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x + y = 5/4

x = .5*y

this makes:

y = 2*x

substituting in original equation gets:

x + 2*x = 5/4

combine like terms to get:

3*x = 5/4

divide both sides by 3 to get:

x = 5/12

y = 2x so:

y = 5/6

substitute in original equation of x + y = 5/12 to get:

5/12 + 5/6 = 5/12 + 10/12 = 15/12 = 5/4

since x = 5/12 and y = 5/6, the product of the two numbers is:

5/12 * 5/6 = 25/72

that would be selection A.

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d = distance the boat travelled.

boat travels for 2 hours against a 4 km/h current.

the whole trip took 5.5 hours which means the boat drifted back for 3.5 hours.

distance the boat travels against the current is equal to:

d = (b-4)*2

distance the boat drifts with the current is equal to:

d = (4)*3.5 = 14 miles

since the distance getting to the point where the boat broke down and getting back is the same distance, the formula for the boat getting to that point becomes:

14 = (b-4)*2

remove parentheses to get:

14 = 2b-8

add 8 to both sides to get:

2b = 22

divide both sides by 2 to get:

b = 11

Speed of the boat should be 11 miles per hour.

plug into the original equations to confirm this value is good.

first equation is boat getting to the breakdown point:

(b-4)*2 = d

replace d with 14 and b with 11 to get:

(11-4)*2 = 14 becomes: 7*2 = 14 becomes 14 = 14 confirming equation 1 is true.

second equation is boat drifting back to the starting point:

d = 4*3.5 becomes:

14 = 4*3.5 becomes 14 = 14 confirming e4quation 2 is true.

values look good.

speed of the boat in still water before breaking down is 11 miles per hour.

basic formula used is:

rate * time = distance

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|3| = 3
|-8| = 8
3 + 8 = 11

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The definition of logarithms states:

log(b,x) = y if and only if b^y = x

and:

b^y = x if and only if log(b,x) = y

Since 1/2 = .5, your formula becomes:

64^(.5) = 8

If we let:

64 = b
.5 = y
8 = x

Then, the definition of logarithms applied to your formula states:

64^(.5) = 8 if and only if log(64,8) = .5.

Algebra.com formula generator make it look like this:

if and only if

If you put 64^(.5) in your calculator, you will get 8.

If you put log(8)/log(64) in your calculator, you will get .5

log(8)/log(64) is equivalent to which means log of 8 to the base of 64.

The procedure uses the change of base of logarithm formula of:



This formula states that if you have a logarithm of a number to the base of b, and you want to convert it to a logarithm of the same number to the base of c, you take the logarithm of the number to the base of c and you divide it by the logarithm of the base of b to the base of c.

In your problem, we converted your exponential equation to the logarithmic form to get:

log of 8 to the base of 64 = .5

to prove that was true using the calculator, we had to convert the logarithm to the base of 64 to a logarithm to the base of 10 because the LOG function of the calculator works to the base of 10.

The formula we used was the change of base formula.

We took the logarithm of 8 to the base of 64 and converted it to the logarithm of 8 to the base as shown below:

log of 8 to the base of 64 = log of 8 to the base of 10 divided by the log of 64 to the base of 10.

this became:

making the result = .5

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If you mean a quadratic equation, the formula is:

x =

standard form of the quadratic equation is:

where:

a is the coefficient of the term.

b is the coefficient of the x term.

c is the constant.

You solve for x.

The y value is 0 because you set the equation equal to 0.

Your roots are:

(x,y) = (x,0) where you replace x with the values you calculated for x.

The roots can be real or imaginary.

If the roots are real, the graph of the equation crosses the x-axis at y = 0.

If the roots are not real (contain an imaginary part which is a negative square root) then the graph of the equation will not cross the x-axis.

The quadratic formula works on all quadratic equations whether the roots are real or imaginary.

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This is what I came up with.

