Line Handout #1f: Find the equation of the line parallel to 3x +y=-3 through the point (3,2)

The quotient and remainder when -x⁴+6x³+7x²+3 is divided by x-7 is 12 and 73, respectively.

Using synthetic division, the quotient and remainder when -x⁴+6x³+7x²+3 is divided by x-7 is 12 and 73, respectively.


To solve this problem, start by writing out the synthetic division form, with the coefficients in the same order as the original problem:



  |-7 |6 7 3
 |1 -7
 |0 12
 |-73


Take the first coefficient of the divisor (-7) and multiply it by the first coefficient of the dividend (-1). The result is 7. Place this number at the top of the first column of the table.


Take the divisor’s first coefficient (-7) and add it to the dividend’s second coefficient (6). Place the result (–1) in the second column of the table.


Take the first coefficient of the divisor (-7) and multiply it by the second coefficient of the dividend (6). Place the result (-42) in the third column of the table.


Now add the first two numbers in the second column of the table (7 and -1). Place the result (6) in the fourth column.

The last number in the table (-73) is the remainder. The second number in the table (12) is the quotient.

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Answer:

x = 5

Step-by-step explanation:

 5x - 7 = 3x + 3

- 3x        -3x

2x - 7 =         3

      +7           +7

 2x =              10

Divide 2 from both sides

x = 5

Answer: 10

Step-by-step explanation:

2x - 7 = 3

add 7 to both sides = 2x = 10

now to solve:

once u have 2x = 10

divide both sides by 2

x=5

The correct answer is B. 25/3.

We can use the equation of a parabola with a focus at (h, k) and a directrix at y = k + p to find the depth of the reflector. The equation is:
(y - k)² = 4p(x - h)

Since the focus is at the light center, we can set h = 0 and k = 0. The distance of the focus from the vertex is 3/4 cm, so p = 3/4. The diameter of the reflector is 10 cm, so the x-coordinate of the vertex is 5 cm. We can plug in these values to find the depth of the reflector:
(y - 0)² = 4(3/4)(x - 0)
y² = 3x
y = √(3x)

When x = 5, we can find the depth of the reflector:

y = √(3*5)
y = √15
y = 3.87 cm
The depth of the reflector is 3.87 cm, or 25/3 cm.

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When a car headlight reflector is cut by a plane along its axis, the section obtained is a parabola. This parabola is such that the light center is at the focus. The distance of focus from the vertex is 3/4 cm and the diameter of the reflector is 10 cm. The depth of the reflector comes out to be CD = VC - VF = (10/3) - (3/4) = 27/4 cm

The vertex of the parabola is the midpoint of the diameter of the reflector. Let V be the vertex of the parabola and let F be the focus. The distance between V and F is given as 3/4 cm.The reflector is such that light rays from the source (headlamp) placed at the focus of the parabola are reflected by the parabola in such a way that the rays are parallel to the axis of the parabola. This is known as the reflecting property of the parabola.

This is equal to CD.Let P be the point on the parabola, as shown in the diagram below, such that PF is equal to the diameter of the reflector. Then, by the definition of the parabola, the distance from P to the vertex C is the same as the distance from the focus F to P, i.e., PF = PC. Since PF is equal to the diameter of the reflector, it is given that PF = 10 cm.Therefore, PC = 10 cm. It is also given that VF = 3/4 cm. Therefore, VC = PC - PV = 10 - 20/3 = 10/3 cm.

Hence, the depth of the reflector is CD = VC - VF = (10/3) - (3/4) = 27/4 cm. Therefore, the depth of the reflector is 27/4 cm, which is the correct option among the given choices.

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The system of inequalities that describes the graph is given as follows:

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

For the lower bound of the inequality, we have that:

Hence:

y ≥ -4/3x + 8

4x/3 + y ≥ 8

4x + 3y ≥ 24.

For the upper bound of the inequality, we have that:

Hence:

y ≤ -5x/8 + 5

5x/8 + y ≤ 5

5x + 8y ≤ 40.

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The solution to the system of equations is:
x = -6.552
y = -1.655
z = 1.276

To solve the system of equations using the inverse of the coefficient matrix, we will first need to find the inverse of the coefficient matrix using the inversion algorithm. The coefficient matrix is:

A = [[2, 6, 6], [2, 7, 1], [2, 7, 7]]

To find the inverse of A, we will use the inversion algorithm:

1. Find the determinant of A:

|A| = 2(7*7 - 1*7) - 6(2*7 - 1*2) + 6(2*7 - 2*7) = 14 - 72 + 0 = -58

2. Find the matrix of minors:

M = [[(7*7 - 1*7), -(2*7 - 1*2), (2*7 - 2*7)], [-(6*7 - 6*1), (2*7 - 6*2), -(2*6 - 6*2)], [(6*1 - 7*6), -(2*1 - 7*2), (2*6 - 7*6)]]

M = [[42, -12, 0], [-36, 2, -12], [-36, 12, -30]]

