Part4: (Surn and difference identities) Show all your work. a) Fadel calculated sin(135) as shown below: V2 V2+2 sin(135) = sin(90 + 45)° = sin 90° + sin 45º = 1 + 2 Is Fadel correct in his calculation, if NOT, Correct the calculation to get the correct answer.(3marks) DA b) Use the formula sin(A + B) = sin A cos B + cos A sin B to show sin 2x = 2 sin x cos x (Hint: Take A=B=x) (2marks) c) Use the formula cos(A + B) = cos A cos B-sin A sin B to show cos 2x = 1 - 2 sin²x. (5marks) d) Use the formula sin(A + B) = sin A cos B + cos A sin B AND the answers of parts b and c to show that sin 3x = 3 sin x-4 sin³ x

Answer:

#6) A = 72 sq units

Step-by-step explanation:

area of ΔAOB = 6

area of ΔCOB = 6

area of ΔCOD = 30

area of ΔCOD = 30

Answer:Answer: (a) = P = 2(x + 2x-5) = 2(3x-5) = 6x - 10.

Step-by-step explanation:

Part a : Polynomial represents the perimeter of the garden is 6W-10

Part b : Polynomial represents the area of the garden is [tex]2w^{2}-5w[/tex]

Part c : Perimeter = 38 feet and area = 88 feet square

A polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables

A perimeter is a closed path that encompasses, surrounds

Area is the quantity that expresses the extent of a region on the plane or on a curved surface

Given,

Length is five less than two times the width

Consider

L is the length and W is the width

Then,

L=2W-5

Part a

Perimeter = 2(L+W)

Substitute the value of L

P=2(L+W)

P=2((2W-5)+W)

P=2(2W-5+W)

P=2(3W-5)

Perimeter= 6W-10

Part b

Area of the garden = L×W

Area = (2W-5)W

Area = [tex]2w^{2}-5w[/tex]

Part c

Given width W = 8 feet

Perimeter = 6W-10

P=6×8-10

Perimeter =38 feet

Area = [tex]2w^{2}-5w[/tex]

Area=[tex]2(8^{2})-5(8)[/tex]

Area = 88 feet square

Hence,

Part a : Polynomial represents the perimeter of the garden is 6W-10

Part b : Polynomial represents the area of the garden is [tex]2w^{2}-5w[/tex]

Part c : Perimeter = 38 feet and area = 88 feet square

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Answer:

61

Step-by-step explanation:

add the numbers up and divide them by 7

Answer:

Step-by-step explanation:

added up the data = 427

divide by the number of days = 427/7 = 61

Answer:

all I know is the answer for 5 so it is 2.0

Step-by-step explanation:

Answer:

4. 79.17%

5. 2.5

Step-by-step explanation:

4.  Q3 - Q1 = 4.75 - 0 = 4.75

      [tex]\frac{4.75}{6} *100[/tex] = 79.17

5.  Q3 - Q1 = 4.75 - 2.25 = 2.5

The box and whisker plot of the prices of some DVDs:

Minimum: 9.99

First Quartile: 12.99

Median: 15.99

Third Quartile: 19.99

Maximum: 26.99

The box and whisker plot shows that the median price of a DVD is $15.99. The prices range from $9.99 to $26.99. There are two outliers, one at $9.99 and one at $26.99.

The box and whisker plot can be used to identify the distribution of the data. In this case, the data is slightly skewed to the right, meaning that there are more DVDs priced at the lower end of the range than at the higher end.

The box and whisker plot can also be used to compare different sets of data. For example, we could compare the prices of DVDs from different stores or from different years.

Overall, the box and whisker plot is a useful tool for visualizing and summarizing data.

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So the answer is not among the options provided. To find the equation that represents the relationship between the length of the shorter side (x) and the perimeter of the rectangle,

we can use the given information.

Let's denote the length of the shorter side as x. According to the problem, the other side is 6 feet longer, so the length of the longer side would be x + 6.

The perimeter of a rectangle is given by the sum of all its sides. In this case, the perimeter is 100 feet.

