State under which condition the original and the simplified expressions are equivalent.
The expressions are equivalent if...
a) a≠0
b) a≠x
c) a≠-x
[tex]\frac{a^2}{ax-x^2} +\frac{x}{x-a}[/tex]

Answer:

x-intercept = 1 and y-intercept = -3

Step-by-step explanation:

The graph of the function is attached with this answer.

I have used some computer program to draw graph but you can draw a rough graph manually on a graph sheet. For that

To Calculate Some Important Points :

These are the points where the the first derivative of the function becomes zero. This means that at these points the graph takes turn, if it was increasing behind this point then it will start decreasing after this point or the other way. The second derivative of the function at these points are either positive or negative (positive for local minima and negative for local maxima).

To calculate the x-intercept, first you need to analyse the graph to know how many x-intercepts are there. According to this graph only one intercept is there, it means that only one real root of this cubic equation is there (a cubic equation has 3 roots in which either one is real and two are imaginary or all the three are real). To calculate roots of a cubic equation there is no specific way. Generally, the first root is through hit and trial method. So, let's start with the simplest number which is x=0

[tex](0)^{3}+2(0)^{2}-3 \neq 0[/tex]

∴ 0 is not a root.

Now, let x=1

[tex](1)^{3}+2(1)^{2}-3=0[/tex]

∴ 1 is a root.

Since 1 is the only real root of the equation, therefore (1,0) is the only x-intercept of the graph.

To calculate y-intercept, simply put x=0 in the equation which is

[tex]f(0)=(0)^{3}+2(0)^{2}-3=-3\\\therefore f(0)=-3[/tex]

Therefore the y intercept is (0,-3).

* Cubic Polynomial : Polynomials which have a degree (highest power of the variable) of 3 are called cubic polynomials.

** Local Maxima : Points at which the left and right neighbours have less function value are called local maxima.

*** Local Minima : Points at which the left and right neighbours have more function value are called local minima.

Answer:

x = 1 and the y = -3

Step-by-step explanation:

here below hope this helps

The formula can be used to describe the sequence is [tex]a_{n}=\frac{-2}{3}(6)^{n-1}[/tex]

Step-by-step explanation:

The formula of the nth term of the geometric sequence is [tex]a_{n}=a(r)^{n-1}[/tex] , where

∵ The sequence is [tex]\frac{-2}{3}[/tex] , -4 , -24 , -144 , .......

∵ The 1st term is [tex]\frac{-2}{3}[/tex]

∵ The 2nd term is -4

∴ [tex]\frac{-4}{\frac{-2}{3}}=6[/tex]

∵ The 3rd term is -24

∴ [tex]\frac{-24}{-4}=6[/tex]

∵ The 4th term is -144

∴ [tex]\frac{-144}{-24}=6[/tex]

∵  [tex]\frac{a_{2}}{a_{1}}[/tex] = [tex]\frac{a_{3}}{a_{2}}[/tex] =  [tex]\frac{a_{4}}{a_{3}}[/tex] = 6

∴ There is a constant ratio between each two consecutive terms

∴ The sequence is a geometric sequence

∵ The formula of the nth term of the geometric sequence is [tex]a_{n}=a(r)^{n-1}[/tex]

∵ a = [tex]\frac{-2}{3}[/tex]

∵ r = 6

∴ The formula of the sequence is [tex]a_{n}=\frac{-2}{3}(6)^{n-1}[/tex]

The formula can be used to describe the sequence is [tex]a_{n}=\frac{-2}{3}(6)^{n-1}[/tex]

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Answer:

c - f(x) = -2/3(6)^x − 1

Step-by-step explanation:

Edge 2020

The explicit formula for the sequence 1/2, -4, 32, -256 is an = 1/2 * (-8)^(n-1), where n represents the position of each term in the sequence.

To find an explicit formula for the sequence 1/2, -4, 32, -256, we need to look for a pattern in the sequence. Upon examination, we notice that each term is obtained by multiplying the previous term by -8.

