Variables and Inequalities

The coefficients in the expression W - 8x + 20y + 6z are -8 for x, 20 for y, and 6 for z. W is not considered a variable with a coefficient but would implicitly have a coefficient of 1.

The coefficients in the expression W - 8x + 20y + 6z are the numerical factors that are multiplied by the variables. In this case:

Coefficients are important as they determine the rate at which the value of the expression changes with respect to changes in the variable values.

Note that W in this expression is not preceded by a coefficient, and if it is supposed to have a coefficient, it is implicitly 1.

However, as the question does not consider W as a variable with a coefficient, we exclude it from the list of coefficients.

Answer:

The greatest number of pairs of skis and snowboards the group can rent are 21 pairs of skis and 9 pairs of snowboards.

Step-by-step explanation:

Given:

A group can rent pairs of skis and snowboards by the week. They get a reduced rate if they rent 7 pairs of skis for every 3 snowboards rented. The reduced ski rate is $45.50 per pair of skis per week, and the reduced snowboard rate is $110 per snowboard per week.

The sales tax on each rental is 16%.

The group has $2,500 available to spend on ski and snowboard rentals.

If the ratio of pairs of skis to snowboards is 7:3.

Now, to find the greatest number of pairs of skis and snowboards rentals.

So, the rent of 7 pairs of skis = [tex]7\times 45.50=\$318.50.[/tex]

And the rent of 3 pairs of snowboards = [tex]3\times 110=\$330.[/tex]

So, total rental amount of 7 pairs of skis and 3 pairs of snowboards:

[tex]318.50 + 330 = 648.50.[/tex]

Now, to get the rental amount after sales tax:

648.50 + 16% of $648.50.

[tex]=648.50 +\frac{16}{100}\times 648.50.[/tex]

[tex]=648.50+103.76[/tex]

[tex]=\$752.26.[/tex]

The total rental amount after sales tax = $752.26.

As the group has available $2,500.

So, the sets according to the given ratio:

[tex]2500\div 752.26 = 3.32.[/tex]

[tex]=3\ sets.[/tex]

Thus, there are 3 sets of the ratio 7:3.

So, the rental price according to sets are:

The rent of Skis:

[tex]7\times 3 = 21[/tex] 

[tex]21\times 45.50= 955.50[/tex]

The rent of snowboards:

[tex]3\times 3 = 9[/tex]

[tex]9\times 110 = 990[/tex]

So. the total rental amount of skis and snowboards according to sets are:

[tex]955.50 + 990 = 1,945.50[/tex]

Now, the amount of rent after sales tax:

$1945.50 + 16% of $1945.50.

[tex]=1945.50+\frac{16}{100}\times 1945.50[/tex]

[tex]=1945.50+311.28=\$2256.78.[/tex]

Thus, the total cost = $2256.78.

Now, to get the greatest number of pairs of skis to snowboards that can be rent:

[tex]2,500 - 2,256.78 = 243.22[/tex]

The cost of 21 pair of skis and 9 pairs of snowboards is $2256.78 and the group has available only $2500 to spend.

Thus, they can rent only 21 pairs of skis and 9 pairs of snowboards.

Therefore, the greatest number of pairs of skis and snowboards the group can rent are 21 pairs of skis and 9 pairs of snowboards.

Answer:

the first blank is 10

and the second blank is 5

Step-by-step explanation:

Answer:

[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)\neq 0[/tex]  is proved for the sum of pth, qth and rth terms of an arithmetic progression are a, b,and c respectively.

Step-by-step explanation:

Given that the sum of pth, qth and rth terms of an arithmetic progression are a, b and c respectively.

First term of given arithmetic progression is A

and common difference is D

ie., [tex]a_{1}=A[/tex] and common difference=D

The nth term can be written as

[tex]a_{n}=A+(n-1)D[/tex]

pth term of given arithmetic progression is a

[tex]a_{p}=A+(p-1)D=a[/tex]

qth term of given arithmetic progression is b

[tex]a_{q}=A+(q-1)D=b[/tex] and

rth term of given arithmetic progression is c

[tex]a_{r}=A+(r-1)D=c[/tex]

We have to prove that

[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)=0[/tex]

Now to prove LHS=RHS

Now take LHS

[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)[/tex]

[tex]=\frac{A+(p-1)D}{p}\times (q-r)+\frac{A+(q-1)D}{q}\times (r-p)+\frac{A+(r-1)D}{r}\times (p-q)[/tex]

[tex]=\frac{A+pD-D}{p}\times (q-r)+\frac{A+qD-D}{q}\times (r-p)+\frac{A+rD-D}{r}\times (p-q)[/tex]

