PLEASE HELP 100 POINTS

Select the correct answer.
The length, l, of a rectangle is modeled by the equation l = w + 4, where w is the width of the rectangle in centimeters.

Two equations have been determined that represent the area of the rectangle, A, in square centimeters:

The first equation was created using the formula for the area of a rectangle: A = w2 + 4w.
The second equation models the relationship between the rectangle's area and width: A = 4w + 45.
Which statement describes the solution(s) of the system?

A.
There are two solutions, and neither are viable.
B.
There are two solutions, but only one is viable.
C.
There are two solutions, and both are viable.
D.
There is only one solution, and it is viable.

The computed mean of 52.4 and the actual mean of 52.7 suggest a close relationship in terms of central tendency.

A computed mean is a statistical measure calculated by summing up a set of values and dividing by the number of observations. In this case, the computed mean of 52.4 implies that when the values are averaged, the result is 52.4.

The actual mean of 52.7 refers to the true average of the population or data set being analyzed. Since it is higher than the computed mean, it indicates that the sample used for computation might have slightly underestimated the true population mean.

However, the difference between the computed mean and the actual mean is relatively small, with only a 0.3 unit discrepancy.

Given the proximity of these two values, it suggests that the computed mean is a reasonably accurate estimate of the actual mean.

However, it's important to note that without additional information, such as the sample size or the variability of the data, it is difficult to draw definitive conclusions about the relationship between the computed mean and the actual mean.

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Answer:

Step-by-step explanation:

The given expression is: 14x^(2n+1) + 7x^(n+3) - 21^(n+2)

Unfortunately, it seems there is a missing exponent for the term "21" in the expression. Please provide the correct exponent for 21, and I'll be happy to help you further simplify the expression.

The expression (3√2) / (3√6) with a rationalized denominator is 3√9 / 6. Option C is the correct answer.

To rationalize the denominator in the expression (3√2) / (3√6), we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of √6 is -√6, so we multiply the expression by (-√6) / (-√6):

(3√2 / 3√6) * (-√6 / -√6)

This simplifies to:

-3√12 / (-3√36)

Further simplifying, we have:

-3√12 / (-3 * 6)

-3√12 / -18

Finally, we can cancel out the common factor of 3:

- 3√9 / - 6.

Simplifying further, we get:

3√9 / 6.

Option C is the correct answer.

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Answer:

Hope this helps and have a nice day

Step-by-step explanation:

To find the value of x, we can use the formula:

Time = Distance / Speed

Let's calculate the time taken to travel from Q to P and back to Q.

From Q to P:

Distance = 12 km

Speed = 6 km/h

Time taken from Q to P = Distance / Speed = 12 km / 6 km/h = 2 hours

From P to Q:

Distance = 12 km

Speed = (6 + x) km/h

Time taken from P to Q = Distance / Speed = 12 km / (6 + x) km/h

Given that the total time taken for the round trip is 3 hours 20 minutes, we can convert it to hours:

Total time = 3 hours + (20 minutes / 60) hours = 3 + (1/3) hours = 10/3 hours

According to the problem, the total time is the sum of the time from Q to P and from P to Q:

Total time = Time taken from Q to P + Time taken from P to Q

Substituting the values:

10/3 hours = 2 hours + 12 km / (6 + x) km/h

Simplifying the equation:

10/3 = 2 + 12 / (6 + x)

Multiply both sides by (6 + x) to eliminate the denominator:

10(6 + x) = 2(6 + x) + 12

60 + 10x = 12 + 2x + 12

Collecting like terms:

8x = 24

Dividing both sides by 8:

x = 3

Therefore, the value of x is 3.