The first term, a1, is 1/2. The second term, a2, is 1/2 times -8, which is -4. If we continue this pattern, the nth term of the sequence can be described by the formula an = 1/2 * (-8)^(n-1).

Therefore, the explicit formula for the nth term of this sequence is an = 1/2 * (-8)^(n-1).

Final answer:

The explicit formula for the sequence 1/2, -4, 32, -256 is an = 1/2 × (-8)(n-1), where n represents the nth term. This is derived from the pattern that each term is -8 times the previous term, indicating a geometric sequence.

Explanation:

To find an explicit formula for the sequence 1/2, -4, 32, -256, we need to look for a pattern in the terms of the sequence. Upon inspection, we see that each term is -8 times the previous term. This is a geometric sequence with a common ratio of -8.

The formula for the nth term of a geometric sequence is given by an = a1 × r(n-1), where a1 is the first term and 'r' is the common ratio.

Applying this to the given sequence with a1 = 1/2 and r = -8, the explicit formula for the nth term (an) is:

These rates are not in proportion. Juan walked faster.

Explanation:

First, we want to make these ratios to have the same denominator so we can compare the numerators. To do this, do 3x5 and 8x5 to get 40:15. Then, we do 5x3 and 14x3 to get 42:15. Now we can just compare the numbers 40 and 42, and since 42 < 40, we know Juan walked faster.

The rates of 8/3 miles per hour and 14/5 miles per hour are not in proportion.

To determine whether the rates of Anna and Juan are in proportion, we can compare the rates of their walking speeds by forming a ratio and seeing if the ratios are equivalent. Anna walked 8 miles in 3 hours, which gives us a rate of 8/3 miles per hour. Juan walked 14 miles in 5 hours, resulting in a rate of 14/5 miles per hour.

To check the proportion, we set up a ratio of Anna's speed to Juan's speed: (8/3) / (14/5). If these rates are proportional, the cross-products of the ratios will be equal. However, when you cross-multiply, you get 8 * 5 = 40 and 3 * 14 = 42. Since 40 is not equal to 42, the rates are not in proportion.

Answer:

Step-by-step explanation:3(180-x)=8(90-x)

540-3x=720-8x

540=720-5x

-180=-5x

36=x

x=36

the angle is 36,supplement is 144,complement is 54.

The angle is 36, the supplement is 144, complement is 54.

Two angles whose sum is 180° are called supplementary angles.

Two angles whose sum is 90° are called complementary angles.

Given;

The three times the measure of a supplement of an angle is 8 times the measure of a complement of the angle.

So, 3(180-x)=8(90-x)

Solve for x;

540-3x=720-8x

540=720-5x

-180=-5x

36=x

x=36

Hence, the angle is 36, the supplement is 144, complement is 54.

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Answer:

1400 cm squared

Step-by-step explanation:

40*35=1400cm squared

Answer:

multiply 35(26)

Step-by-step explanation:

35(26)=910

and 910/26=35 ,

so w=910

Answer:

910

Step-by-step explanation:

Your equation is w ÷ 26 = 35

To solve for 'w,' you have to get 'w' alone on one side. To do that, you multiply both sides by 26, getting rid of the '÷ 26' on the left side of the equation.

w ÷ 26 = 35

w = 910

So your answer is 910  ^-^

Answer:

10

Step-by-step explanation:

To solve this, we have to do 4 divided by 2/5, because there are stops every 2/5 mile of a 4-mile track.

To make dividing easier, you can write 4 as a fraction, which would be 4/1

Now, divide:

4/1 ÷ 2/5

Remember that when you divide a fraction by another fraction, you can multiply the first fraction by the reciprocal of the second fraction and get the same fraction.

The reciprocal of 2/5 is 5/2.

So 4/1 x 5/2 = 20/2, which can be simplified as 10.

So there are 10 stops. ^-^

Answer:

Step-by-step explanation:

he whole assignment seems to deal only with one simple concept, a direct variation

Sometimes text or your notes have definitions that are difficult to understand.