[tex]=\frac{Aq+pqD-Dq-Ar-prD+rD}{p}+\frac{Ar+rqD-Dr-Ap-pqD+pD}{q}+\frac{Ap+prD-Dp-Aq-qrD+qD}{r}[/tex]

[tex]=\frac{[Aq+pqD-Dq-Ar-prD+rD]\times qr+[Ar+rqD-Dr-Ap-pqD+pD]\times pr+[Ap+prD-Dp-Aq-qrD+qD]\times pq}{pqr}[/tex]

[tex]=\frac{Arq^{2}+pq^{2} rD-Dq^{2} r-Aqr^{2}-pqr^{2} D+qr^{2} D+Apr^{2}+pr^{2} qD-pDr^{2} -Ap^{2}r-p^{2} rqD+p^{2} rD+Ap^{2} q+p^{2} qrD-Dp^{2} q-Aq^{2} p-q^{2} prD+q^{2}pD}{pqr}[/tex]

[tex]=\frac{Arq^{2}-Dq^{2}r-Aqr^{2}+qr^{2}D+Apr^{2}-pDr^{2}-Ap^{2}r+p^{2}rD+Ap^{2}q-Dp^{2}q-Aq^{2}p+q^{2}pD}{pqr}[/tex]

[tex]=\frac{Arq^{2}-Dq^{2}r-Aqr^{2}+qr^{2}D+Apr^{2} -pDr^{2}-Ap^{2}r+p^{2}rD+Ap^{2}q-Dp^{2}q-Aq^{2}p+q^{2}pD}{pqr}[/tex]

[tex]\neq 0[/tex]

ie., [tex]RHS\neq 0[/tex]

Therefore [tex]LHS\neq RHS[/tex]

ie.,[tex]\frac{a}{p}\times (q-r)+\frac{b}{q}\times (r-p)+\frac{c}{r}\times (p-q)\neq 0[/tex]  

Hence proved

Answer:

The set has 256 subsets

Step-by-step explanation:

The set is [a,b,c,d,e,f,h,i]

Number of subset can be calculated from the formula

[tex] {2}^{n} [/tex]

where n is the number of element in the set

n=8

This implies that

[tex] {2}^{n} = {2}^{8} =256[/tex]

Answer:

[tex]-\sqrt{15}[/tex]

Step-by-step explanation:

we know that

To put factor back into the square root, we have to put squared value

we have

[tex]-5\sqrt{0.6}[/tex]

Remember that

[tex]5=\sqrt{5^2}[/tex]

substitute in the expression above

[tex]-5\sqrt{0.6}=-\sqrt{(5^2)(0.6)}=-\sqrt{25*0.6}=-\sqrt{15}[/tex]

Answer:

C. 64 pie m^3

Answer: 64[tex]\pi[/tex][tex]m^{3}[/tex]

Step-by-step explanation:

The volume of cylinder is given as

V = [tex]\pi[/tex][tex]r^{2}[/tex]h

r = 4m

h = 4m

Substituting into the formula , we have

V = [tex]\pi[/tex]  x [tex]4^{2}[/tex] x 4

V = [tex]\pi[/tex] x 16 x 4

V = [tex]\pi[/tex] x 64

Therefore : V = 64[tex]\pi[/tex][tex]m^{3}[/tex]

Answer:

Step-by-step explanation:

A horizontal line has the equation "y = " something, while a vertical line has the equation "x = " something.  If our line is horizontal through (3, 6), it will be horizontal through the y coordinate (and vertical through the x coordinate).  Our equation is y = 6

(The vertical equation through this point would be x = 3, just so you understand what I meant above.)

Area of sector is 17.584 meters

Solution:

Given that we have to find the approximate area of a sector given O= 56 degrees with a diameter of 12m

Diameter = 12 m

Radius = Diameter / 2 = 6 m

An angle of  56 degrees is the fraction [tex]\frac{56}{360}[/tex] of the whole rotation

A sector of a circle with a sector angle of 56 degrees is therefore also the fraction [tex]\frac{56}{360}[/tex] of the circle

The area of the sector will therefore also be  [tex]\frac{56}{360}[/tex] of the area

[tex]\text{ sector area } = \frac{56}{360} \times \pi r^2\\\\\text{ sector area } = \frac{56}{360} \times 3.14 \times 6^2\\\\\text{ sector area } = 17.584[/tex]

Thus area of sector is 17.584 meters

Answer:

yes it is!

