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the difference of two numbers is 6. Their sum is 30. Find the two numbers.
please show all work
Let x and y be the two numbers. Then we have:
1.) x - y = 6 and
2.) x + y = 30
From 1.) we have x = y + 6. Substituting y + 6 for x in 2.) we have:
(y+6) + y = 30
2y = 24
y = 12
Since x = y + 6 we have x = 12 + 6 = 18

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Find the value of k so that the line has slope m:
6x + ky = 10 and m = -2
The line in slope-intercept form is:
6x + ky = 10
ky = -6x + 10
y = (-6/k)x + 10/k
Since -6/k is the slope of the line we have
-6/k = -2
-2k = -6
k = 3

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Can you please help me with these?
Factor, if possible. (If not possible, enter PRIME.)
y^4-256 =
(y^2 - 16)*(y^2 + 16) =
(y-4)(y+4)*(y^2 + 16)

and
Factor, if possible. (If not possible, enter PRIME.)
3u^2-12 =
3*(u^2 - 4) =
3*(u-2)(u+2)


Andy

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Factor the expression into a product of two binomials.
5x^2(x-10)+ ( x-10) =
(5x^2 + 1)*(x-10)

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what is the perimeter of a rectangle that has an area of 12,150 square feet? the ratio of length to width is 3:2.
Let the two sides of the rectangle be 3x and 2x respectively. Then we have:
3x*2x = 12150
6x^2 = 12150
x^2 = 2025
x = 45
The two sides then are 2*45 = 90 and 3*45 = 135

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Is AB the same as A U B? Please resubmit. Thanks.

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The National Association for Women in Science asked recent high school grads if they had taken certain science classes. Of those surveyed, 75 said they had taken physics, 121 said they had taken chemistry and 11 said they had taken both. Fifteen said they had taken neither. How many recent high school grads were surveyed?
Thank you

Let P be the number of those who took physics
Let C be the number of those who took chemistry
P = 75
C = 121
Since the number of students who took both courses is 11, the number who took physics only is 75 - 11 - 64 and C only is 121 - 11 = 110.
Since 15 took neither we have for the total number of students:
Physics only + Chemistry only + both + neither =
64 + 110 + 11 + 15 = 200.

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[SQRT (8) + SQRT (16)] / [SQRT (8) + SQRT (81)] =
(sqrt(4*2) + 4)/(sqrt(4*2) + 9) =
2*sqrt(2) + 4 / 2*sqrt(2) + 9
Multiply numerator and denominator by 2*sqrt(2) - 9:
(2*sqrt(2) + 4)*(2*sqrt(2) - 9)/ (2*sqrt(2) + 9)*(2*sqrt(2) - 9)) =
(4*sqrt(2)^2 - 18*sqrt(2) + 8*sqrt(2) - 36)/(4*sqrt(2)^2 - 81) =
(4*2 - 10*sqrt(2) - 36)/(4*2 - 81) =
(8 - 10sqrt(2) - 36)/ (8 - 81)
(-10*sqrt(2) - 28)/-73 =
(28 + 10 sqrt(2))/73

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log(x + 4) + log(x + 3) = log 2
So we have:
log(x+4)*(x+3) = log 2
And:
(x+4)*(x+3) = 2
x^2 + 7x + 12 = 2
x^2 + 7x + 10 = 0
(x+2)*(x+5) = 0
x+2 = 0 or
x+5 = 0
So
x = -2 or x = -5

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7P2/5P4 =
(7/5)*(p2/p4) =
(7/5)*p^(2-4) =
(7/5)*(p^-2) =
(7/5)*(1/p^2) =
7/(5*p^2)

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12, 42, 92, 162,...
The first digit in each case is 1^2=1, 2^2=4 3^2=9 and 4^2=16.
The second digit is always 2.
So the next number should be 5^2 followed by 2, that is 252.