The equation to find the length x of the shorter side can be written as:

2x + 2(x + 6) = 100

Simplifying the equation, we have:

2x + 2x + 12 = 100

4x + 12 = 100

4x = 100 - 12

4x = 88

x = 88/4

x = 22

therefore, the correct equation to find the length x of the shorter side is:

2x + 2(x + 6) = 100

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The equation to find the length x of the shorter side is x + (x - 6) = 50.

Option (d) is the correct answer.

Given that the perimeter of a rectangle is 100 feet, and one side is 6 feet longer than the other side.

Let's suppose that x represents the length of the shorter side of the rectangle.

Therefore, the length of the longer side of the rectangle is (x + 6).

Using the formula of the perimeter of a rectangle, we get:

Perimeter = 2 × (Length + Width)

According to the problem,

Perimeter = 100

Length = (x + 6)

Width = x

Substituting the values,

Perimeter = 2 × (Length + Width)

100 = 2 × [(x + 6) + x]

50= 2x + 6

x = (50 - 6)/2

x = 22

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Answer:

100degrees

Step-by-step explanation:

Find the diagram attached

From the diagram

<PQT =<TRS = 40

Since the triangle STR is isosceles, here,

<TSR =<TRS = 40

Also the sum of angle in a triangle is 189, hence:

Arc SR+<TSR +<TRS = 18₩

ArcSR +40+40=180

ArcSR +80=180

ArcSR = 180-80

ArcSR = 100degrees

Hence the measure of SR is 100degrees

The measure of arc SR is 100°. The correct option is the third option - 100°

From the question, we are to determine the measure of arc SR.

The measure of arc SR = <STR

Now, we will determine the measure of <STR

In the diagram, T is the center of the circle.

∴ TP and TQ are radii.

Then, we can conclude that ΔPQT is an isosceles triangle.

Recall: Base angles of an isosceles triangle are equal.

∴ <PQT = <TPQ = 40°

Now, consider ΔPQT

<QTP + <TPQ + <PQT = 180° (Sum of angles in a triangle)

<QTP + 40° + 40° = 180°

<QTP + 80° = 180°

<QTP = 180° - 80°

<QTP = 100°

Also, in the diagram, <QTP and <STR are vertically opposite angles

NOTE: Vertically opposite angles are equal

That is, <QTP = <STR

∴ <STR = 100°

Hence, the measure of arc SR is 100°. The correct option is the third option 100°

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Here is the complete and correct question:

Line PR and Line QS are diameters of circle T. What is the measure of Arc SR?

50°

80°

100°

120°

Please find the attached image

The probability that an individual distance is greater than 211.80 cm is 0.1292.

The problem statement is:

The overhead reach distances of adult females are normally distributed with mean of 202.5 cm and standard deviation of 8.3 cm Find the probability that an individual distance is greater than 211.80 cm.

We need to find the z-score first as follows:$\begin{aligned}z&=\frac{x-\mu}{\sigma} \\z&=\frac{211.80-202.5}{8.3} \\z&=1.122\end{aligned}$

Using the standard normal table,

The standard normal distribution table is a compilation of areas from the standard normal distribution, more commonly known as a bell curve, which provides the area of the region located under the bell curve and to the left of a given z-score to represent probabilities of occurrence in a given population.

The probability is given by:P(Z > 1.122) = 0.1292

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The given information is Population Mean (µ) is 202.5 cm

Standard deviation (σ) is 8.3 cm

The distance we have to find is x = 211.80 cm.

Hence, the probability that an individual distance is greater than 211.80 cm is 0.1314.

The formula used for finding the probability is: Now we need to find z score to use the standard normal distribution tables. The formula for finding z score is:

z = (x - µ) / σ

Substitute the values in the above formula, we get

z = (211.80 - 202.5) / 8.3

z = 1.12

Now use this z value to look up in the standard normal distribution tables to find the probability.

P(z > 1.12) = 1 - P(z < 1.12)

From standard normal distribution tables, the probability of P(z < 1.12) is 0.8686.

Therefore, P(z > 1.12) = 1 - P(z < 1.12)

= 1 - 0.8686

= 0.1314

Hence, the probability that an individual distance is greater than 211.80 cm is 0.1314.

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The probability of selecting 2 out of 4 patients with heart disease from a group of 15 patients, where 5 of them have the disease, can be calculated using the combination formula. The probability is approximately 0.595.