Step-by-step explanation:

if you add 8+10=18

11+10+21

8+11=19

if the numbers do not repeat or add up to one of the numbers you've got as an option it is a right triangle.

Answer:

18

Step-by-step explanation:

18x9/0-12.6 x 14 x 13.2 = 18

f(x)=x2+7

f(4)=4*2+7

4*2=8

8+7=15

Answer:

The value of r is $0.95 and b is $6.5 , The graph of given equation plotted as shown

Step-by-step explanation:

Given as :

The two line equation are

2 r + b = $8.40              .........A    and

3 r + b = $9.35               .........B

Now, Solving equations A and B

i.e (3 r + b) - (2 r + b) = $9.35 - $8.40

Or, (3 r - 2 r) + (b - b) = 0.95

or, r + 0 = 0.95

∴ r = $0.95

Now, put the value og r into eq A

∵ 2 r + b = $8.40  

So, 2× $0.95 + b = $8.40  

Or, $1.9 + b = $8.40  

Or, b = $8.40 - $1.9

∴ b = $6.5

So, The value of r = $0.95 and b = $6.5

Now, for plotting the equation on graph

let put r on x axis

and put b on y-axis

Hence, The value of r is $0.95 and b is $6.5 , The graph of given equation plotted as shown . Answer

Answer:

x=-1 and x=5

Step-by-step explanation:

we have

[tex]x^{2} -4x=5[/tex]

Solve it by completing the square method

We take the coefficient of x that is -4, divide it by 2 and then square it

so

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

square it

[tex](-2)^2 = 4[/tex]

Adds 4 both sides

[tex]x^{2} -4x+4=5+4[/tex]

Combine like terms right side

[tex]x^{2} -4x+4=9[/tex]

Rewrite as perfect squares

[tex](x-2)^{2}=9[/tex]

take square root both sides

[tex]x-2=\pm3[/tex]

[tex]x=2\pm3[/tex]

therefore

[tex]x=2+3=5[/tex]

[tex]x=2-3=-1[/tex]

Answer:

x = -2 (multiplicity 2).

Step-by-step explanation:

3x^2+12x+12 = 0

3(x^2 + 4x + 4) = 0

3(x + 2)(x + 2) = 0

x = -2 (multiplicity 2).

2 roots - both equal.

Answer:

3(x+2) (x+2)

Step-by-step explanation:

Given 3x²+12x+12

We expand the equation

That is,

3x²+6x+6x+12

We add parentheses to factor out

(3x²+6x)+(6x+12)

3x(x+2)+6(x+2) > Taking out the

common factors.

(3x+6) (x+2)

= 3(x+2) (x+2)

Answer:

Table C represents an exponential function.

Answer:

[tex]\frac{1}{36}[/tex]

Step-by-step explanation:

We will find the probability of rolling a 5 and then probability of rolling a 3 and then multiply the numbers to get the probability that he will roll a 5 and then a 3 in two rolls.

There are 6 total possibilities (1 through 6).

There are one 5s, so the probability of rolling a 5 would be:

P(roll 5) = 1/6

Also, there is one 3, so rolling a 3 would have the same probability.

P(roll 3) = 1/6

Hence,

Probability of rolling a 5 and then a 3 is:

P(5,3) = 1/6 * 1/6 = 1/36

Answer:

Step-by-step explanation:

y=-2x+4

-5x+10y=5

A:Parallel

B:Perpendicular

C: Neither

Tell whether the lines for each pair of equations are parallel, perpendicular, or neither.

