in how many ways can we seat $8$ people around a table if alice and bob won't sit next to each other? (two seatings are the same if one is a rotation of the other.)

To find the volume of the solid region bounded by the paraboloids y = x^2 and z = 4 - x^2, we need to set up triple iterated integrals in terms of x, y, and z.

One way to do this is to integrate over x first, then y, then z, or vice versa. Here are six different triple iterated integrals we can use:

1. ∫∫∫d dz dy dx
2. ∫∫∫d dx dy dz
3. ∫∫∫d dx dz dy
4. ∫∫∫d dy dx dz
5. ∫∫∫d dy dz dx
6. ∫∫∫d dz dx dy

Let's evaluate the first integral:

∫∫∫d dz dy dx

We start by finding the limits of integration for z. The paraboloid z = 4 - x^2 is above the paraboloid y = x^2, so the lower limit for z is y - x^2, and the upper limit is 4 - x^2.

Next, we find the limits of integration for y. The paraboloid y = x^2 is a function of x, so the limits are given by the x-values that bound the region d. Since the paraboloids intersect at x = -2 and x = 2, the limits for y are x^2 and 4 - x^2.

Finally, we find the limits of integration for x. The region d is symmetric about the yz-plane, so we can integrate over x from 0 to 2 and multiply by 2 to get the full volume. Therefore, the limits for x are 0 and 2.

Putting it all together, we have:

∫∫∫d dz dy dx = ∫0^2 ∫x^2^(4-x^2) ∫y-x^2^(4-x^2) dz dy dx

Evaluating this integral is a bit messy, but it can be done with some algebraic manipulation and trigonometric substitutions. The answer turns out to be: 64/15

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The complete factorization of the polynomial x³ + 2x² -x - 2 is equal to (x + 2)(x + 1)(x - 1)

An equation is an expression that shows how numbers and variables are related to each other using mathematical operators.

A polynomial is an expression consisting of coefficients and variables forming an equation.

Given the polynomial:

x³ + 2x² -x - 2

Factorizing:

= (x + 2)(x² - 1)

= (x + 2)(x + 1)(x - 1)

The polynomial x³ + 2x² -x - 2 is equal to (x + 2)(x + 1)(x - 1)

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Answer:

The specific values mentioned in the content (e.g., λ = 1, μ = 0.5) are needed to perform the calculations accurately.

Step-by-step explanation:

To find the pdf (probability density function) of Z, let's start by understanding the variables involved:

X and Y are independent random variables with exponential pdfs (probability density functions). The pdf for X is given by

fX(x) =\(\lambda e^_(-\lambda x)\),

where λ = 1 for X.

Similarly, the pdf for Y is

fY(y) = \(\mu e^_(-\mu y)\),

where μ = 0.5 for Y.

(a) Finding the pdf of Z = X + Y:

To find the pdf of Z, we need to determine the distribution of the sum of two random variables. Since X and Y are independent, the sum Z = X + Y will follow the convolution of their individual pdfs.

Let's denote the pdf of Z as fZ(z). To find fZ(z), we convolve fX(x) and fY(y) using the convolution integral:

fZ(z) = ∫[fX(x) * fY(z - x)] dx

Plugging in the pdfs of X and Y, we have:

fZ(z) = \(\int[e^{(-\lambda x)} * \mu e^{(-\mu(z - x))}] dx\)

Simplifying the expression and integrating, we obtain the pdf of Z.

(b) Finding the pdf of Z = |X - Y|:

To find the pdf of Z, we need to determine the distribution of the absolute difference between X and Y. Since X and Y are independent, we can consider the cases where X > Y and Y > X separately.

For X > Y:

Z = X - Y, so the pdf can be obtained by finding the distribution of X - Y and taking its absolute value.

For Y > X:

Z = Y - X, so the pdf can be obtained by finding the distribution of Y - X and taking its absolute value.

In both cases, we need to perform the convolution of the individual pdfs, similar to part (a), but with a slight modification for taking the absolute value.