B. Explanation:

To calculate the probability, we need to use the concept of combinations. The formula for calculating combinations is given by:

C(n, k) = n! / (k!(n-k)!)

Where n is the total number of elements and k is the number of elements we want to choose.

In this case, we have a total of 15 patients, out of which 5 have the heart disease. We want to choose 2 patients with heart disease from a group of 4 patients.

The probability can be calculated as:

P(2 patients with heart disease) = C(5, 2) / C(15, 4)

C(5, 2) represents the number of ways to choose 2 patients with the heart disease from the group of 5 patients, and C(15, 4) represents the total number of ways to choose 4 patients from the group of 15 patients.

Using the combination formula, we can calculate C(5, 2) and C(15, 4) as follows:

C(5, 2) = 5! / (2!(5-2)!) = 10

C(15, 4) = 15! / (4!(15-4)!) = 1365

Substituting these values into the probability formula:

P(2 patients with heart disease) = 10 / 1365 ≈ 0.007

Therefore, the probability of selecting 2 out of 4 patients with the heart disease from the given group is approximately 0.595.

Moving on to the second part of the question, to find the probability that the clinic will have 4 patients in the next hour, we need to determine the average number of patients per hour and use the Poisson distribution.

The average number of patients per hour is given as 3. The Poisson distribution formula is:

P(x; λ) = (e^(-λ) * λ^x) / x!

Where P(x; λ) is the probability of x events occurring in a given interval, λ is the average rate of events, e is the base of the natural logarithm, and x! denotes the factorial of x.

In this case, we want to find P(4; 3), which represents the probability of having 4 patients when the average rate is 3.

Substituting the values into the formula:

P(4; 3) = (e^(-3) * 3^4) / 4!

Calculating the values:

P(4; 3) ≈ 0.168

Therefore, the probability that the clinic will have 4 patients in the next hour is approximately 0.168.

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The radius of the simple curve is not provided and b. The stationing of the PC and PT is not provided.

The given information is insufficient to determine the radius of the simple curve or the stationing of the PC (Point of Curvature) and PT (Point of Tangency).

To determine the radius of the simple curve, additional information is needed, such as the angle between the two tangent lines or the length of the third tangent line AB. Without this information, we cannot calculate the radius.

Similarly, the stationing of the PC and PT requires more details, such as the length of the curve or the degree of curvature. The information provided in the question does not include these parameters, making it impossible to determine the stationing of the PC and PT.

Therefore, based on the given information, we cannot determine the radius of the simple curve or the stationing of the PC and PT.

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Step-by-step explanation:

You can apply cosinus theory for finding are

Area=cos40°3.4(ft)*2.7(ft)/2 like thia

Quotient: The result of dividing two numbers

Explanation: Just some simple dividing and rounding

Quotient - 102.756098

Rounding - 102.76

Answer: 102.76

Answer:

ED = 105

Step-by-step explanation:

Answer:

ED=105 and x=12

Step-by-step explanation:

The exact value of the integral [tex]\int\limits^0_{\pi/6}[/tex] 2/cos²θ dθ is √3/3 using the Fundamental Theorem of Calculus.

To calculate the integral ∫[0 to π/6] 2/cos²θ dθ using the Fundamental Theorem of Calculus, we need to find the antiderivative of the integrand and evaluate it at the upper and lower limits of integration.

The antiderivative of 2/cos²θ is tan(θ), so applying the Fundamental Theorem of Calculus:

[tex]\int\limits^0_{\pi/6}[/tex] 2/cos²θ dθ = [tan(θ)] evaluated from θ = 0 to θ = π/6

Substituting the upper and lower limits of integration:

= tan(π/6) - tan(0)

Since tan(0) = 0, we have:

= tan(π/6)

Using the value of tan(π/6) = √3/3, the exact value of the integral is √3/3.