y=(-1/5)x+6

-2x+10y=5

A:Parallel

B:Perpendicular

C:Neither

Understanding parallel and perpendicular lines on the coordinate plane is essential in geometry. Here's how you can determine the relationships between lines in this context.
**Parallel Lines:**
Two lines are parallel if they are always the same distance apart (equidistant) and will never meet. In the coordinate plane, parallel lines can be identified by having the same slope.
To show that two lines are parallel, you must find that they have the same slope. For example, if you have two lines described by the equations:
1. \(y = m_1x + b_1\)
2. \(y = m_2x + b_2\)
If \(m_1 = m_2\), then these two lines are parallel.
**Perpendicular Lines:**
Two lines are perpendicular if they intersect at a right angle (90 degrees). On the coordinate plane, two non-vertical lines are perpendicular if the product of their slopes is -1. This means that the slope of one line is the negative reciprocal of the other.
To show that two lines are perpendicular, you would find that the slopes of the two lines multiplied together equal -1. For instance, if you have two lines defined by:
1. \(y = m_1x + b_1\)
2. \(y = m_2x + b_2\)
These lines are perpendicular if \(m_1 \cdot m_2 = -1\), which means \(m_1 = -\frac{1}{m_2}\) or \(m_2 = -\frac{1}{m_1}\).
**Example 1: Check if lines are parallel**
Suppose you are given two lines with equations:
1. \(y = 2x + 3\)
2. \(y = 2x - 4\)
Both lines have a slope of 2 (since the coefficient of \(x\) in both equations is 2). Since the slopes are equal, these lines are parallel.
**Example 2: Check if lines are perpendicular**
Imagine you are given two lines:
1. \(y = \frac{1}{3}x + 2\)
2. \(y = -3x + 1\)
To determine if they're perpendicular, find the product of the slopes: \(\frac{1}{3} \cdot -3 = -1\). Since the product is -1, these lines are perpendicular.
In summary, for parallel lines, check if the slopes are equal. For perpendicular lines, check if the product of the slopes is -1. Of course, these rules presume you're working with non-vertical lines, since vertical lines have undefined slopes and horizontal lines have a slope of 0. Vertical and horizontal lines are also perpendicular to each other.

Answer:

3780 feet

Step-by-step explanation:

49/7*60*3(be careful that it is 3 minutes,not seconds)

=1260

Now,we have to convert yards into feet.

1260 yards = 3780 feet

Because feet is three times yard.

Hope this answer helps! ;)

Answer:

Distance ran by Sara in 3 minutes is 3780 feet.

Step-by-step explanation:

We know that,

1 yard = 3 feet

So, 49 yards = 49 × 3 feet = 147 feet

Now, it is given that Sara can run 49 yards in 7 seconds. So, we can say that, Sara can run 147 feet in 7 seconds

So, distance ran by Sara in 7 seconds = 49 yards = 147 feet

Now, distance ran by Sara in 1 second = (147 ÷ 7) feet = 21 feet

Now, 1 min = 60 sec, then 3 min = 3 × 60 sec = 180 sec

So, distance ran by Sara in 3 min or 180 sec = 21 × 180 = 3780 feet

∴ distance ran by Sara in 3 minutes is 3780 feet.

Answer:

25%

Step-by-step explanation:

180+60=240

60/240=0.25

0.25*100

25%

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Using ants on a stretching ruler to illustrate Hubble's Law shows how objects further away move faster relative to an observer, much like galaxies in the universe, while highlighting the organization and communication in ant colonies.

The scenario described involves ants on various points of a stretchable ruler, illustrating a concept similar to Hubble's Law in cosmology. One ant measures how the distance to the other ants increases as the ruler stretches, showing that ants further away appear to move faster relative to the observing ant. This phenomenon occurs not because the ants are moving of their own volition, but because the ruler itself is stretching, much like space-time expands in the universe.

Furthermore, ants in a colony communicate using pheromones, and their large groups, called colonies, are highly organized with different roles such as workers, soldiers, and a queen. The example of ants traveling on a ruler and cooperating in a colony also demonstrates principles of coordination, communication, and perception from a relative standpoint.

Answer:

$162÷24=$6.75

$6.75-per each

five pairs of socks-$6.75x5=$33.75

Final answer:

To determine the cost of five pairs of socks when the coach spent $162 for 24 pairs, we find the cost per pair ($6.75) and multiply it by 5, resulting in $33.75 for five pairs.

Explanation:

The student's question is, "A soccer coach spent 162 dollars on 24 pairs of socks for his players. How much did five pairs of socks cost?" To solve this, we start by determining the cost per pair of socks. We divide the total cost by the number of pairs to get the price for each pair:

Cost per pair = Total cost / Number of pairs
= $162 / 24 pairs
= $6.75 per pair

Now, to find the cost for five pairs of socks, we multiply the cost per pair by the number of pairs desired:

Cost for five pairs = Cost per pair x 5
= $6.75 x 5
= $33.75

Therefore, five pairs of socks cost $33.75.

Answer:

OA there is not enough information

Step-by-step explanation:

only the angles are congruent in both squares

Answer:

OA there is not enough information to determine whether the squares are congruent.

Step-by-step explanation:

Answer: is c 3/2

Step-by-step explanation:

The answer is option C) 3/2.

To find out how many 1/4 cup servings are in 3/8 cups of pudding, we can use the formula:

First, let's convert 3 8/10 cups to an improper fraction.

To do this, we multiply the whole number (3) by the denominator (10) and add the numerator (8) to get the numerator of the improper fraction. The denominator remains the same.