3. Find the matrix of cofactors:

C = [[42, 12, 0], [36, 2, 12], [-36, -12, -30]]

4. Find the adjugate matrix:

adj(A) = [[42, 36, -36], [12, 2, -12], [0, 12, -30]]

5. Find the inverse of A:

A^-1 = (1/|A|)adj(A) = (1/-58)[[42, 36, -36], [12, 2, -12], [0, 12, -30]]

A^-1 = [[-0.724, -0.621, 0.621], [-0.207, -0.034, 0.207], [0, -0.207, 0.517]]

Now, we can use the inverse of the coefficient matrix to solve the system of equations. The system of equations can be written in matrix form as:

AX = B

Where A is the coefficient matrix, X is the matrix of unknowns, and B is the matrix of constants:

A = [[2, 6, 6], [2, 7, 1], [2, 7, 7]]
X = [[x], [y], [z]]
B = [[6], [7], [8]]

Multiplying both sides of the equation by the inverse of A, we get:

A^-1AX = A^-1B

IX = A^-1B

X = A^-1B

Substituting the values of A^-1 and B, we get:

X = [[-0.724, -0.621, 0.621], [-0.207, -0.034, 0.207], [0, -0.207, 0.517]] * [[6], [7], [8]]

X = [[-6.552], [-1.655], [1.276]]

Therefore, the solution to the system of equations is:

x = -6.552
y = -1.655
z = 1.276

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The difference between the two groups is likely due to chance and the two groups likely have the same mean.

To conduct the appropriate test of significance for a research situation, you must first identify whether you need to use a one-tailed or two-tailed test. A one-tailed test is used when you are looking to see if the mean of one group is greater or less than the mean of another group. A two-tailed test is used when you are testing the null hypothesis that the two groups have the same mean.

Once you have identified the type of test, you must determine if it is a one or two sample test. A one sample test is used when you are comparing the mean of one group to a known population mean, whereas a two sample test is used when you are comparing the means of two separate groups.

Next, you must select the appropriate alpha level. Alpha levels are a measure of the significance of the test and refer to the probability of rejecting the null hypothesis when it is actually true. Generally, an alpha level of 0.05 is used in research settings.

Once you have selected the appropriate test and alpha level, you can make a decision about the test significance. This is done by comparing the calculated test statistic to the critical value in the test statistic table. If the calculated test statistic is larger than the critical value, the test is statistically significant and the null hypothesis can be rejected. If the calculated test statistic is smaller than the critical value, the test is not statistically significant and the null hypothesis cannot be rejected.

Finally, you must interpret the results of the test. If the test is statistically significant, it means that the difference between the groups is not likely due to chance, and therefore the two groups are likely to have different means. If the test is not statistically significant, it means that the difference between the two groups is likely due to chance and the two groups likely have the same mean.

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We have verified that A^-1A = I and AA^-1 = I for both matrices.

To compute the inverse of a matrix, we can use the formula: A^-1 = 1/det(A) * adj(A), where det(A) is the determinant of the matrix A and adj(A) is the adjoint of the matrix A.

(a) A = [1 1; 1 -4]

det(A) = (1*-4) - (1*1) = -4 - 1 = -5

adj(A) = [-4 -1; -1 1]

A^-1 = 1/-5 * [-4 -1; -1 1] = [4/5 1/5; 1/5 -1/5]

(b) A = [2 1 1; 3 1 2; -1 3 -1]

det(A) = (2*(1*-1) - 1*(2*3) + 1*(-1*3)) - (1*(1*-1) - 1*(2*-1) + 1*(3*3)) = -6 - 3 - 3 - 9 = -21

adj(A) = [-1 -2 7; 5 1 -8; 2 5 -7]

A^-1 = 1/-21 * [-1 -2 7; 5 1 -8; 2 5 -7] = [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21]

To verify that A^-1A = I and AA^-1 = I, we can simply multiply the matrices and check if the result is the identity matrix.

For (a):

A^-1A = [4/5 1/5; 1/5 -1/5] * [1 1; 1 -4] = [1 0; 0 1] = I

AA^-1 = [1 1; 1 -4] * [4/5 1/5; 1/5 -1/5] = [1 0; 0 1] = I

For (b):

A^-1A = [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21] * [2 1 1; 3 1 2; -1 3 -1] = [1 0 0; 0 1 0; 0 0 1] = I

AA^-1 = [2 1 1; 3 1 2; -1 3 -1] * [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21] = [1 0 0; 0 1 0; 0 0 1] = I

Therefore, we have verified that A^-1A = I and AA^-1 = I for both matrices.

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The correct matches for the vocabulary would be :

Apathy is a lack of interest or passion for anything. The line "Ahmein doesn't care if he does well or not" describes apathy since Ahmein is apathetic about whether he succeeds or fails.

Being competent is having the knowledge and skills required to complete a task successfully which describes Javier as he is aware that he can construct a birdhouse.

Expectations are convictions or presumptions regarding future events. Value is a term used to describe something's importance or worth.

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Answer:

1. apathy = Trudy doesn't care about doing well in school.