Here is the definition that is easy to understand:

A direct variation gives you an equation that contains no constants, and there are no exponents on the two variables.

e.g. y = 8x is a direct variation,

but y = 8x + 4 is NOT, since I see a constant of 4

a direct variation can be written in the form

y = kx , where k is the constant of variation

#1, not a variation, I see a +1 sitting there as a constant

#2, yes it is a direct variation:

-12x = 6y

6y = -12x

y = -2x

so the constant of variation is -2 , if y varies directly with x

or

-12x = 6y

x = -(1/2)x

now the constant of variation is -1/2 , if x varies directly as y

so it depends how you express the equation, either in terms of x or in terms of y

#3,

.7x - 1.4y = 0

.7x = 1.4y

1.4y = .7x

y = (.7/1.4)x = (1/2)x

so , yes it is, and the constant of variation is 1/2

From #4 on, you have to first find the equation.

I will do #4, you try the remaining.

#4, they tell us it is a direct variation, so it must look like

y = kx

They give us x=2 when y = -10

let's plug in those values:

-10 = k(2)

2k = -10

k = -5

aaahhhh, so y = -5x is our equation

#6, same way, just some fractions to deal with

#7, to graph, pick any x, and find the y to get ordered pairs,

e.g. let x = 3, then y = (1/3)(3) = 1, --> point(3,1)

let x = 9, then y = (1/3)(9) = 3 ---> point (9,3)

plot those two points, and join with a straight line

Notice how I picked multiple of 3's for my x's so I could take 1/3 of the x without ending up with fractions?

#8 , same as #7, pick multiples of 2 for your x's

#9 and #10 are the same type

#9

assume it is a direct variation, then its equation must be

y = kx

use the first given data: x = 3, y = 5.4

then 5.4 = k(3)

3k = 4.5

k = 1.5

equation is y = 1.5x

do the other points satisfy our equation?

for (7, 12.6)

is 12.6 = 1.5(7) ?

12.6 = 10.5 ??? NO, so the given data is NOT a direct variation

#10 same way

Answer:1. 2y=5x+1 answer is A Not a direct variation

2. -12x=6y answer is D Direct variation, constant of variation is -2

3. 0.7x-1.4y=0 answer is B Direct variation, constant is 1/2

4. y=-10 when x=2 answer is A y=-5x

5. y=7 1/2 when x=3 answer is C y=5/2 x

6. y=10.4 when x=4 answer is D y=2.6x

7. y=1/3 x answer is A the graph that reads (6, 2) where the green arrow is on the top right portion and on the bottom left is (-6, -2)

8 y=-1/2 x answer is C graph

9. the table x is first 3, 7, 12 y is next 5.4, 12.6, 21.6 answer is B Direct variation, y=1.8x

10. the table x is first -2, 3, 8 y is next 1, 6, 11 answer is A not a direct variation

p.s. this is for the direct variation practice

hope i helped

Step-by-step explanation:

Answer:

C) Multiplicative Inverse Property

Step-by-step explanation:

Because 1/10 is the inverse of 10.

For example, 15*(1/15)=1.

Answer:

C

Step-by-step explanation:

For this case we must solve the following quadratic equation:

[tex]3x ^ 2-2x + 5 = 0[/tex]

Where:

[tex]a = 3\\b = -2\\c = 5[/tex]

The roots are given by:

[tex]x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2a}[/tex]

Substituting the values we have:

[tex]x = \frac {- (- 2) \pm \sqrt {(- 2) ^ 2-4 (3) (5)}} {2 (3)}\\x = \frac {- (- 2) \pm \sqrt {(- 2) ^ 2-4 (3) (5)}} {2 (3)}\\x = \frac {2 \pm \sqrt {4-60}} {6}\\x = \frac {2 \pm \sqrt {-56}} {6}[/tex]

By definition we have to:

[tex]i ^ 2 = -1\\x = \frac {2 \pm \sqrt {56i ^ 2}} {6}\\x = \frac {2 \pm i \sqrt {56}} {6}\\x = \frac {2 \pm i \sqrt {2 ^ 2 * 14}} {6}\\x = \frac {2 \pm 2i \sqrt {14}} {6}\\x = \frac {1 \pm i \sqrt {14}} {3}[/tex]

We have two complex roots:

[tex]x_ {1} = \frac {1+ i \sqrt {14}} {3}\\x_ {2} = \frac {1- i \sqrt {14}} {3}[/tex]

Answer:

[tex]x_ {1} = \frac {1+ i \sqrt {14}} {3}\\x_ {2} = \frac {1- i \sqrt {14}} {3}[/tex]

Answer:

Below.