ST = total number of seniors
FT = total number of seniors playing football
BT = total number of seniors playing baseball
LT = total number of seniors playing lacrosse
F = number of seniors playing football only.
B = number of seniors playing baseball only.
L = number of seniors playing lacrosse only.
FL = number of seniors playing football and lacrosse only.
FB = number of seniors playing football and baseball only.
BL = number of seniors playing baseball and lacrosse only.
FBL = number of seniors playing football and baseball and lacrosse.

We know that all seniors play on at least one sport.

The total number of seniors can be accounted for by the equation:

ST = F + B + L + FB + FL + BL + FBL

What we know so far:

ST = 85
FT = 74
BT = 26
LT = ???
F = ???
B = ???
L = ???
FB = 18
FL = 17
BL = 13
FBL = 0

We know that FT = 74

Since FT = F + FB + FL + FBL, we get 73 = F + 18 + 17 + 0 = F + 35.

This makes F = 39 which means that 39 seniors that play football only.

We assign 39 people to F.

Our tally is now:

ST = 85
FT = 74
BT = 26
LT = ???
F = 39 (removed ???)
B = ???
L = ???
FB = 18
FL = 17
BL = 13
FBL = 0

We know that BT = 26

We also know that BT = B + FB + BL + FBL

Since FB = 18 and BL = 13, and BT = 26 and FBL = 0, then the equation becomes:

26 = B + 18 + 13 + 0 which becomes:

26 = B + 31

We have 5 more people than total people on the team.

Since the team is composed of FB and BL only, and we are assuming that the numbers shown are correct, then we must have people who play Baseball AND Football.

These would be classified as FBL because they play on all 3 sports.

This means we have 5 people who being counted twice.

We subtract 5 people from FB and 5 people from BL and we add 5 people to FBL to get:

BT = B + FB + BL + FLB becomes:

26 = B + 13 + 8 + 5 because our category of FBL just became 5 from 0.

Baseball equation now becomes:

26 = B + 13 + 8 + 5 = B + 26

This becomes 26 = B + 26 which makes B = 0 which now accounts for all of the members of the baseball team.

The tally for Baseball is now:
B = 0
FB = 13 (down from 18)
BL = 8 (down from 13)
FBL = 5
BT = 0 + 13 + 8 + 5 = 26 (checks out ok because BT was given as 26).

Our overall tally now becomes:

ST = 85
FT = 74
BT = 26
LT = ???
F = 39 (removed ???)
B = 0 (removed ???)
L = ???
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5 (up from 0)

The equation for FT has to change because we dropped 5 FB and added 5 FBL.

The tally for Football is now:
F = 39
FB = 13 (down from 18)
FL = 17
FBL = 5
FT = 39 + 13 + 17 + 5 = 74 (checks out ok)

This does change the overall tally any further. It just corrects the account for Football.

We check the tally for total seniors again to see if anything is missing.
F = 39
B = 0
L = ???
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5
ST = 39 + 0 + L 13 17 + 8 + 5 = L + 82

We are missing 3 seniors.

The only category that is not completed is the number of seniors who only play lacrosse, and since all seniors must be playing at least one sport, these 3 seniors must be playing lacrosse.

We assign them to lacrosse by making L = 3.

Out tally for seniors now becomes:
F = 39
B = 0
L = 3
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5
ST = 39 + 0 + 3 + 13 + 17 + 8 + 5 = 85 (checks out ok)

The tally for the lacrosse team now becomes:
L = 3
FL = 17
BL = 8 (down from 13)
FLB = 5
LT = 3 + 17 + 8 + 5 = 33

LT is now equal to 33.

Our overall tally now becomes:

ST = 85
FT = 74
BT = 26
LT = 33 (removed ???)
F = 39 (removed ???)
B = 0 (removed ???)
L = 3 (removed ???)
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5 (up from 0)

Our tally for seniors now becomes:
F = 39
B = 0
L = 3 (up from 0)
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5 (up from 0)

ST = 39 + 3 + 13 + 17 = 8 = 5 = 85 (checks out ok).

All the seniors are accounted for.

The number of seniors who play on all three sports is 5.

The only part that is questionable is the statement "Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team."