By evaluating the convolutions and considering both cases (X > Y and Y > X), we can derive the pdf of Z - |X - Y|.

The specific values mentioned in the content (e.g., λ = 1, μ = 0.5) are needed to perform the calculations accurately.

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Answer:

Around 43 turns.

Step-by-step explanation:

The values of t that satisfy the equation 2t - 1/(t + 3) = -2 are t = -2 + sqrt(6) and t = -2 - sqrt(6).

To find the value of t that satisfies the equation 2t - 1/(t + 3) = -2, we can start by simplifying the left-hand side of the equation.

Multiplying both sides by (t + 3) to eliminate the denominator, we get:

2t(t + 3) - 1 = -2(t + 3)

Expanding and simplifying, we get:

2t^2 + 6t - 1 = -2t - 6

Rearranging and simplifying, we get:

2t^2 + 8t + 5 = 0

To solve for t, we can use the quadratic formula:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

Plugging in the values for a, b, and c from our equation, we get:

t = (-8 ± sqrt(8^2 - 4(2)(5))) / 2(2)

t = (-8 ± sqrt(24)) / 4

t = (-8 ± 2sqrt(6)) / 4

Simplifying, we get two solutions:

t = -2 + sqrt(6) or t = -2 - sqrt(6)

Therefore, the values of t that satisfy the equation 2t - 1/(t + 3) = -2 are t = -2 + sqrt(6) and t = -2 - sqrt(6).

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The confidence interval for the population variance is [17.81,19.79 ].

What is standard deviation?

Standard Deviation is a measure that shows what quantity variation (such as unfold, dispersion, spread,) from the mean exists. the quality deviation indicates a “typical” deviation from the mean. it's a well-liked live of variability as a result of it returns to the first units of live of the info set.

Main body:

N = 25

mean - 18.8

standard deviation = 3

z value of 90% in z score = 1.645

C.I. = mean± z*s/√n

C.I. = 18.8 ± 1.645*3/√25

C.I. = 18.8 ±0.99

C.I. =[17.81,19.79 ]

Hence the confidence interval for the population variance is [17.81,19.79 ].

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The domain is the set of all t-value (inputs) used by the function/graph.

The least t-value is 0.  The greatest is 4.

The domain is [0,4].

The domain of graph for function h(t) is [0,4].

The range of values that we are permitted to enter into our function is known as the domain of a function. The x values of a function like f(x) are part of this collection. A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.

We have,

a diagram below shows the graph of h(t), which models the height in t seconds.

we know that the domain are the input values.

Here the domain is the set of all t-value used in the graph.

So, from the graph we can see that the least t-value is 0 and the greatest is 4.

Thus, the domain of graph is [0,4].

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Answer:

12=-3(-7.5)-10.5

Step-by-step explanation:

first find the slope

m=y2-y1/x2-x1

0-12=(-12)

-3.5-(-7.5)=4

-12/4=(-3)

now write the equation

y=mx+b

12=-3(-7.5)+b

now solve

-3*-7.5=22.5

22.5-22.5=0

12-22.5=(-10.5)

b=-10.5

hope this helps :3

if it did pls mark brainliest

Answer:

Vertex form: \(y = 3(x - 4)^2 - 2\)

Vertex: (4, -2)

Step-by-step explanation:

Hello!

We have to complete the square, and then simplify.

The equation in vertex form is \(y = 3(x - 4)^2 - 2\)

The vertex is (4,-2)

Vertex form is f(x) = a(x - h)² + k, where (h, k) is the vertex.

Answer:

$63

Step-by-step explanation:

Ordered pairs on an unit circle..

Exact values of the ordered pairs of the unit circle is represented in pictorial form above in degrees and radians.

Hope this helps.

The probability that you will get through the door without returning for more keys is 0.03.

Probability is the occurence of likely events. It is the area of mathematics that deals with numerical estimates of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1.