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(6, 10.2), (1.5, 3.45), (x, 2.85)
The slope-intercept form of a line is:
y = m*x + b where m is the slope and b is the value of y where the line crosses the x-axis (that is when x = 0 ).
m = (10.2 - 3.45)/(6 - 1.5)
m = 6.75/4.5 = 1.5
We have then
1.) y = 1.5*x + b.
Since the line passes through the point (6,10.2) when x = 6, y = 10.2. Substituting these values in 1.) we have:
10.2 = 1.5*6 + b
10.2 = 9 + b
b = 1.2
The equation of the line is then
2.) y = 1.5*x + 1.2
To find x for the third point on the line we substitute 2.85 for y in 2.):
2.85 = 1.5*x + 1.2
1.5*x = 2.85 - 1.2
1.5*x = 1.65
x = 1.65/1.5 = ?

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The sum of the digits of a two digit counting number is 10. If the digits are reversed, the new number is two less than three times the original number. What is the original number?
Let x be the tens digit and y the units digit.
The original number then is 10*x + y. The new number is 10*y + x.
We have then:
1.) x + y = 10 and
2.) 3*(10*x + y) - 2 = 10*y + x
From 1.) we have x = 10 - y. Substituting 10 - y for x in 2.) we have:
3*(10*(10-y) + y) - 2 = 10*y + (10 - y)
3*(100 - 10y + y) - 2 = 10*y + (10 - y)
3*(100 - 9*y) - 2 = 10*y + (10 - y)
300 - 27*y - 2 = 10*y + 10 - y
298 - 27*y = 9*y + 10
36*y = 288
y = 8
Substituting 8 for y in 1.) above we have:
x + y = 10
x + 8 = 10
x = 2
The original number then is 28.

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how do you work out 4 times the square root of 2 plus 6 times the sqaure root of 50 ??
4*sqrt(2) + 6*sqrt(50) =
4*sqrt(2) + 6*sqrt(25*2) =
4*sqrt(2) + 6*sqrt(25)*sqrt(2)=
4*sqrt(2) + 6*5*sqrt(2) =
4*sqrt(2) + 30*sqrt(2) = 34*sqrt(2)

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I dont even know where to start with this problem. The boat can travel 112 miles downstream in 4 hours but requires 6 hours to travel 48 miles upstream. What is the speed of the boat in still water, and what is the speed of the current?
We will need to use the formula distance = speed*time (d=s*t). Let c be the speed of the current and b the speed the speed the boat can travel in still water. The speed going downstream (with the current) is b+c and the speed going upstream (against the current ) is b-c. We have two equations:
Upstream:
d = s*t
48 = (b-c)* 6
Divide both sides above by 6:
8 = b - c
1.) b = 8 + c
Downstream:
d = s*t
112 = (b+c) * 4
Divide both sides above by 4:
2.) 28 = b + c
Substituting 8 + c for b (from 1.) in equation 2.) we have:
28 = (8 + c) + c
28 = 8 + 2*c
2*c = 20
c = 10
Sustituting 10 for c in 1.) we have b = 8 + 10 = 18.

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The measure of one angle is seven less than twice the measure of the other angle.what are the measures of both angles.
More information is needed. Are the angles complementary? Supplementary?

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[((x+1)/(x-1)+1)/((x+1)/(x-1)-1)]^5
Let's look at the expression that is being raised to the 5th power. A common denominator for the terms is x-1. Rewriting 1 as (x-1)/(x-1):
((x+1)/(x-1)) + (x-1)/(x-1) / ((x+1)/(x-1) - (x-1)/(x-1))
Combining terms we have:
(((x+1) + (x+1))/(x-1)) / (((x+1) - (x-1))/(x-1)) =
((2x+2)/(x-1)) / (2/(x-1))
Multiplying numerator and denominator above by (x-1) gives:
(2x+2)/2 = 2*(x+1)/2 = x+1
So the original problem reduces to (x+1)^5
You could expand out (x+1)^5.