What that means exactly is a mystery to me. I have computed the number of seniors who play all three sports. The number has to be 5 in order to make all other numbers check out. At least twice this number are members of the lacrosse team so I would have to say that the statement is satisfied.

The total accounting for all the seniors is now:

ST = 85       (given)
FT = 74       (given)
BT = 26       (given) 
LT = 33       (calculated)
F = 39        (calculated
FB = 13       (calculated down 5 from 18 given)
FL = 17       (given)
FBL = 5       (calculated)
BL = 8        (calculated down 5 from 13 given)
B = 0         (calculated)
L = 33        (calculated)

Tally for Football is now:
F = 39
FB = 13
FL = 17
FBL = 5
FT = 39 + 13 + 17 + 5 = 74 (checks out ok)

Tally for Baseball is now:
B = 0
FB = 13
BL = 8
FBL = 5
BT = 0 + 13 + 8 + 5 = 26 (checks out ok)
Tally for Lacross is now:
LT = L + FL + BL + FBL
L = 3
FL = 17
BL = 8
FBL = 5
LT = 3 + 17 + 8 + 5 = 33 (checks out ok)

Tally for Seniors is now.
F = 39
B = 0
L = 3
FB = 13
FL = 17
BL = 8
FBL = 5
ST = 39 + 0 + 3 + 13 + 17 + 8 + 5 = 85 (checks out ok)

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T = 5+3+4 = 12
This means the total number of marbles is equal to 5 red plus 3 green plus 4 blue which equals a total of 12.

G = 3

This means the number of green marbles you are starting out with is 3.

P(G) is now 3/12 = 1/4

P(G) means the probability that you will draw a green marble is equal to .25 right now.

We want:

P(G) = 3/4

We need to add X number of green marbles.

P(G) = 3/4 = (X+3) / (12 + X)

This means that we need to add x number of green marbles to the total. This will change both the numerator of the equation (number of green marbles) and the denominator of the equation (number of total marbles).

We need to solve for X.

(X + 3) / (X + 12) = 3/4

Multiply both sides by (X+12) to get:

(X+3 = 3/4 * (X + 12)

Simplify to get:

X + 3 = (3/4)*X + (3/4)*12

Simplify further to get:

X + 3 = (3/4)*X + 9

Subtract 3 from both sides of this equation to get:

X = (3/4)*X + 9 - 3

Subtract (3/4)*X from both sides of this equation to get:

X - (3/4)*X = 9 - 3

Combine like terms to get:

(1/4)*X = 6

Multiply both sides of the equation by 4 to get:

X = 6 * 4 = 24

Number of green marbles should be 24.

Add 24 green marbles to 3 and you get 27.

Add 24 to 12 and you get 36

27/36 = .75 = 3/4

the probability of getting a green marble now is .75 because 75% of the marbles out of the total number of marbles are green.

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5x - 6x = -17
3x - 4y + 5z = -1
2z = -6

looks like you have 3 equations that have to be solved simultaneously.

looks like your third equation has already solved for z so we can start there.

2z = -6 becomes z = -3 when you divided both sides of the equation by 2.

z = -3 is the value of one of your variables.

you are left with:

5x - 6x = -17
3x - 4y + 5z = -1

that first equation looks like it's all in x so we can solve for x from that equation.

5x - 6x = -17 becomes -x = -17 after you combine like terms.

multiply both sides of this equation by -1 to get:

x = 17

you now have two values out of the 3.

you have:

x = 17
z = -3

you have only one equation left to solve.

that is:

3x - 4y + 5z = -1

replace x with 17 and z with -3 in that equation to get:

3*(17) - 4*y + 5*(-3) = -1

this becomes:

51 - 4*y - 15 = -1

combine like terms to get:

36 - 4*y = -1

add 4*y to both sides of the equation and add 1 to both sides of the equation to get:

36 + 1 = 4*y

combine like terms to get:

37 = 4*y

divide both sides of the equation by 4 to get:

9.25 = y

this is the same as:

y = 9.25

the answer to your equation is:

x = 17
y = 9.25
z = -3

substitute in your original equations to confirm these answers are good.