Step-by-step explanation:

Let the distance in miles be d and the time taken = t hours.

Then the direct proportion is d = kt  where k is some constant.

In this case k = d/t = 18.

d = 18t is the equation relating  distance and time.

In this case the 18 represents Milo's speed in  miles per hour.

Answer:

A) -4

Step-by-step explanation:

f(x)=10-2x

f(7)=10-2(7)

f(7)=10-14

f(7)=-4

fx)= 10-2x

f(7)=10-2(7)

do the bracket first

-2(7)=-14

f(7)=10-14

f(7)=-4

answer:

-4

Answer: 40 000

Step-by-step explanation:

Population doubles in first 55 years =

5000 * 2 = 10 000

In 220 years from now, 220/55 = 4

Hence, 10 000 * 4 = 40 000 population

Answer:

  12/5

Step-by-step explanation:

The tangent ratio is the ratio of the opposite side (GH) to the adjacent side (FG). The mnemonic SOH CAH TOA can help you remember this:

  Tan = Opposite/Adjacent

In this triangle, ...

  tan(F) = GH/FG = 12/5

Answer:

[tex]f(x)=\frac{1}{2}x-4[/tex]

Step-by-step explanation:

Given:

The range is 4 less than half of the domain.

The domain of a function is its input values represented by 'x'. The range of a function is its output values represented by 'y' values.

Domain is 'x'. So, half of domain is [tex]\frac{1}{2}x[/tex].

Now, range is 4 less than half of domain. Hence,

[tex]y=\frac{1}{2}x-4[/tex]

So, the correct option is the last option.

[tex]\( \cos(2x) = \frac{1}{2} \).[/tex]

Given that [tex]\( \sin(x) = -\frac{1}{2} \) and \( \tan(x) \)[/tex] is negative, we can find \[tex]( \cos(2x) \)[/tex]using the trigonometric identities.

First, let's find the value of [tex]\( \cos(x) \)[/tex] using the Pythagorean identity:

[tex]\[ \cos^2(x) = 1 - \sin^2(x) \][/tex]

Given [tex]\( \sin(x) = -\frac{1}{2} \),[/tex] we have:

[tex]\[ \cos^2(x) = 1 - \left(-\frac{1}{2}\right)^2 \][/tex]

[tex]\[ \cos^2(x) = 1 - \frac{1}{4} \][/tex]

[tex]\[ \cos^2(x) = \frac{3}{4} \][/tex]

Taking the square root of both sides, since [tex]\( \cos(x) \)[/tex] is positive in the first and fourth quadrants:

[tex]\[ \cos(x) = \pm \frac{\sqrt{3}}{2} \][/tex]

Given that [tex]\( \tan(x) \)[/tex] is negative, we know that ( x ) lies in either the second or fourth quadrant. In the second quadrant, both [tex]\( \sin(x) \) and \( \cos(x) \)[/tex] are negative. In the fourth quadrant, [tex]\( \sin(x) \)[/tex] is negative but [tex]\( \cos(x) \) i[/tex]s positive.

Since [tex]\( \cos(x) = \pm \frac{\sqrt{3}}{2} \),[/tex] we conclude that [tex]\( \cos(x) = -\frac{\sqrt{3}}{2} \)[/tex]  (since [tex]\( \cos(x) \)[/tex] is negative in the second quadrant).