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The question is -

Use the Fundamental Theorem to calculate the following integral exactly:

[tex]\int\limits^0_{\pi/6}[/tex] 2/cos²∅ d∅ = _______

1.

a. 8 x 7 x 1/2 = 56 x 1/2 = 28

b. 24 x 14 x 1/2 = 336 x 1/2 = 168

c. 5 x 12 x 1/2 = 60 x 1/2 = 30

d. 4 x 4.8 x 1/2 = 19.2 x 1/2 = 9.6

2.

two triangles: 2(4 x 8 x 1/2) = 4 x 8 = 32

rectangle: 15 x 8 = 120

total area: 32 + 120 = 152

The half-life of the radioactive goo is 48 minutes. To determine the half-life of the radioactive goo, we use the formula N(t) = N₀ * (1/2)^(t / T).

where N(t) is the amount of the radioactive substance at time t, N₀ is the initial amount, T is the half-life, and t is the time elapsed. Given N₀ = 120 grams and N(240) = 3.75 grams, we substitute these values into the formula and solve for T. Plugging in the given values, we have: 3.75 = 120 * (1/2)^(240 / T)

To find the half-life T, we need to isolate it on one side of the equation. We can begin by dividing both sides of the equation by 120:

3.75 / 120 = (1/2)^(240 / T)

0.03125 = (1/2)^(240 / T)

Next, we can take the logarithm base 2 of both sides to eliminate the exponential term: log₂(0.03125) = log₂[(1/2)^(240 / T)]

-5 = (240 / T) * log₂(1/2)

Simplifying further, we know that log₂(1/2) is equal to -1: -5 = (240 / T) * (-1)

To solve for T, we can multiply both sides by -T/240: 5T/240 = 1

Multiplying both sides by 240/5, we find: T = 48

Therefore, the half-life of the radioactive goo is 48 minutes.

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The average braking distance for small cars traveling at 65 miles per hour significantly differs from the advertised value of 120 feet.

In this case, we want to determine if the average braking distance is significantly different from 120 feet. Since the researcher wants to detect any difference, whether it is shorter or longer than 120 feet, the alternative hypothesis will be two-tailed.

H0: The average braking distance for small cars traveling at 65 miles per hour is 120 feet.

Ha: The average braking distance for small cars traveling at 65 miles per hour is not equal to 120 feet.

To conduct the hypothesis test, we will use the sample data provided by the researcher. The sample size is 34, and the sample average braking distance is 115 feet. The population standard deviation is given as 20 feet.

The formula for the test statistic (z-score) is:

z = (sample average - hypothesized population average) / (population standard deviation / √sample size)

Plugging in the values from the problem:

z = (115 - 120) / (20 / √34)

z = -5 / (20 / √34)

Using Table 1 or a statistical calculator, we can determine the critical z-value corresponding to a significance level of 0.01. Since we have a two-tailed test, we need to split the significance level in half. Each tail will have an alpha of 0.005 (0.01/2).

Looking up the z-value for α/2 = 0.005, we find it to be approximately 2.576.

Now we compare the calculated z-value to the critical z-value:

If the calculated z-value falls outside the range defined by the critical z-values, we reject the null hypothesis. Otherwise, if the calculated z-value falls within the range, we fail to reject the null hypothesis.

In our case, the calculated z-value is -5 / (20 / √34), which we need to compare to -2.576 and +2.576.

If the calculated z-value is less than -2.576 or greater than +2.576, we reject the null hypothesis. Otherwise, if the calculated z-value is between -2.576 and +2.576, we fail to reject the null hypothesis.

By performing the calculation, we find that the calculated z-value falls outside the range defined by -2.576 and +2.576. Therefore, we can reject the null hypothesis.

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The probability that all four tosses land on heads given that the first toss lands on heads is 12.5%.

Let's denote the event "first toss lands on heads" as A, and the event "all four tosses land on heads" as B. We need to find the probability of B given that A has occurred, denoted as P(B|A).

The probability of event B occurring is simply the probability of getting four heads in four tosses, which is (1/2)^4 = 1/16.

The probability of event A occurring is 1/2 since there are two equally likely outcomes for the first toss (heads or tails).

To find the conditional probability P(B|A), we use the formula:

P(B|A) = P(A ∩ B) / P(A)

Since event B implies event A (if all four tosses land on heads, then the first toss must land on heads), we have P(A ∩ B) = P(B). Therefore:

P(B|A) = P(B) / P(A) = (1/16) / (1/2) = 1/8.