5x - 6x = -17
3x - 4y + 5z = -1
2z = -6

these become:

5*(17) - 6*(17) = -17
3*(17) - 4*(9.25) + 5*(-3) = -1
2*(-3) = -6

simplify and combine like terms to get:

-17 = -17
51 - 37 - 15 = -1
-6 = -6

simplify further to get:

-17 = -17
-1 = -1
-6 = -6

all equations confirm so the answer is good.

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slope intercept form is y = mx + b where m is the slope and b is the y intercept.

for your equation, do the following:

3x - y - 2 = 0

add y to both sides of the equation to get:

3x - 2 = y

this is the same as:

y = 3x - 2

that's the slope intercept form of your equation.

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g. x/2+2x/3=5
x/2 + 2x/3 = 5
multiply both sides of the equation by 6 to get:
3x + 4x = 30
combine like terms to get:
7x = 30
divide both sides of the equation by 7 to get:
x = 30/7

substitute in original equation to get:

x/2 + 2x/3 = 5 becomes:

(30/7)/2 + (2*(30/7))/3 = 5

multiply both sides of the equation by 6 to get:

3*(30/7) + 2*(2*(30/7)) = 6*5

simplify to get:

90/7 + 120/7 = 30

combine like terms to get:

210/7 = 30

simplify to get:

30 = 30 confirming that x = 30/7 is good.

It's easiest to remove fractions by multiplying both sides of the equation by a number that will eliminate them.

Just be careful to multiply both sides of the equation by the same amount to preserve the equality.

One little slip messes the whole thing up.

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

t. 5x/6-(3-x)/8=(4x+3)/12

I think this is:

(5x/6) - (3-x)/8 = (4x+3)/12

multiply both sides of the equation by 12 to get:
10x - (12*(3-x))/8 = 4x+3

multiply both sides of the equation by 8 to get:

80x - 12*(3-x) = 8*(4x+3)

simplify by removing parentheses to get:

80x - 36 + 12x = 32x + 24

combine like terms to get:

92x - 36 = 32x + 24

subtract 32x from both sides of the equation to get:

92x - 32x - 36 = 24

add 36 to both sides of the equation to get:

92x - 32x = 24 + 36

combine like terms to get:

60x = 60

divide both sides of the equation by 60 to get:

x = 1

substitute in the original equation to get:

(5x/6) - (3-x)/8 = (4x+3)/12 becomes:

(5/6) - (2/8) = (7/12)

multiply both sides of the equation by 12 to get:

10 - (24/8) = 7

simplify to get:

10 - 3 = 7

combine like terms to get:

7 = 7 confirming the value of x = 1 is good.

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

s. 1/3(a+5)=7/2

multiply both sides of the equation by 3 to get:

a + 5 = 21/2

multiply both sides of the equation by 2 to get:

2a + 10 = 21

subtract 10 from both sides of the equation to get:

2a = 11

divide both sides of the equation by 2 to get:

a = 11/2

substitute in the original equation to get:

1/3(a+5)=7/2 becomes:

(1/3)*((11/2) + 5) = 7/2

Multiply both sides of the equation by 6 to get:

6*(1/3)*((11/2)+5) = 6*7/2

simplify to get:

2*((11/2+5) = 3*7

simplify further to get:

11 + 10 = 21

combine like terms to get:

21 = 21 confirming the value of a = 11/2 is good.

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

b. (3k-8)/14+5/7=(k+1)/2

I think this is:

((3k-8)/14) + (5/7) = (k+1)/2

assuming that it is, then the equation would be solved as follows:

multiply both sides of the equation by 14 to get:

(3k-8) + 10 = 7*(k+1)

simplify by removing parentheses to get:

3k-8+10 = 7k+7

combine like terms to get:

3k+2 = 7k+7

subtract 3k from both sides of the equation to get:

2 = 7k+7-3k

subtract 7 from both sides of the equation to get:

2-7 = 7k-3k

combine like terms to get:

-5 = 4k

this is the same as:

4k = -5

divide both sides of the equation by 4 to get:

k = -5/4

substitute in the original equation to get:

(3k-8)/14) + (5/7) = (k+1)/2 becomes:

(3*(-5/4) -8)/14 + (5/7) = ((-5/4)+1)/2

Simplify to get:

((-15/4)-8)/14 + (5/7) = (-1/4)/2

multiply both sides of the equation by 14 to get:

((-15/4) -8) + 10 = 7*(-1/4)

remove parentheses to get:

(-15/4) - 8 + 10 = (-7/4)

Multiply both sides of the equation by 4 to get:

-15 - 32 + 40 = -7

combine like terms to get:

-7 = -7 confirming that x = -5/4 is a good answer.

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

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350 * (1.06)^t = future value of this account.

at the end of 1 year, the account will have 350 * (1.06)^1 = 371

at the end of 2 years, the account will have 350 * (1.06)2 = 393.26

that comes out as 350 * 1.06 * 1.06

since 350 * 1.06 = 371 * 1.06 = 393.26, the interest earned each year is combined with the principal and then interest is earned on the sum.

at the end of 10 years, the account will have 350 * (1.06)^10 = 626.80

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x^3 - 3x = 0

you can factor out the x to get:

x * (x^2-3) = 0

Either x is 0 or (x^2-3) is 0

x = 0 is one of the solutions.

x^2 - 3 = 0

add 3 to each side to get:

x^2 = 3

take square root of each side to get:

x = +/- sqrt(3)

those are your solutions.

x = 0
x = sqrt(3)
x = -sqrt(3)

a graph of your equation looks like this:



+/- sqrt(3) is roughly +/- 1.7632 as shown on the graph.

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Interest rate per month is calculated as .08/12 + 1 = 1.0066666667

In the beginning you deposit 15,000

Interest is compounded monthly has been assumed though not stated.

Payments are made at the end of each month has been assumed though not stated.

The money invested is multiplied by (1 plus the interest rate) and then the payment is added.

Here's how it works.

x = (1 + i) = 1.0066666667

This is rounded here but not in the calculator.

All the numbers below are truncated to the nearest dollar but the complete number is stored in the calculator and used in the calculations.

you invest 15000.
at the end of the first month you have:
15000 * x = 15100 + 5000 = 20100
at the end of the second month you have:
20100 * x = 20234 + 5000 = 25234
at the end of the third month you have:
25234 * x = 25402 + 5000 = 30402
at the end of the fourth month you have:
30402 * x = 30604 + 5000 = 35604
at the end of the fifth month you have:
35604 * x = 35842 + 5000 = 40842

The complete number is:

40842.27423

That's $40,842.27

I confirmed using a financial calculator that this number is correct.

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x + (x + 2) = 58

2x+2 = 58

2x = 56

x = 28

28 + 30 = 58

house number are 28 and 30

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A square give you the most area for the same perimeter.

If the rectangle is composed of 4 squares than it should provide the maximum area because the overall area is equal to the sum of the individual areas.

consider:

x * x = x^2
x + x = 2x

Subtract 1 from the left x and add 1 to the right x and you get:

(x-1) * (x+1) = x^2 - 1
(x-1) + (x+1) = 2x

perimeter remains the same but the area is reduced by 1.

Subtract 2 from the left x and add 2 to the right x and you get:

(x-2) * (x+2) = x^2 - 4
(x-2) + (x+2) = 2x

Perimeter remains the same but the area is reduced by 4.

Square is the maximum area for the same perimeter.

She has 100 meters of fence.

She will need 4 squares.

That's a total of 16 sides.

4 of those sides are on the river bank so she can subtract 4 sides from the requirement to get 12 sides.

6 of those sides are inside the perimeter of the large rectangle so that 2 sides can be collapsed into 1. This reduces the number of sides required by another 3 making the total number of sides required equal to 9

100 meters of fence divided by 9 equals 11.111111111 meters per side.

The exact measurement for each side is (100/9).