Probability door is locked = 0.7

The probability that you will get through the door without returning for more keys will be:

= (1 - P(locked)) × P(key)

= (1 - 0.7) × 0.1

= 0.3 × 0.1

= 0.03

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Answer:

D. 36

Step-by-step explanation:

because to find volume we should multiply all the side

Answer:

d.36in    

Step-by-step explanation:

Therefore, the maximum displacement of the particle is 4 units, and it occurs at time t = 1.

To find the maximum displacement, we need to first determine the particle's velocity and acceleration.

The velocity of the particle is given by the derivative of its position function:

\(v(t) = y'(t) = 3t^2 - 12t + 9\)

The acceleration of the particle is given by the derivative of its velocity function: a(t) = v'(t) = 6t - 12

Now, to find the maximum displacement, we need to find the time at which the particle comes to rest.

This occurs when its velocity is zero:

\(3t^2 - 12t + 9 = 0\)

Simplifying this equation, we get:

\(t^2 - 4t + 3 = 0\)

This quadratic equation factors as:

(t - 1)(t - 3) = 0

So the particle comes to rest at t = 1 or t = 3.

Next, we need to determine whether the particle is at a maximum or minimum at each of these times.

To do this, we look at the sign of the acceleration:

When t = 1, a(1) = 6(1) - 12 = -6, which is negative.

Therefore, the particle is at a maximum at t = 1.

When t = 3, a(3) = 6(3) - 12 = 6, which is positive.

Therefore, the particle is at a minimum at t = 3.

Finally, we need to find the displacement of the particle at each of these times:

\(y(1) = 1^3 - 6(1)^2 + 9(1) = 4\)

\(y(3) = 3^3 - 6(3)^2 + 9(3) = 0.\)

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Answer:

(this is the simplified answer)

Step-by-step explanation:

There are 4 terms in the simplified equation making it a quadrinomial

The result is Quadrinomial cubic

Start by simplifying the polynomial using the FOIL method:

Multiply one by one:

First multiply the term:

\((2a-5)(a^2-1)=(2a)(a^2-1) -5(a^2-1)\)

And, split out the terms, we get

\((2a-5)(a^2-1)=(2a)(a^2)+(2a)(-1)+(-5)(a^2)+(-5)(-2)\\\\(2a-5)(a^2-1)=2a^3-2a-5a^2+5\)

Now, the degree is equivalent to the highest exponent in the polynomial. So, the degree for this polynomial is 3 meaning its a cubic.

There are 4 terms in the simplified equation making it a quadrinomial

Hence, The result is Quadrinomial cubic .

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Answer:

Rounding the adjusted forecast to two decimal places, the adjusted forecast in year 2022 for Period-2 is 12136.37.

Step-by-step explanation:

To find the adjusted forecast in 2022 for Period-2, we'll need to use the given seasonality index and the trend line equation.

The trend line equation is:

Fₜ = 179 + 4t

First, we need to determine the value of 't' for 2022 Period-2. Since Period-1 corresponds to 2021, and each period represents a year, we can calculate the value of 't' for 2022 Period-2 as follows:

2022 Period-2 = 2022 + 1 = 2023

Now, we can substitute the value of 't' into the trend line equation:

Fₜ = 179 + 4t

Fₜ = 179 + 4 * 2023

Fₜ = 179 + 8092

Fₜ = 8271

The unadjusted forecast for 2022 Period-2 is 8271.

To adjust the forecast, we multiply it by the seasonality index for Period-2, which is given as 1.47:

Adjusted Forecast = Unadjusted Forecast * Seasonality Index

Adjusted Forecast = 8271 * 1.47

Adjusted Forecast = 12136.37

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The range of time values that describes the entire interval over which she would be interpolating is 0 to 18 minutes.

Interpolating is a mathematical technique which involves estimating values between two known points. It is commonly used in science and engineering to estimate values of a function or equation at points between two known points. Interpolation can also be used to determine values of a dataset at points where the data is missing.

Interpolating involves estimating values between two known points, so she would need to consider all points between 0 and 18 minutes.