(3/8) ÷ (1/4) = (3/8) × (4/1) = 12/8 = 3/2

Therefore, there are 3/2 or 1 1/2 servings of 1/4 cup in 3/8 cups of pudding.

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Answer:

5/6

Step-by-step explanation:

1/3+3/6=2/6+3/6=5/6

Answer:

answer b

Step-by-step explanation:

The coordinates of p are (3, -2). Option D is correct.

A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.

From the graph, the coordinates of A = (9,-8) and the coordinates of B =(-6, 7) p divides the line segmentin 2/3 thus,

We can use the section formula to find the coordinates of the point P that divides the line segment AB into the ratio 2:3.

The section formula states that if a line segment AB is divided by a point P into two line segments, with lengths in the ratio m:n, then the coordinates of P are:

[tex][Px = (nA_x + mB_x)/(m+n), \\Py = (nA_y + mB_y)/(m+n)][/tex]

In this case, we want to divide the line segment AB into the ratio 2:3, so we have m:n = 2:3. This means that m = 2 and n = 3.

Substituting the given values, we get:

[tex]P_x = (3*9 + 2(-6))/(2+3) = 3\\Py = (3*(-8) + 2*7)/(2+3) = -2[/tex]

Therefore, the coordinates of P are (3, -2).

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Answer:

x^14

Step-by-step explanation:

(x^7)^2=x^7×2

=x^14

Answer:

x = [tex]\frac{4}{35}[/tex]

Step-by-step explanation:

Given

32x + 1 = - 3x + 5 ( add 3x to both sides )

35x + 1 = 5 ( subtract 1 from both sides )

35x = 4 ( divide both sides by 35 )

x = [tex]\frac{4}{35}[/tex]

Answer:

[tex]x = \frac{ - 6}{29} [/tex]

Step-by-step explanation:

[tex]32x + 1 - 3x + 5[/tex]

Collect like terms

[tex]32x - 3x + 1 + 5[/tex]

[tex] 29x + 6[/tex]

For simplifying, equate the expression to zero. Thus,

[tex]29x + 6 = 0[/tex]

Add -6 to both sides of the equation

[tex]29x + 6 - 6 = 0 - 6[/tex]

[tex]29x = - 6[/tex]

Divide through by 29

[tex] \frac{29x}{29} = \frac{ - 6}{29} [/tex]

[tex]x = \frac{ - 6}{29} [/tex]

Answer:

The y-coordinate of the original point is negative and the y-coordiinate of the new point is positive.

The transformation is:

[tex](-7.5, 6)[/tex] →  [tex](-7.5, -6)[/tex]

Step-by-step explanation:

For this exercise it is important to remember the definition of "Reflection".

A Reflection is a transformation that consists in folding an figure over a line called "Line of reflection",

In tranformations, the original figure is called "Pre-image" and the figure obtained after the transdformation is called "Image".

Given a point [tex](x,y)[/tex], if this is reflected across the x-axis, the x-coordinate of the point does not change, but the sign of the y-coordinate is transformed into its opposite. Then:

 [tex](x,y)[/tex] →  [tex](x,-y)[/tex]

So, given the new point:

[tex](-7.5, 6)[/tex]

You can determine that the original point is:

[tex](-7.5, -6)[/tex]

Answer:

Option C. 7.2 units

Step-by-step explanation:

we know that

triangle PAM is similar to triangle RAE by AA Similarity Theorem

Remember that

If two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent

so

[tex]\frac{PA}{RA}=\frac{AM}{AE}[/tex]

substitute the given values

[tex]\frac{20+x}{20}=\frac{25+9}{25}[/tex]

solve for x

[tex]\frac{20+x}{20}=\frac{34}{25}[/tex]

[tex]20+x=\frac{34}{25}(20)[/tex]

[tex]x=\frac{34}{25}(20)-20[/tex]

[tex]x=7.2\ units[/tex]

The value of 'x' is 7.2 units and this can be determined by using the properties of the triangle and also by using the arithmetic operations.

Given :

According to the given data, the side RE is parallel to side PM therefore it can be said that:

[tex]\rm \dfrac{PA}{RA}=\dfrac{MA}{EA}[/tex]

Now, substitute the values of PA, MA, RA, and EA in the above expression.

[tex]\rm \dfrac{x+20}{20}=\dfrac{9+25}{25}[/tex]

Cross multiply in the above equation.

[tex]25 \times (x + 20) = 20\times (34)[/tex]

25x + 500 = 680

Subtract 680 by 500 in the above equation.

25x = 180

DIvide 180 by 25 in the above equation.

x = 7.2

Therefore, the correct option is C).

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