With each side measuring (100/9), then the area of each of the squares will be (100/9)^2 = (10000/81)

4 * (10000/81) = (40000/81) = 493.8271605 square meters.

her fence will look like this:

                       ---- ---- ---- ----
                      |    |    |    |    |
                      |    |    |    |    |

That's 9 sides of (100/9) meters each consuming all of the 100 meters of fence.

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I'm no sure what you're doing, but if you want to move the point to the right 4 units and down 3 units, then the new point will be at (1,-8)

(x,y) = (-3,-5)

(x+4,y-3) = (-3+4,-5-3) = (1,-8)

Along the x-axis, going to the right is adding units and going to the left is subtracting units.

Along the y-axis, going up is adding units and going down is subtracting units.

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y = k*x

if y = 38 and x = 114, this becomes:

38 = k*114

k = 38/114

k = (1/3)

This is saying that y = (1/3) * x

38 = (1/3) * 114

38 = 38 confirming the value for k is good.

they are directly proportional which means that an increase in one yields an increase in the other by the stated proportion.

suppose y = 1

formula states that 1 = (1/3)*x which makes x = 3.

An increase of 1 in y will result in an increase of 3 in x.

2 = (1/3)*x makes x = 6

y increased by 1
x increased by 3

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ln(x) = 11

you want to find x?

In general,

ln(x) = y if and only if e^y = x

if y = 11, this means that:

e^11 = x

e^11 = 59874.14171 which means that:

x = 59874.14171

This means that:

ln(x) = 11 becomes:

ln(59874.14171) = 11 which becomes:

11 = 11 confirming the value of x is good.

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Formula for continuous compounding is:

FV = PA * e^(rt)

FV = future value
PA = present amount
e = scientific constant of 2.718281828.....
r = annual interest rate
t = time in years

In your equation,

FV = 2
PA = 1
e = 2.71828182818.....
r = r
t = 12

Equation becomes:

2 = 1 * e^(12r)

This becomes:

2 = e^(12r)

Take natural log of both sides (natural log is log to the base of e)

ln(2) = ln(e^(12r)

By laws of logarithms, this becomes:

ln(2) = 12r*ln(e)

ln(e) = 1

Equation becomes:

ln(2) = 12r

Divide both sides by 12 to get:

ln(2)/12 = r

Solve for r to get:

r = .057762265 * 100% = 5.7762265%

Using continuous compounding, an annual interest rate of 5.7762265% will double your money in 12 years.

Confirm by plugging into the original equation.

2 = 1 * e^(.057762265*12)

Simplify to get:

2 = 2 confirming the value of r is good.

You might try some of the LESSONS FROM ALGEBRA.COM if you have the time. There's a few in the financial area that might help you.

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A = rate of A = 1/12 of the job in 1 hour.
B = rate of B = 1/16 of the job in 1 hour.

A * (1/12) * 12 = 1 means A takes 12 hours to complete 1 job.

B * (1/16) * 16 = 1 means B takes 16 hous to complete 1 job.

A * 9 = (1/12)*9 = 9/12 means A completes 9/12 of the job in 9 hours.

1 - 9/12 = 3/12 means 3/12 of the job remains to be done.

B * x = 3/12 means B will complete 3/12 of the job in x hours.

x = (3/12) / B means divide both sides of the equation by B to solve for x.

x = (3/12) / (1/16) means dividing by B is the same as dividing by (1/16)

x = (3/12) * (16) means dividing by (1/16) is the same as multiplying by 16.

x = (3 * 16) / 12 means (3*16)/12 is equivalent to (3*16)/12.

x = 16 / 4 means simplify the expression.

x = 4 means solve for x to get the answer.

It will take B 4 more hours to complete the job.

That's a total of 9 + 4 = 13 hours.

9 * (1/12) + 4 * (1/16) = (9/12) + (1/4) = (3/4) + (1/4) = 1 means confirm the answer is good by carrying out the complete calculation with the solved value for x.