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Answer:

For the sample of soil given a) the porosity is 100.4%; b) the specific yield is 12.2%; c) the specific retention is 14.6%; d) the void ratio is 0.5342; e) the initial moisture content is 2.3%; and f) the initial degree of saturation is 41.97%.

a) The porosity of soil can be defined as the ratio of the void space in the soil to the total volume of the soil.

The total volume of the soil = Initial volume of soil = 100 c.m³

Weight of water added to the soil = 234.6 g – 220 g = 14.6 g

Volume of water added to the soil = 14.6 c.m³

Volume of soil occupied by water = Weight of water added to the soil / Density of water = 14.6 / 1 = 14.6 c.m³

Porosity = Void volume / Total volume of soil

Void volume = Volume of water added to the soil + Volume of voids in the soil

Void volume = 14.6 + (Initial volume of soil – Volume of soil occupied by water) = 14.6 + (100 – 14.6) = 100.4 c.m³

Porosity = 100.4 / 100 = 1.004 or 100.4%

Therefore, the porosity of soil is 100.4%.

b) Specific yield can be defined as the ratio of the volume of water that can be removed from the soil due to the gravitational forces to the total volume of the soil.

Specific yield = Volume of water removed / Total volume of soil

Initially, the weight of the oven dried soil is 220 g. After allowing it to drain by gravity, the weight of soil is 222.4 g. Therefore, the weight of water that can be removed by gravity from the soil = 234.6 g – 222.4 g = 12.2 g

Volume of water that can be removed by gravity from the soil = 12.2 c.m³

Specific yield = 12.2 / 100 = 0.122 or 12.2%

Therefore, the specific yield of soil is 12.2%.

c) Specific retention can be defined as the ratio of the volume of water retained by the soil due to the capillary forces to the total volume of the soil.

Specific retention = Volume of water retained / Total volume of soil

Initially, the weight of the oven dried soil is 220 g. After adding water to the soil, the weight of soil is 234.6 g. Therefore, the weight of water retained by the soil = 234.6 g – 220 g = 14.6 g

Volume of water retained by the soil = 14.6 c.m³

Specific retention = 14.6 / 100 = 0.146 or 14.6%

Therefore, the specific retention of soil is 14.6%.

d) Void ratio can be defined as the ratio of the volume of voids in the soil to the volume of solids in the soil.

Void ratio = Volume of voids / Volume of solids

Initially, the weight of the oven dried soil is 220 g. The density of solids in the soil can be calculated as,

Density of soil solids = Weight of oven dried soil / Volume of solids

Density of soil solids = 220 / (100 – (14.6 / 1)) = 2.384 g/c.m³

Volume of voids in the soil = (Density of soil solids / Density of water) × Volume of water added

Volume of voids in the soil = (2.384 / 1) × 14.6 = 34.8256 c.m³

Volume of solids in the soil = Initial volume of soil – Volume of voids in the soil

Volume of solids in the soil = 100 – 34.8256 = 65.1744 c.m³

Void ratio = Volume of voids / Volume of solids

Void ratio = 34.8256 / 65.1744 = 0.5342

Therefore, the void ratio of soil is 0.5342.

e) Initial moisture content can be defined as the ratio of the weight of water in the soil to the weight of oven dried soil.

Initial moisture content = Weight of water / Weight of oven dried soil

Initial weight of soil = 225.1 g

Weight of oven dried soil = 220 g

Therefore, the weight of water in the soil initially = 225.1 – 220 = 5.1 g

Initial moisture content = 5.1 / 220 = 0.023 or 2.3%

Therefore, the initial moisture content of soil is 2.3%.

f) Initial degree of saturation can be defined as the ratio of the volume of water in the soil to the volume of voids in the soil.

Initial degree of saturation = Volume of water / Volume of voids

Volume of water = Weight of water / Density of water

Volume of water = 14.6 / 1 = 14.6 c.m³

Volume of voids in the soil = 34.8256 c.m³

Initial degree of saturation = 14.6 / 34.8256 = 0.4197 or 41.97%

Therefore, the initial degree of saturation of soil is 41.97%.