Converting the probability to a percentage, we have:

P(B|A) = (1/8) × 100 = 12.5%.

Therefore, the probability that all four tosses land on heads given that the first toss lands on heads is 12.5%.

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Answer:

60 and 90

Step-by-step explanation:

Mario has gross biweekly earnings of $784.21. By claiming 1 more withholding allowance, Mario would have $13 more in his take home pay. so the correct answer is b. 4.

By claiming 1 more withholding allowance, Mario would have $13 more in his take-home pay. This suggests that each withholding allowance reduces Mario's taxable income by $13. To find out how many withholding allowances Mario currently claims, we can calculate the difference between his gross earnings and take-home pay without any allowances. If we subtract $13 from the take-home pay ($784.21 - $13 = $771.21) and compare it to the original gross earnings, we can determine how many withholding allowances Mario currently claims. In this case, the difference is $13, indicating that Mario currently claims 4 withholding allowances.

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The critical points function relative extrema and saddle points.

(a) f(x, y) = x - x2 - y2 =f(x, y) = 0: x - x² - y²= 0

(b) f(x, y) = (x + 1)(y – 2)=: x + 1 = 0 and y - 2 = 0.

(c) f(x, y) = sin(xy)= cos(xy),

(d) f(x, y) = xy(x - 1)= (0, 0) and (1, y)

(a) For the function f(x, y) = x - x² - y²

To find the critical points, to find where the gradient is zero or undefined. The gradient of f(x, y) is given by (∂f/∂x, ∂f/∂y):

∂f/∂x = 1 - 2x

∂f/∂y = -2y

Setting both partial derivatives to zero,

1 - 2x = 0 -> x = 1/2

-2y = 0 -> y = 0

The only critical point is (1/2, 0).

To determine the nature of the critical point, examine the second-order partial derivatives:

∂²f/∂x² = -2

∂²f/∂y² = -2

∂²f/∂x∂y = 0

The determinant of the Hessian matrix is Δ = ∂²f/∂x² * ∂²f/∂y² - (∂²f/∂x∂y)² = (-2)(-2) - (0)² = 4.

Since Δ > 0 and ∂²f/∂x² = -2 < 0, the critical point (1/2, 0) is a local maximum.

To sketch the level sets,  set f(x, y) to different constant values and plot the corresponding curves. For example:

f(x, y) = -1: x - x² - y² = -1

This equation represents a circle with radius 1 centered at (1/2, 0).

f(x, y) = 0: x - x² - y² = 0

This equation represents a parabolic shape that opens downward.

(b) For the function f(x, y) = (x + 1)(y - 2):

To find the critical points, we set both partial derivatives to zero:

∂f/∂x = y - 2 = 0 -> y = 2

∂f/∂y = x + 1 = 0 -> x = -1

The only critical point is (-1, 2).

To determine the nature of the critical point, we can examine the second-order partial derivatives:

∂²f/∂x² = 0

∂²f/∂y² = 0

∂²f/∂x∂y = 1

Since the second-order partial derivatives are all zero, we cannot determine the nature of the critical point based on them. We need further analysis.

To sketch the level sets,  set f(x, y) to different constant values and plot the corresponding curves. For example:

f(x, y) = 0: (x + 1)(y - 2) = 0

This equation represents two lines: x + 1 = 0 and y - 2 = 0.

(c) For the function f(x, y) = sin(xy):

To find the critical points, both partial derivatives to zero:

∂f/∂x = ycos(xy) = 0 -> y = 0 or cos(xy) = 0

∂f/∂y = xcos(xy) = 0 -> x = 0 or cos(xy) = 0

From y = 0 or x = 0, the critical points (0, 0).

When cos(xy) = 0,  xy = (2n + 1)π/2 for n being an integer. In this case,  infinitely many critical points.

To determine the nature of the critical points, we can examine the second-order partial derivatives:

∂²f/∂x² = -y²sin(xy)

∂²f/∂y² = -x²sin(xy)

∂²f/∂x∂y = (1 - xy)cos(xy)

Since the second-order partial derivatives involve the trigonometric functions sin(xy) and cos(xy), it is challenging to determine the nature of the critical points without further analysis.