Answer:

x = 3

Step-by-step explanation:

speed  = distance / time

time = distance / speed

Total time from P to Q to P:

T = 3h 20min

P to Q :

s = 6 km/h

d = 12 km

t = d/s

= 12/6

t = 2 h

time remaining t₁ = T - t

= 3h 20min - 2h

=  1 hr 20 min

= 60 + 20 min

= 80 min

t₁ = 80/60 hr

Q to P:

d₁ = 12km

t₁ = 80/60 hr

s₁ = d/t₁

[tex]= \frac{12}{\frac{80}{60} }\\ \\= \frac{12*60}{80}[/tex]

= 9

s₁ = 9 km/h

From question, s₁ = (6 + x)km/h

⇒ 6 + x = 9

⇒ x = 3

Answer:

Comparing the centers of the data sets:

- The median for Data Set A is greater than the median for Data Set B.

- The mean for Data Set A is greater than the mean for Data Set B.

Comparing the variability of the data sets:

- The range of Data Set A is 22, while the range of Data Set B is 27. Therefore, the range of Data Set B is greater.

- The standard deviation of Data Set A is greater than the standard deviation of Data Set B, indicating higher variability in Data Set A.

Step-by-step explanation:

a linear relationship or function is described in general as

y = f(x) = ax + b

Because the variable term has the variable x only with the exponent 1, this makes this a straight line - hence the name "linear".

here f(x) is P(x) :

P(x) = ax + b

now we are using both given points (ordered pairs) to calculate a and b :

20 = a×170 + b

2820 = a×220 + b

to eliminate first one variable we subtract equation 1 from equation 2 :

2800 = a×50

a = 2800/50 = 280/5 = 56

now, we use that in any of the 2 original equations to get b :

20 = 56×170 + b

b = 20 - 56×170 = 20 - 9520 = -9500

so,

P(x) = 56x - 9500

Answer:

8

Step-by-step explanation:

let x be the number,

according to the question,

-29 + 28 = -7 + x

1 + 7 = x

thus, x = 8

The range of the given graph is expressed as:

Option A: {-∞, ∞}

The range of a function is defined as the set of all the possible output values of y. The formula to find the range of a function is y = f(x).

In a relation, it is only a function if every x value corresponds to only one y value,

Now, looking at the given graph, we see that At x = 0, the function is also y = 0.

However, between 0 and π intervals, we see that the graph approaches positive and negative infinity and as such we can tell that the range is expressed as: {-∞, ∞}

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Answer: Logarithmic function

Step-by-step explanation:  y=logax and it's a reflection of an exponential curve that curves up and a logarithmic function curves down.

The solution to the system of equations using the inverse matrix method is x = -1, y = 2, z = 3.

To solve the system of equations using the inverse matrix method, we need to represent the system in matrix form.

The given system of equations can be written as:

| 5 -4 1 | | x | = | 12 |

| 1 7 -1 | [tex]\times[/tex]| y | = | -9 |

| 2 3 3 | | z | | 8 |

Let's denote the coefficient matrix on the left side as A, the variable matrix as X, and the constant matrix as B.

Then the equation can be written as AX = B.

Now, to solve for X, we need to find the inverse of matrix A.

If A is invertible, we can calculate X as [tex]X = A^{(-1)} \times B.[/tex]

To find the inverse of matrix A, we can use the formula:

[tex]A^{(-1)} = (1 / det(A)) \times adj(A)[/tex]

Where det(A) is the determinant of A and adj(A) is the adjugate of A.

Calculating the determinant of A:

[tex]det(A) = 5 \times (7 \times 3 - (-1) \times 3) - (-4) \times (1 \times 3 - (-1) \times 2) + 1 \times (1 \times (-1) - 7\times 2)[/tex]

= 15 + 10 + (-13)

= 12.

Next, we need to find the adjugate of A, which is obtained by taking the transpose of the cofactor matrix of A.