9/12J =

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She can do 6 steps one at a time only 1 way
She can do 4 steps one at a time and 1 step 2 at a time 5 ways
21111
12111
11211
11121
11112
She can do 2 steps one at a time and 2 steps 2 at a time 6 ways
1122
1212
1221
2112
2121
2211
She can do 3 steps 2 at a time only 1 way
222

Total number of ways equals 1 + 5 + 6 + 1 = 13

The formula is the number of possible combinations without taking order into account.

0 steps 2 at a time and 6 steps 1 at a time winds up being 6! / 0!6! = 1/1 = 1

1 step 2 at a time and 4 steps 1 at a time winds up being 5! / 1!4! = 5/1 = 5

2 steps 2 at a time and 2 steps 1 at a time winds up being 4! / 2!2! = 12/2 = 6

3 steps 3 at a time and 0 steps 1 at a time winds up being 3! / 3!0! = 1

The combination formula is (n!) / (x! * (n-x)!)

Each time she does 2 steps at a time, the total number of steps she has to take is reduced by 1.
6 steps one at a time and 0 steps 2 at a time becomes a total of 6 steps that she has to take.
4 steps one at a time and 1 step 2 at a time becomes a total of 5 steps that she has to take.
2 steps one at a time and 2 steps 2 at a time becomes a total of 4 steps that she has to take.
0 steps one at a time and 3 steps 2 at a time becomes a total of 3 steps that she has to take.

It's hard to see.

You have to work it out and then the pattern becomes apparent.

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the standard form of the equation for a straight line is:

ax + by = c

where:

a = 1
b = 0
c = 5

the equation becomes:

1x + 0y = 5

this reduces to x = 5

the value of x is always 5.

y can be any value because it is not dependent on the value of x since it is no longer a part of the equation.

the slope is equal to (y2-y1)/(x2-x1) which winds up being anything divided by 0 which makes it undefined.

this is the equation of a vertical line at x = 5.

the slope is undefined.

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Should be:

1 to 9 = 9 - 1 + 1 = 8 + 1 = 9 * 1 digit = 9
10 to 99 = 99 - 10 + 1 = 89 + 1 = 90 * 2 digits = 180
100 to 728 = 728 - 100 + 1 = 628 + 1 = 629 * 3 digits = 1887

total = 9 + 180 + 1887 = 2076

As far as I can tell.

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The perimeter of an isosceles triangle is equal to 2*s + b where s equals one of the equal sides of the triangle and b equals the base.

If the base is equal to one of the sides, then you have an equilateral triangle.

The perimeter of this triangle has to be greater than 21 units and is not a prime number.

Possible perimeters are therefore:

22,23,24,25,26,27,28,29,30,...

The length of each side of the triangle has to be a prime number.

I presume this includes the base of the triangle as well as the 2 equal sides.

Since 7*3 = 21, one of the sides of this triangle has to be greater than 7 at least.

We'll start with the next highest prime number which is 11.

If 11 is the base of the triangle, then the smallest side could be 7 which would make the total perimeter equal to 14 + 11 = 25.

25 is not a prime.

25 looks good.

If 11 was a side of the triangle, then the perimeter of the triangle could be 22 + 2 = 24

24 is not a prime.

24 looks good.

None of the other combinations produce a perimeter less than 24.

23 would be no good because it's a prime number.

The smallest possible perimeter that meets the requirements is 22 but no combinations I was able to find yielded a total of 22.

I'd go with 24 which is a combination of 2 times 11 plus 2.

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It would be 3.75 units to the right of the 0 point.

It would be three quarters of the way between the 3 and the 4.

Each integer would be divided into 4 parts.

from 3 to 4, you would have the measuyrements of:

3.00     3.25     3.50     3.75    4.00

If what is shown below is your number line, then 3.75 would be as shown.

 |- - - -|- - - -|- - - -|- - - -|- - - -|- - - -|- - - -|- - - -|- - -|-|- - - -|
-5      -4      -3      -2      -1       0       1       2       3     | 4       5
                                                                       |
                                                                    3.75     

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You plot some selected points for x and then calculate the y value and then plot the points on the graph and then draw a straight line through the points.