Now, using the double angle identity for cosine:

[tex]\[ \cos(2x) = 2\cos^2(x) - 1 \][/tex]

Substituting [tex]\( \cos(x) = -\frac{\sqrt{3}}{2} \):[/tex]

[tex]\[ \cos(2x) = 2\left(-\frac{\sqrt{3}}{2}\right)^2 - 1 \][/tex]

[tex]\[ \cos(2x) = 2\left(\frac{3}{4}\right) - 1 \][/tex]

[tex]\[ \cos(2x) = \frac{3}{2} - 1 \][/tex]

[tex]\[ \cos(2x) = \frac{3}{2} - \frac{2}{2} \][/tex]

[tex]\[ \cos(2x) = \frac{1}{2} \][/tex]

So, [tex]\( \cos(2x) = \frac{1}{2} \).[/tex]

Answer:

[tex]y=0.95x+2.5[/tex]

Step-by-step explanation:

we know that

The linear equation in slope intercept form is equal to

[tex]y=mx+b[/tex]

where

m is the slope

b is the y-intercept

Find the slope

The formula to calculate the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

take the points

(0,2.5) and (100,97.5)

substitute in the formula

[tex]m=\frac{97.5-2.5}{100-0}[/tex]

[tex]m=\frac{95}{100}[/tex]

[tex]m=0.95[/tex]

Find the y-intercept b

we have

[tex]m=0.95[/tex]

[tex]point\ (0,2.5)[/tex]

substitute i the linear equation  [tex]y=mx+b[/tex]

[tex]2.5=0.95(0)+b[/tex]

solve for b

[tex]b=2.5[/tex]

Note It was not necessary to calculate the value of b because the value is given in the table

Remember that the y-intercept is the value of y when the value of x is equal to zero (the y-intercept is the point (0,2.5))

The linear equation is

[tex]y=0.95x+2.5[/tex]

To write the equation of a linear relationship in slope-intercept form, find the slope and y-intercept. Use the formula y = mx + b, where m is the slope and b is the y-intercept. Calculate the slope using the formula m = (y2 - y1) / (x2 - x1). Substitute the slope and one set of coordinates into the slope-intercept form equation and solve for b.

To write the equation of the linear relationship in slope-intercept form, we need to find the slope (m) and the y-intercept (b). We can use the formula y = mx + b, where m is the slope and b is the y-intercept.

To find the slope, we can use the formula: m = (y2 - y1) / (x2 - x1). Choosing any two points from the given data, we can calculate the slope.

Let's use the points (0, 2.5) and (100, 97.5):
m = (97.5 - 2.5) / (100 - 0) = 95 / 100 = 0.95

Next, we can substitute the slope and one set of coordinates into the slope-intercept form equation and solve for b:

2.5 = 0.95(0) + b
b = 2.5

Therefore, the equation of the linear relationship is: y = 0.95x + 2.5

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Answer:

4/7

Step-by-step explanation:

32/56=4/7

Answer:

The number of boys = 32

The number of girls = 56

therefore, to fine the possible ratio, you divided 32 and 56 to their lowest terms.

i.e 32: 56. 32÷8 : 56÷8 = 4:7

Answer:

The truck have travelled 345.5 inches after 5 rotations of the tires

Step-by-step explanation:

Given:

Diameter of the tyre= 22 inches

Number of rotations= 5

To find:

Distance travelled after 5 rotations=?

Solution:

We have given with diameter,

So let radius be r

r= [tex]\frac{diameter}{2}[/tex]

[tex]r=\frac{22}{2}[/tex]

r=11 inches

The distance covered by one rotation is given by circumference

Circumference =[tex]2\pi r[/tex]

Substituting the values, we get

Circumference =[tex]2\times\pi\times r[/tex]

Circumference =[tex]2\times\pi\times 11[/tex]

Circumference =[tex]2\times\3.14\times 11[/tex]

Circumference =[tex]6.28\times 11[/tex]

Circumference = 69.11

Now for 5 rotation,

Distance travelled = [tex]5\times(\text{circumference value})[/tex]

Distance travelled = [tex]5\times(69.11)[/tex]

Distance travelled = [tex]5\times(69.11)[/tex]

The truck will travel 345.5 inches.

Answer:

x=2, y=4. (2, 4).