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Answer:

24 months.

Step-by-step explanation:

168 sellouts.

7 times per month.

So to find the amount of months, all we have to do is divide the sellouts by the times per month.

168/7=24.

24 months.

The graph of f(x) = 3* has a y-intercept at (0, 3), an x-intercept at (1, 0), and a horizontal asymptote at y = 4. The exponential function f(x) = a* is increasing if a > 1 and decreasing if 0 < a < 1.

The y-intercept of the graph occurs when x = 0, so substituting x = 0 into the function f(x) = 3*, we get f(0) = 3*0 = 3. Therefore, the y-intercept is (0, 3).

The x-intercept of the graph occurs when y = 0, so substituting y = 0 into the function f(x) = 3*, we get 0 = 3*x. Solving for x, we find x = 1. Therefore, the x-intercept is (1, 0).

The horizontal asymptote represents the value that the function approaches as x approaches positive or negative infinity. In this case, the horizontal asymptote is y = 4, indicating that as x becomes extremely large or extremely small, the function approaches a value of 4.

For the exponential function f(x) = a*, the value of a determines whether the function is increasing or decreasing. If a > 1, the function is increasing as x increases. If 0 < a < 1, the function is decreasing as x increases.

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\( \huge \boxed{\mathbb{QUESTION} \downarrow}\)

\( \large \boxed{\mathbb{ANSWER\: WITH\: EXPLANATION} \downarrow}\)

➜ Area of the lot = 2x² + 7x + 3 cm²

➜ Width of the garden = 2x + 1 cm.

➜ Length of the garden = y

﹋﹋﹋﹋﹋

✪ Area of a rectangle = length × width

⇒ length = area ÷ width

⇒ y = 2x² + 7x + 3 ÷ 2x + 1

﹋﹋﹋﹋﹋

WORKING ↷

\( \tt \: y = \frac{ {2x}^{2} + 7x + 3}{2x + 1} \\ \\ \sf \: Factorise \: {2x}^{2} + 7x + 3. \\ \\ \tt \: y = \frac{\left(x+3\right)\left(2x+1\right)}{2x+1} \\ \\ \sf \: Cancel \: out \:( 2x + 1 )\\ \\ \large\boxed{\boxed{\bf y = x+3 }}\)

✯ Length of the garden = x + 3 cm.

▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨

➜ Area of the frame = 4x² - 4xy + y² cm²

➜ Length of the side of the frame = s

﹋﹋﹋﹋﹋

✪ Area of a square = side²

⇒ 4x² - 4xy + y² = s²

﹋﹋﹋﹋﹋

WORKING ↷

\( \tt {4x}^{2} - 4xy + {y}^{2} = {s}^{2} \\ \\ \sf \: Use \: the \: algebraic \: identity \downarrow \: \\ \sf {a}^{2} - 2ab + {b}^{2} = (a - b) ^{2} ... \\ \sf \: a = 2x \: and \: b = y \\ \\ \tt \left(2x-y\right)^{2} = {s}^{2} \\ \\ \sf \: Squaring \: on \: both \: the \: sides \\ \\ \tt \sqrt{(2x - y) ^{2} } = \sqrt{ {(s)}^{2} } \\ \large\boxed{\boxed{\bf \: (2x - y) = s}}\)

✯ Length of the side of the frame = 2x - y cm.

▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨

➜ Length of the rectangle = x + 5 cm

➜ Width of the garden = x + 3 cm.

➜ Area of the garden = a

﹋﹋﹋﹋﹋

✪ Area of a rectangle = length × width

⇒ a = (x + 5) × (x + 3)

﹋﹋﹋﹋﹋

WORKING ↷

\( \tt \: a = (x + 5) \times (x + 3) \\ \\ \sf \: multiply \: (x + 5) \: with \: (x + 3) \\ \\ \tt \: a = (x + 5) \times (x + 3) \\ \tt \: a = x(x + 3) + 5(x + 3) \\ \tt \: a = {x}^{2} + 3x + 5x + 3 \\ \large \boxed{\boxed{ \bf \: a = {x}^{2} + 8x + 3}}\)

✯ Area of the rectangle = x² + 8x + 3 cm².

▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨

➜ Length of the side of a square = x + 6 cm

➜ Area of the square = a

﹋﹋﹋﹋﹋

✪ Area of a square = side × side

⇒ Area of a square = side²

⇒ a = (x + 6)²

﹋﹋﹋﹋﹋

WORKING ↷

\( \tt \: a = (x + 6) ^{2} \\ \\ \sf \: Use \: the \: algebraic \: identity \downarrow \: \\ \sf (a + b) ^{2} = {a}^{2} + 2ab + {b}^{2} ... \\ \sf \: a = x \: and \: b = 6 \\ \\ \tt \: a = (x + 6) ^{2} \\ \tt \: a = {x}^{2} + 2 \times x \times 6 + {6}^{2} \\ \large \boxed{\boxed{ \bf \: a = {x}^{2} + 12x + 36}}\)

✯ Area of the square = x² + 12x + 36 cm².

▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨▧▨

The set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers.

What are integers ?

An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of integers are: -5, 1, 5, 8, 97, and 3,043. Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .

According to the Property of Closure, A set has the closure property under a particular operation if the result of the operation is always an element in the set.  If a set has the closure property under a particular operation, then we say that the set is “closed under the operation.”

Hence ,  the set of integers is closed under the operation of addition because the sum of any two integers is always another integer and is therefore in the set of integers.

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The expected number of new cases per 100,000 in Michigan according to the multiple regression model is 930

In statistics, multiple regression model refers a statistical technique that can be used to analyze the relationship between a single dependent variable and several independent variables.

Here we have given that if Michigan has a strictness level equal to 50 and has more than 55 available hospitals,

And we need to find the expected number of new cases per 100,000 in Michigan according to the multiple regression model.

While we looking into the given question, we have identified the value of

Strictness level = 50

Available hospital = 55

New cases per day = 100,000

Then the expected number is calculated by using the multiple regression model is written as,

=> 50x + 55y ≤ 100,000

When we solve it graphically, then we get, the graph like the following,

As per the given graph, the expected number is 930.

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The rectangle model with three sections labeled three-fourths and one section labeled three-eighths represents how many days' worth of seed Jay has left.

Based on the given information, Jay has a fraction of 3/4 pound of bird seed, and he needs a fraction of 3/8 pound to feed the birds daily. We need to determine the rectangle model that shows how many days' worth of seed Jay has left.

The correct rectangle model is:

Rectangle model divided into four equal sections, three sections are labeled three-fourths and one section is labeled three-eighths, equaling three days.

This is because Jay initially has 3/4 pound of seed, and each day he needs 3/8 pound. By dividing the rectangle into four equal sections, where three sections are labeled three-fourths (3/4) and one section is labeled three-eighths (3/8), it represents that Jay has enough seed to feed the birds for three days.

Therefore, how many days of seed Jay has left is shown by a rectangle with three portions labelled "three-fourths" and one section labelled "three-eighths."

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To find the equation of a line with a slope of Four-thirds that passes through point (3, 5), we can use the point-slope form of a linear equation:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the given point.

Plugging in the values, we get:

y - 5 = Four-thirds(x - 3)

Simplifying, we can write:

y - 5 = Four-thirds x - 4

Adding 5 to both sides, we get:

y = Four-thirds x + 1

Therefore, the statement that is true is:

The slope-intercept equation is y = four-thirds x + 1.

Answer:

B) Y=4/3X+1

Step-by-step explanation:

Y-Y1=M(X-X1)

Y-5=4/3(X-3)

Y-5+5=4/3(X-3)+5

Y=4/3X-4+5

Y=4/3X+1

Hope this helps <3