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C(20,2) = 20!/((20-2)!*2!) = 20!/(18!*2!)

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Question is unclear. Please resubmit.

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which expression is equivalent to log(7)2401 - log(3)243
Since 7*7*7*7 = 7^4 = 2401, log(7)2401 = 4
Also 3*3*3*3*3 = 3^5 = 243, so log(3)243 = 5
The expression above then reduces to 4 - 5 = -1

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A chemist has one solution that is 40% alcohol and another that is 55% alcohol. How much of each must she use to make 15 liters of solution that is 50% alcohol?
Let x be the amount of 40% alcohol and y the amount of 55% alcohol. Then we have:
1.) x + y = 15 and
2.) .40x + .55y = .50*15
From 1.) we know that x = 15-y. Substituting 15-y for x in 2.) gives us:
.40*(15-y) + .55*y = 7.5
6 - .4y + .55y = 7.5
.15y = 1.5
y = 10
Substituting 10 for y in equation 1.) we have x + 10 = 15, so x =5.

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sqrt(x) < 4
Squaring both sides we have:
x < 16
Note that sqrt(x) is not defined for x < 0 so the answer is actually:
0 <= x < 16

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amount used equals the rate of use multiplied by the time so:
amount = 10*2 = 20 gallons.

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(x-1) (x-3)=-4
x^2 - 4x + 3 = -4
x^2 - 4x + 7 = 0
This is a quadratic equation with a = 1, b = -4 and c = 7. Use the quadratic formula to find the solutions.

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a force of 72 n acting upon a given object results in an acceleration of 8 m/s squared.
if the magnitude of the force acting upon the same object is changed to 63 n, what will the objects acceleration become?
would it be 7 m/s squared
Force = mass * acceleration (F = m*a)
Let m be the mass of the object.
We know that 72 = m*8 so m = 72/8 = 9.
If the acceleration is 63 then we have:
63 = m*a
63 = 9*a
a = 63/9 = 7 m/s sq

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Each odd number can be written as 2*k + 1 for some value of k. Let's say the five numbers are:
2*k1 + 1
2*k2 + 1
2*k3 + 1
2*k4 + 1
2*k5 + 1
The sum of these then is 2*(k1+k2+k3+k4+k5) + 5 =
2*(k1+k2+k3+k4+k5+2) + 1 which is an odd number so can't be 32 (which is even)

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How do you calculate the value of the following expression:11/6 + 5/16 = 88/48 +15/48 = 103/48
11/6 + 5/16 = 88/48 + 15/48 = 103/48 is correct :)

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What is the y-intercept of x = y + 6?
The y-intercept is the value of y when x is zero so:
0 = y + 6
y = -6

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x^2 + 6x + 23 = 0
Rewrie the above as:
x^2 + 6x = -23
To complete the square on the loeft side we need to divide the coefficient on the x term by 2, square it and add the result to both sides:
The coefficient on the x term is 6, so we have (6/2)^2 = 3^2 = 9. Adding 9 to both sides we have:
x^2 + 6x + 9 = -23 + 9
x^2 + 6x + 9 = -14
(x+3)*(x+3) = -14
(x+3)^2 = -14
Taking the square root of both sides:
(x+3)= +sqrt(-14) and
(x+3)= -sqrt(-14)
Note that sqrt(-14) is a complex number, namely i*sqrt(14).
We have then:
x+3 = i*sqrt(14)
x = -3 + i*sqrt914)
and
x+3 = -i*sqrt(14)
x = -3 - i*sqrt(14)

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Are you sure there are two variables, m AND x?

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f(x)=4x
h(x)=-7x-5
find h(f(3))
please tell me what i did wrong this is how i tried to solve the above problem
4(3)(-7(3)-5)
f(3) = 4*3 = 12
h(f(3)) = h(12) =
(-7*12) - 5 =
-84 - 5 = -89