Cofactor matrix of A:

| (73-(-1)3) -(13-(-1)2) (1(-1)-72) |

| (-(53-(-1)2) (53-12) (5[tex]\times[/tex] (-1)-(-1)2) |

| ((5(-1)-72) (-(5(-1)-12) (57-(-1)[tex]\times[/tex](-1)) |

Transpose of the cofactor matrix:

| 20 -7 -19 |

| 13 13 -3 |

| -19 13 36 |

Finally, we can calculate the inverse of A:

A^(-1) = (1 / det(A)) [tex]\times[/tex] adj(A)

= (1 / 12) [tex]\times[/tex] | 20 -7 -19 |

| 13 13 -3 |

| -19 13 36 |

Multiplying[tex]A^{(-1)[/tex] with B, we can solve for X:

[tex]X = A^{(-1)}\times B[/tex]

= | 20 -7 -19 | | 12 |

| 13 13 -3 | [tex]\times[/tex] | -9 |

| -19 13 36 | | 8 |

Performing the matrix multiplication, we can find the values of x, y, and z.

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Let's use the fact that the sum of the angles of a triangle is always 180 degrees to solve this problem. Let the two equal angles be x, then the third angle is x + 45.Let's add all the angles together:x + x + x + 45 = 180Simplifying this equation, we get:3x + 45 = 180Now, we need to isolate the variable on one side of the equation. We can do this by subtracting 45 from both sides of the equation:3x = 135Finally, we can solve for x by dividing both sides of the equation by 3:x = 45Therefore, the value of x is 45 degrees.

Answer:

Step-by-step explanation:

180 = x + x + x + 45

180 - 45 = 3x

135 = 3x

x = 135 : 3

x = 45°

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

check

180 = 45 + 45 + 45 + 45

180 = 180

same value the answer is good

Based on the provided data sets, it can be observed that both Data Set K and Data Set L have identical values. Therefore, their centers and spreads are also identical.

The center of a data set can be measured using various statistical measures such as the mean, median, or mode. Since the data sets have the same values, all these measures will yield the same result for both sets.

In this case, the center of Data Set K is equal to the center of Data Set L.

Similarly, the spread of a data set refers to the measure of variability or dispersion within the data. Common measures of spread include the range, variance, and standard deviation.

However, since the data sets are exactly the same, all these measures will yield identical results for both sets. Thus, the spread of Data Set K is the same as the spread of Data Set L.

In summary, both the center and the spread of Data Set K are the same as those of Data Set L. Therefore, there is no difference between the two data sets in terms of their central tendency or variability.

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Answer:

[tex]\textsf{The parabola has a vertex at $\left(\:\boxed{-3}\:,\boxed{-7}\:\right)$, has a p-value of $\boxed{-1}$ and it}[/tex]

[tex]\textsf{$\boxed{\sf op\:\!ens\;to\;the\;left}$\:.}[/tex]

Step-by-step explanation:

The given directrix of the parabola is x = -2, which is a vertical line.

The directrix is perpendicular to the axis of symmetry. Therefore, this means that the parabola has a horizontal axis of symmetry.

The focus of a parabola is a fixed point located inside the curve. The x-coordinate of the given focus is x = -4. As this is to the left of the directrix, it means that the parabola opens to the left.

The standard form of a horizontal parabola is:

[tex]\boxed{(y-k)^2=4p(x-h)}[/tex]

where:

As the focus is (-4, -7), then:

[tex]\begin{aligned}(h+p, k)&=(-4,-7)\\\\\implies k&=-7\\\implies h+p&=-4\end{aligned}[/tex]

As the directrix is x = -2, then:

[tex]h - p=-2[/tex]

To find the value of h, sum the equations involved h and p to eliminate p:

[tex]\begin{array}{crcccr}&h &+& p& =& -4\\+&h& -& p& = &-2\\\cline{2-6}&2h&&& =& -6\\\cline{2-6}\\\implies &h&&&=&-3\end{array}[/tex]

To find the value of p, substitute the found value of h into one of the equations:

[tex]\begin{aligned}-3 - p&=-2\\p&=-3+2\\p&=-1\end{aligned}[/tex]    

Therefore, the values of h, k and p are:

The parabola has a vertex at (-3, -7), has a p-value of -1 and it opens to the left.