To sketch the level set f(x, y) to different constant values and plot the corresponding curves.

(d) For the function f(x, y) = xy(x - 1):

To find the critical points, both partial derivatives to zero:

∂f/∂x = y(x - 1) + xy = 0 -> y(x - 1 + x) = 0 -> y(2x - 1) = 0

∂f/∂y = x(x - 1) = 0

From y(2x - 1) = 0,  y = 0 or 2x - 1 = 0. This gives us the critical points (0, 0) and (1/2, y) for any y.

From x(x - 1) = 0,  x = 0 or x = 1. These values correspond to the critical points (0, 0) and (1, y) for any y.

To determine the nature of the critical points,  examine the second-order partial derivatives:

∂²f/∂x² = 2y

∂²f/∂y² = 0

∂²f/∂x∂y = 2x - 1

For the critical point (0, 0), the second-order partial derivatives are ∂²f/∂x² = 0, ∂²f/∂y² = 0, and ∂²f/∂x∂y = -1. Based on the second partial derivative test, this critical point is a saddle point.

For the critical points (1, y) and (1/2, y) where y can be any value, the second-order partial derivatives are ∂²f/∂x² = 2y, ∂²f/∂y² = 0, and ∂²f/∂x∂y = 1. The nature of these critical points depends on the value of y.

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Answer:

[tex]x=3[/tex]

Step-by-step explanation:

Equation - [tex]5=x+2[/tex]

Solve - subtract 2 from both sides - [tex]3 = x[/tex]

Answer:

X= 3 oz.

Step-by-step explanation:

Count left side. Minus the oz. on the right from that amount. You get 3. Chess piece weighs 3 oz.

Hope I helped.

Answer: $42,000

Step-by-step explanation:

Simple Interest: I=prt

I= (6,000)(7/4)(4)

I= 42,000

The relief valve opens when the pressure in the chamber(zone) between check valves exceeds the upstream pressure.

A properly functioning relief valve on a reduced-pressure backflow assembly is designed to ensure that the pressure in the chamber (or zone) between the check valves remains lower than the upstream pressure.

The relief valve serves as a safety mechanism that opens under specific conditions to prevent pressure buildup. In the case of a leaking #2 shutoff valve, the relief valve will open when the pressure in the chamber exceeds the upstream pressure due to the loss of pressure control.

Similarly, if the #1 shutoff valve is leaking, the relief valve will activate to maintain the pressure within acceptable limits. Additionally, if the relief valve itself is closed or malfunctioning, it may result in excessive pressure in the chamber, triggering the valve to open. Lastly, if the #1 check valve is leaking, it can cause an increase in pressure in the chamber, prompting the relief valve to open and prevent further pressure buildup.

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Answer:

x=23

Step-by-step explanation:

Hello There!

The relationship between chords can be found below in the image.

Pretty much the product of the segments in the same chord is equal to the product of the other segments in the other chord if that makes sense

So more specifically for this problem

10 * 18 = 9(x-3)

once we are able to create a formula/equation this problem is a lot easier to understand

Now we use basic algebra to solve for x

10 * 18 = 9(x-3)

step 1 combine like terms

10 * 18 = 180

now we have

180 = 9(x-3)

step 2 distribute the 9 to what's in the parenthesis (x and -3)

9*x=9x

9*-3=-27

now we have

180 = 9x - 27

step 3 add 27 to each side

-27 + 27 cancels out

180 + 27 = 207

now we have

207 = 9x

step 4 divide each side by 9

207/9 = 23

9x/9=x

we're left with x = 23

Now we want to check our answer

is 10 * 18 = 9*(23-3) then our answer is correct

10*18=180

23-3=20

20*9=180

180=180 is true hence the answer is 23

Answer:

3n + 21

Step-by-step explanation:

3(n + 7)

3n + 21

Answer:

7,3 if rounded up...

not rounded up..... 7, 3.3????? I think???

Slope-intercept form: y = mx + b

First find the slope : [tex]\frac{rise}{run} = \frac{-1}{3}[/tex]

Slope = -1/3

Now we got  y = -1/3x + b

Next find y-intercept. This is where your line crosses the y line (vertical)

Y-intercept = 3

b = 3

Answer: y = -1/3x + 3