Step-by-step explanation:

x+y=6

3x-2y=-2

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

y=6-x

3x-2(6-x)=-2

3x-12+2x=-2

5x-12=-2

5x=-2+12

5x=10

x=10/5

x=2

y=6-(2)=6-2=4

The number 0.251 is a rational number because it can be expressed as the fraction 251/1000, which has both integers as its numerator and denominator, with the denominator not being zero.

To determine whether 0.251 is a rational number, one must understand the definition of rational numbers.

Rational numbers include all numbers that can be expressed as a fraction where both the numerator (the top number) and the denominator (the bottom number) are integers, and the denominator is not zero.

The number 0.251 meets this criterion because it can be written as the fraction 251/1000, where 251 and 1000 are both integers, and 1000 is not zero.

If we look at different numerical expressions, such as ones resulting from operations on fractions or converting numbers into scientific notation, the underlying principle is that the operations maintain equality and the representations still refer to rational numbers as long as they can be expressed as a ratio of integers.

Answer:

x = 6

Step-by-step explanation:

The formula of slope of a straight line passing through two given points ([tex]x_{1},y_{1}[/tex]), and ([tex]x_{2},y_{2}[/tex]) is given by  

[tex]m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex].

As, the straight line passes through the points (x,-4) and (2,8) and has slope - 3.

So, we can write [tex]\frac{8 - (- 4)}{2 - x} = - 3[/tex]

⇒ 12 = 3x - 6

⇒ 3x = 18

⇒ x = 6 (Answer)

A graph of a right-angled triangle, ΔABC with side lengths of 3, 4, and 5 units is shown in the picture below.

In Mathematics and Euclidean Geometry, a triangle is a two-dimensional geometric shape that comprises three side lengths, three vertices and three angles only.

Generally speaking, there are five major types of triangle based on the length of their side lengths and angles, and these include the following;

In this scenario, the lengths of three of the sides of the triangle and the measure of the included angle are;

AB = 3 units with vertices A (0, 3)

BC = 4 units with vertices B (0, 0).

AC = 5 units with vertices C (4, 0)

m∠B = 90 degrees.

In conclusion, we would use an online graphing tool to plot this right-angled triangle as shown in the image attached below.

Complete Question:

Drag the vertices of the triangle below to draw a right triangle with side lengths 3, 4, and 5.

[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ V(\stackrel{x_1}{-2}~,~\stackrel{y_1}{-6})\qquad W(\stackrel{x_2}{x+2}~,~\stackrel{y_2}{y+3}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{(x+2)-2}{2}~~,~~\cfrac{(y+3)-6}{2} \right)\implies \left( \cfrac{x}{2}~~,~~\cfrac{y-3}{2} \right)[/tex]

Answer:

10 pages per technician per day is the unit metric for the production work.

Step-by-step explanation:

Consider the provided information.

Stiviko's approved work plan included 4 technicians to create 200 web pages  over 5 days.

They can create 200 web pages over 5 days.

[tex]\frac{200}{5}=40[/tex]

That means in one day they can create 40 pages.

There are 4 technicians who work, therefore the unit metric for the production work is: [tex]\frac{40}{4}=10[/tex]

Hence, 10 pages per technician per day is the unit metric for the production work.

Answer:

10 pages per technician per day

Step-by-step explanation:

Answer:

81

Step-by-step explanation:

Answer:

A if you're subtracting amount grown from amount cut

C if your subtracting husband's gain from wife's loss,

E for sure. 100%. a sum or 1.5 is reached through addition

& F if you're subtracting amount used from amount bought.

Step-by-step explanation:

Please comment & ask for further explanation if you would like it :)

Answer: they are A C E and F

Step-by-step explanation:

My siblings explained it to me

Answer:

see explanation

Step-by-step explanation:

All angles of a square are right angles by definition, thus

∠ACD = 90°

The diagonals bisect each other at right angles, thus

∠1 = 90°

The diagonals bisect the angles at the vertices, thus

∠2 - 45°

Step-by-step explanation:

the segments are equal to each other because O is the midpoint of the segment