The parabola has a vertex at (-3, y), has p-value of 1 and it equation is

(x + 3)² = 4y.

To find the equation of the parabola with the given focus and directrix, we can use the standard form equation of a parabola:

(x - h)² = 4p(y - k)

where (h, k) is the vertex of the parabola and "p" is the distance from the vertex to the focus (and also from the vertex to the directrix).

Given:

Focus: (-4, -7)

Directrix: x = -2

1. Finding the vertex:

Since the directrix is a vertical line, the vertex lies on the line that is equidistant from the focus and directrix. In this case, it lies on the line x = (-4 + (-2))/2 = -3.

Therefore, the vertex of the parabola is (-3, y).

2. Finding the p-value:

The distance from the vertex to the focus (and also to the directrix) is the same. In this case, the distance is |-3 - (-4)| = 1.

Therefore, the value of "p" is 1.

3. Writing the equation of the parabola:

Using the vertex (-3, y) and the p-value of 1, we can write the equation of the parabola:

(x - h)² = 4p(y - k)

(x - (-3))² = 4(1)(y - y)

Simplifying, we get:

(x + 3)² = 4(y - y)

(x + 3)² = 4y

So, the equation of the parabola is (x + 3)² = 4y.

The vertex of the parabola is (-3, y) and the p-value is 1.

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Answer:

102

Step-by-step explanation:

we substitute x by 7

7^2+9(7)-10

49+63-10

=102

Trent will need approximately 2.86 bottles of waterproof spray to cover his tent.

To calculate the number of bottles of waterproof spray Trent will need to cover his tent, we first need to find the surface area of the tent.

The surface area of a square-based pyramid is given by the formula:

Surface Area = Base Area + (0.5 x Perimeter of Base x Slant Height)

The base of the pyramid is a square with a side length of 14 feet, so the base area is:

Base Area = (Side Length)^2 = 14^2 = 196 square feet

To find the slant height of the pyramid, we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by one side of the base, the height of the pyramid, and the slant height. The height of the pyramid is given as 8 feet, and half the length of the base is 7 feet.

Using the Pythagorean theorem:

[tex]Slant Height^2 = (Half Base Length)^2 + Height^2[/tex]

[tex]Slant Height^2 = 7^2 + 8^2Slant Height^2 = 49 + 64Slant Height^2 = 113Slant Height ≈ √113 ≈ 10.63 feet[/tex]

Now we can calculate the surface area of the tent:

Surface Area = 196 + (0.5 x 4 x 10.63)

Surface Area = 196 + (2 x 10.63)

Surface Area = 196 + 21.26

Surface Area ≈ 217.26 square feet

Since each bottle of waterproof spray covers 76 square feet, we can divide the total surface area of the tent by the coverage of each bottle to find the number of bottles needed:

Number of Bottles = Surface Area / Coverage per Bottle

Number of Bottles = 217.26 / 76

Number of Bottles ≈ 2.86

Therefore, Trent will need approximately 2.86 bottles of waterproof spray to cover his tent. Since we can't have a fraction of a bottle, he will need to round up to the nearest whole number. Therefore, Trent will need 3 bottles of waterproof spray to fully waterproof his tent.

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Answer: -15/17

Step-by-step explanation:

sin ∅ = 8/17

If you drew a a line in the 2rd quadrant because tan ∅ <0, which means tan∅ is negative.

Tan∅= sin∅/cos∅            

they told you sin∅ is positive which is related to your y.

but cos∅ needs to be negative which is related to x

This happens in the second quadrant x is negative and y is positive.

Now we know which way to draw our line.  Label the opposite of the angle 8 and the hypotenuse 17 because sin∅ = 8/17

Use pythagorean to find adjacent.

