

Substitute x = -3 and z = 4 into  3x + y + 2z  =  -2   --->   3(-3) + y + 2(4)  =  -2

                                                                                             -9 + y + 8   =  -2

                                                                                                            y  =  -1

The result must be in the form:  y  =  mx + b

Use the distributive property on:  (-2/5)(x + 2)

                                       --->    (-2/5)(x) + (-2/5)(2)

                                       --->    (-2/5)x - 4/5

y + 1  =  (-2/5)(x + 2)

becomes:

y + 1  =  (-2/5)x - 4/5

Subtract 1 from both sides:

y  =  (-2/5)x - 4/5 - 1

y  =  (-2/5)x - 9/5

The slope is  -2/5  and the y-intercept is  -9/5.

What number times +3 will give you -6?

What number times -6 will give you +12?

What number times +12 will give you -24?

Can you now find the next number?

A got gold.

C got bronze.

D got silver.

7% = 0.07          "of"  means  "times"

you need to find the answer to:  0.07 x $6500.00

In direct variation, as one variable increases, so does the other.

For example, the more you eat, the more you weigh; the more items you buy, the more you have to pay.

First, get all the variable terms to one side; remove the smaller variable;

remove -7n by adding 7n to both sides:

1 - 7n  =  8 + 2n

1 - 7n + 7n  =  8 + 2n + 7n

1  =  8 + 9n

Remove the 8 by subtracting 8 from both sides:

1 - 8  =  8 - 8 + 9n

-7  =  9n

Divide both sides by 9:
-7/9  =  n