Example:

When x = 0, y = 0

When x = 10, y = 10

Plot (x,y) = (0,0) on the graph.

Plot (x,y) = (10,10) on the graph.

Draw a straight line between them.

There's your graph.

The point (x,y) is x units to the right or left of x = 0, and y units up or down from y = 0.

If x is negative, you move to the left.

If y is negative, you move down.

If x is positive, you move to the right.

If y is positive, you move up.

A graph of your equation looks like:

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area of a circle equals pi*r^2

radius of your two smaller circles are:

24 cm and 7 cm

area of your two smaller circles are:

pi*24^2 + pi*7^2

Sum of these areas is equal to:

pi*(24^2 + 7^2) = pi * (576 + 49) = pi * (625)

pi*625 is the area of your new circle that is equal to the sum of the areas of the two smaller circles.

Since pi*625 is equal to pi*r^2, this means that r^2 = 625 which means that r = sqrt(625) = 25.

Since the radius of your new circle equals 25, the diameter must be equal to 2*25 = 50.

That would be selection D.

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I believe the solution is going to be A (r^2 squared units).

The area of a triangle is equal to the base times the height.

In a semi circle, the diameter is the base of the semi-circle.

This is equal to 2*r (r = the radius)

If the triangle is an isosceles triangle with an angle of 45 degrees at each end, then the height of the triangle is also a radius of the circle.

A = (1/2)*b*h formula for the area of a triangle becomes:

A = (1/2)*2*r*r because:

The base of the triangle is equal to 2*r
The height of the triangle is equal to r

A = (1/2)*2*r*r becomes:

A = r^2

Since the area of the triangle is equal to (1/2)*b*h, and the base remains the same while the height of the triangle is less at any other point on the surface of the semi-circle, then the largest area is when the height equals the radius of the triangle.

Your answer is selection A.

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The two smaller parks have diameters of 16 meters and 12 meters.

The total area of these 2 parks is:

8^2*pi + 6^*pi = pi*(8^2 + 6^2) = pi*(64 + 36) = pi*100 = 314.1592654 meters.

The radius of this new park will be sqrt(A/pi) where A is the sum of the areas of the two smaller parks.

This is derived from the formula for Area as follows:

A = pi*r^2
divide both sides of equation by pi to get:
A/pi = r^2
take the square root of both sides of the equation to get:
r = sqrt(A/pi)

This means that r = sqrt((100*pi)/pi) = sqrt(100) = 10.

The radius of the new park should be 10.

10^2*pi = 100*pi = the sum of the areas of the two smaller parks that was previously calculated above.

radius of the new park is 10 meters is your answer.

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A rectangle is 3 times as long as it is wide. If the length and the width are both decreased by 4 cm, the resulting rectangle has area 28cm squared. Find the dimensions of the original rectangle.

L = 3W
(L-4) * (W-4) = 28

Replace L with 3W to get:

(3W-4) * (W-4) = 28

remove parentheses by multiplying out the factors to get:

3W^2 - 16W + 16 = 28

subtract 28 from both sides to get:

3W^2 - 16W - 12 = 0

this factors out to:

(3W+2) * (W-6) = 0

This makes 3W = -2 or W = 6

W can't be negative so W = 6 is the only valid answer.

if W = 6, then L = 3W makes L = 18

You have L = 18 and W = 6

substitute in the original equation to get:

(L-4) * (W-4) = 28 becomes:

(18-4) * (6-4) = 28 which becomes:

14 * 2 = 28 which becomes:

28 = 28 which is true so the values for L and W must be good.

Your answers are:

L = 18 and W = 6

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5^(3-x) = 1/125

since 5^3 = 125, then 5^(-3) = 1/5^3 = 1/125

we have 5^-3 = 1/125 which means that:

3-x = -3 which means that
x = 6

your equation becomes:

5^(3-6) = 1/125 which becomes:

5^(-3) = 1/125 which becomes:

1/5^3 = 1/125 which becomes:

1/125 = 1/125 which is true so the answer of x = 6 is good.