17² = 8² + a²

225 = a²

a = 15

The adjacent is negative because the adjacent is on the x-axis in the negative direction.

cos ∅ = adj/hyp

cos∅ =  -15/17

Answer: -15/17

Step-by-step explanation:

In the first quadrant, all sine, cosine and tan are all positive. In the second quadrant, only sine is positive. Third quadrant, only tangent is positive and fourth quadrant, only cosine is positive.

Therefore, when sine is positive and tan is negative, the angle can only be in quadrant 2. Then draw the triangle. Draw a triangle with the angle from the origin, with the opposite leg from the angle with value of 8 and the hypotenuse of value 17. Since cosine is what the question is asking for, and we know the data given forms an right triangle, the value of the other leg is 15. It is an 8-15-17 special triangle or use the Pythagorean Theorem.

Finally, taking the cosine of the angle from the origin gives -15/17 since it is in quadrant 2.

Answer:

f(3r - 1) = -6r + 6

Step-by-step explanation:

To find f(3r - 1), we substitute 3r - 1 for x in the expression for f(x) and simplify:

f(x) = 4 - 2x

f(3r - 1) = 4 - 2(3r - 1)

= 4 - 6r + 2

= -6r + 6

So, f(3r - 1) = -6r + 6.

Answer: (5, -1)

Step-by-step explanation:

To rotate a point counterclockwise by 90° about the origin, we swap the x and y coordinates and negate the new x-coordinate. For the point (-1, 5), we swap the x and y coordinates to get (5, -1). The x-coordinate becomes positive, and the y-coordinate becomes negative. Therefore, the coordinates of the image of the point (-1, 5) after a counterclockwise rotation of 90° about the origin are (5, -1).

I think you put down the same answer choice twice and instead meant to say (5, -1) instead of (-5, -1) twice.

Answer:

Step-by-step explanation:

Let's calculate the expression step by step:

93 - (15 × 10) + (160 ÷ 16)

First, we perform the multiplication:

93 - 150 + (160 ÷ 16)

Next, we perform the division:

93 - 150 + 10

Finally, we perform the subtraction and addition:

-57 + 10

The result is:

-47

Therefore, 93 - (15 × 10) + (160 ÷ 16) equals -47.

Answer:

Step-by-step explanation:

The correct answer is A. A rectangle the same size as the base.

When a rectangular pyramid is sliced parallel to the base, the resulting cross-section is a rectangle that is the same size as the base. The parallel slicing ensures that the cross-section maintains the same dimensions as the base of the pyramid. Therefore, option A, a rectangle the same size as the base, represents the cross-section.

When 2 items are selected without replacement from a population of 6 items, there are 15 possible samples that can be formed. Option A.

To determine the number of possible samples when 2 items are selected at random without replacement from a population of 6 items, we can use the concept of combinations.

The number of combinations of selecting k items from a set of n items is given by the formula C(n, k) = n! / (k! * (n-k)!), where n! represents the factorial of n.

In this case, we have a population of 6 items and we want to select 2 items. Therefore, the number of possible samples can be calculated as:

C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5 * 4!) / (2! * 4!) = (6 * 5) / (2 * 1) = 15. Option A is correct.

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Answer: O (4, -4)

Step-by-step explanation:

To solve the system of equations using elimination, we can multiply the second equation by -3 to eliminate the y term:

Original equations:

5x + 3y = 8 (Equation 1)

4x + y = 12 (Equation 2)

Multiply Equation 2 by -3:

-3(4x + y) = -3(12)

-12x - 3y = -36 (Equation 3)

Now we can add Equation 1 and Equation 3 to eliminate the y term:

(5x + 3y) + (-12x - 3y) = 8 + (-36)

Simplifying:

5x - 12x + 3y - 3y = 8 - 36

-7x = -28

Divide both sides by -7:

x = -28 / -7

x = 4

Now substitute the value of x back into either of the original equations, let's use Equation 2:

4(4) + y = 12

16 + y = 12

y = 12 - 16

y = -4

Therefore, the solution to the system of equations is x = 4 and y = -4.

Answer:

Option C is correct.

The kind of growth model is shown in the table is exponential

Step-by-step explanation:

Exponential growth function is in the form of :    ......[1];   where a is the initial value and b> 0.

Consider any two point from the table:

(1 , 5) and ( 2 , 25)

Substitute these in the equation [1] we get;

        ......[2]

     ......[3]

Divide equation [3] by [2] we have;

Simplify:

Now substitute this value in equation [2] we get;

Divide both sides by 5 we get;

Simplify:

1=a or a = 1

Therefore, the table shown the  exponential growth function y=5^x

Answer:

56

Step-by-step explanation:

f(x)=4x^2-7

f'(x)=8x

f'(7)=56

Step-by-step explanation:

a probability is always the ratio of

desired cases / totally possible cases.

so, in this case it is the

area of the triangle / area of the circle.

as everything of the triangle is also a part of the circle.

and so, that fraction of the area of the whole circle that is the area of the triangle in refutation to the area of the whole circle is the probability that a random point inside the circle would be also inside the triangle.

the area of a right-angled triangle is

leg1 × leg2 / 2

in our case

12 × 12 / 2 = 72 units²

the area of a circle is

pi × r²

in our case that is

pi × 12² = 144pi units²

the requested probability is

P = 72 / 144pi = 1/2pi = 0.159154943... ≈ 0.16

The perimeter of the rectangle is 140 inches.

Let's denote the width of the rectangle as w. According to the given information, the length of the rectangle is six times its width, so we can express the length as 6w.

The area of a rectangle is given by the formula A = length × width. Substituting the values we have:

A = (6w) × w

600 = 6w^2

To solve for w, we divide both sides of the equation by 6:

w^2 = 100

Taking the square root of both sides:

w = ±10

Since width cannot be negative in this context, we discard the negative value and consider the positive value, w = 10.

Now that we have the width, we can find the length of the rectangle:

Length = 6w = 6 × 10 = 60

The perimeter of a rectangle is given by the formula P = 2(length + width). Substituting the values:

P = 2(60 + 10)

P = 2(70)

P = 140

Therefore, the perimeter of the rectangle is 140 inches.

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Answer:

45 inches represent 55 miles on the scale drawing.

Step-by-step explanation:

To solve this proportion, we can set up the following ratio:

9 inches / 11 miles = x inches / 55 miles

We can cross-multiply to solve for x:

9 inches * 55 miles = 11 miles * x inches

495 inches = 11 miles * x inches

Now, we can isolate x by dividing both sides by 11 miles:

495 inches / 11 miles = x inches

Simplifying the expression:

45 inches = x inches

1a. Compute NPV by calculating the present value of net cash flows and subtracting the investment amount.

1b. Compute PVI by dividing NPV by the investment amount.

2. Determine IRR by finding the discount rate corresponding to an NPV of zero.

3. Compare NPV, PVI, and IRR to identify the better financial opportunity.

1a. To compute the net present value (NPV) for each project, we need to calculate the present value of the annual net cash flows and subtract the amount to be invested. Using the present value of an annuity of $1 from the table, we can fill in the following values:

Wind Turbines:

Present value of annual net cash flows: $420,000 * 1.736 + $420,000 * 2.487 + $420,000 * 3.170 + $420,000 * 3.791

Less amount to be invested: $1,199,100

Net present value: NPV_Wind_Turbines = Present value of annual net cash flows - Amount to be invested

Biofuel Equipment:

Present value of annual net cash flows: $880,000 * 1.736 + $880,000 * 2.487 + $880,000 * 3.170 + $880,000 * 3.791

Less amount to be invested: $2,278,320

Net present value: NPV_Biofuel_Equipment = Present value of annual net cash flows - Amount to be invested

1b. The present value index (PVI) can be calculated by dividing the NPV by the amount to be invested:

Present Value Index = NPV / Amount to be invested

2. To determine the internal rate of return (IRR) for each project, we need to find the discount rate at which the NPV becomes zero. We can use the present value of an annuity of $1 from the table to calculate the present value factor for an annuity of $1. Then, we can find the discount rate that corresponds to an NPV of zero.

Wind Turbines:

Present value factor for an annuity of $1: Fill in the values from the table

Internal rate of return: IRR_Wind_Turbines = Discount rate corresponding to NPV = 0

Biofuel Equipment:

Present value factor for an annuity of $1: Fill in the values from the table

Internal rate of return: IRR_Biofuel_Equipment = Discount rate corresponding to NPV = 0

3. Based on the calculations of NPV, PVI, and IRR, we can compare the two projects. The project with the higher NPV, PVI, and IRR is considered the better financial opportunity. Both investments meet the minimum return criterion of 10%, but the project with the higher financial indicators is preferred.

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┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈

✦ The number is - 5

┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈┈

[tex]\begin{gathered} \; \sf{\color{pink}{Let \; the \; other \; number \; be \; (x)::}} \\ \end{gathered}[/tex]

Atq,,

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x + ( - 7) \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3 \times \bigg \lgroup \: x - 7 \bigg \rgroup = - 36} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x - 21 = - 36} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 36 + 21} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{3x = - 15} \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{skyblue} :\dashrightarrow \: \tt{x = \dfrac{\cancel{ - 15}}{\cancel{ \: 3}}} \qquad \bigg \lgroup \sf{Cancelling \: by \: 3} \bigg \rgroup \\ \\ \end{gathered}[/tex]

[tex]\begin{gathered} \; \color{pink} :\dashrightarrow \underline{\color{pink}\boxed{\colorbox{black}{x = - 5}}} \: \pmb{\bigstar} \\ \\ \end{gathered}[/tex]

The answer is:

Work/explanation:

The product means we multiply two numbers.

Here, we multiply 3 and a number increased by -7; let that number be z.

So we have

[tex]\sf{3(z+(-7)}[/tex]

simplify:

[tex]\sf{3(z-7)}[/tex]

This equals -36

[tex]\sf{3(z-7)=-36}[/tex]

[tex]\hspace{300}\above2[/tex]

[tex]\frak{solving~for~z}[/tex]

Distribute

[tex]\sf{3z-21=-36}[/tex]

Add 21 on each side

[tex]\sf{3z=-36+21}[/tex]

[tex]\sf{3z=-15}[/tex]

Divide each side by 3

[tex]\boxed{\boxed{\sf{z=-5}}}[/tex]

You will need to purchase approximately 1/42 of a tube, which is less than a full tube. In practical terms, you would need to purchase at least one tube to meet your requirement of 42 ounces of capsaicin arthritis rub.

To determine the number of tubes of capsaicin arthritis rub you will need to purchase, we can divide the total required quantity by the quantity in each tube.

Given that the cost of capsaicin arthritis rub is $21 and you need to buy 42 ounces, we need to find out how many ounces are in each tube.

Let's assume that each tube contains x ounces of capsaicin arthritis rub.

Now we can set up a proportion to solve for x:

42 ounces / x tubes = 1 tube / x ounces

Cross-multiplying gives us:

42x = 1 * x

Simplifying the equation:

42x = x

Dividing both sides of the equation by x (since x cannot be zero):

42 = 1

Since this equation is not true, it means that there is an error in our assumption. We need to revise our assumption.

Let's assume that each tube contains 1 ounce of capsaicin arthritis rub.

Now we can set up a new proportion:

42 ounces / x tubes = 1 tube / 1 ounce

Cross-multiplying gives us:

42x = 1 * 1

Simplifying the equation:

42x = 1

Dividing both sides of the equation by 42:

x = 1/42

As a result, you will need to buy less than a full tube—roughly 1/42 of a tube. In order to get the 42 ounces of capsaicin arthritis rub you need, you would essentially need to buy at least one tube.

for such more question on purchase

https://brainly.com/question/28787564

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