2. competent = Javier knows he is able to build a birdhouse.

3. expectations = fear of failure or success

4. autonomy = Alfonzo likes to make choices on his own.

5. stress = Quincy feels anxious and distressed about a test.

Answer:

the answer is B. 2x(5x+1)

Answer:

D

Step-by-step explanation:

The line touches the graph at 2 different points so it doesnt pass the verticle line test

Based on the given mean and standard deviation of Sarah's finishing times, we can estimate that approximately 68% of her races will have finishing times between 61.5 and 64.5 seconds.

The empirical rule, also known as the 68-95-99.7 rule, is a useful tool for estimating the percentage of data that falls within a certain number of standard deviations from the mean in a normal distribution.

In this case, Sarah's finishing times in the 400-meter dash are normally distributed with a mean of 63 seconds and a standard deviation of 1.5 seconds. To use the empirical rule, we need to first calculate the z-scores for the lower and upper bounds of the range we're interested in.

For the lower bound of 61.5 seconds:

z = (61.5 - 63) / 1.5 = -1

For the upper bound of 64.5 seconds:

z = (64.5 - 63) / 1.5 = 1

These z-scores tell us how many standard deviations are away from the mean each time. Using the empirical rule, we know that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

Since the range we're interested in is within one standard deviation of the mean, we can estimate that approximately 68% of Sarah's finishing times will be between 61.5 and 64.5 seconds.

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The measure of the height of the triangle measured by Manuel is given as follows:

5 feet.

The height of a triangle is the perpendicular distance from the base to the vertex opposite the base, and is used to calculate the area of the triangle.

Hence, the height is measured as the distance from the top vertex of the triangle to it's base.

In this problem, it is stated that:

"Manuel measured the distance from the top vertex of the triangle shown to its base".

He found the distance to be of 5 ft, hence the height of the triangle is of:

5 ft.

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The value of 6 = 3 × 2³ is 24 by using PEMDAS operations, the correct option is (d).

To understand why, we can simplify the equation using the order of operations, which is PEMDAS (parentheses, exponents, multiplication/division, addition/subtraction). To evaluate the expression 3 x 2³, we first need to simplify the exponent, which means multiplying 2 by itself three times:

2³ = 2 x 2 x 2

2³ = 8

Now we can substitute the value of 2³ into the original expression:

3 x 2³ = 3 x 8

3 x 2³ = 24

However, the value of 6 = 3 x 2³, is equal to 24 the correct option is (d).

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The complete question is:

What is the value of 6 = 3 × 2³?

a. 16

b. 64

c. 42

d. 24

Step-by-step explanation:

just ignore the % sign and answer :

is 1/2 equivalent to 50 ? yes or no ?

no, of course.

1/2 = 0.5

50 = 50

clearly they are different.

Sam is confused, because 50% = 1/2 = 0.5.

but 50% is NOT 1/2%

1/2% is half of 1%, which by itself is 1/50 of 50%.

1/2% = 1/50/2 of 50% = 1/100 of 50% =

= 0.01 × 0.5 = 0.005

To average a 90 on all three tests, you must score a 95 on the next test.

To find the score you need to average exactly a 90 on all three tests, you can use the formula for the mean (average) of a set of numbers:

Mean = (Sum of all numbers) / (Total number of numbers)

Let's call the score you need on the next test x. We can plug in the known values and solve for x:

90 = (81 + 94 + x) / 3

Multiply both sides by 3 to get rid of the fraction:

270 = 81 + 94 + x

Subtract 81 and 94 from both sides to isolate x:

95 = x

So, you need to score a 95 on the next test to average exactly a 90 on all three tests.

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Answer: a. To calculate the crude mortality rate in NJ in 2017, we need to divide the total number of deaths (74,846) by the total population (8,888,543) and then multiply by 1,000 to express the result per 1,000 population.

Crude Mortality Rate = (Total deaths / Total population) x 1,000

Crude Mortality Rate = (74,846 / 8,888,543) x 1,000

Crude Mortality Rate = 8.42 per 1,000 population

b. To calculate the cause-specific mortality rate due to diabetes in NJ in 2017, we need to divide the number of deaths due to diabetes (1,908) by the total population and then multiply by 1,000 to express the result per 1,000 population.

Cause-specific Mortality Rate due to Diabetes = (Deaths due to diabetes / Total population) x 1,000

Cause-specific Mortality Rate due to Diabetes = (1,908 / 8,888,543) x 1,000

Cause-specific Mortality Rate due to Diabetes = 0.21 per 1,000 population

c. To calculate the case-fatality percent of diabetes, we would need to know the number of people diagnosed with diabetes who died during the same time period. Specifically, the case-fatality percent of diabetes would be calculated by dividing the number of people with diabetes who died (numerator) by the total number of people with diabetes (denominator) and then multiplying the result by 100 to express the percentage. Therefore, the additional variable required is the number of people with diabetes in the population.

Step-by-step explanation:

Lucky duck probabilities include:

A duck with a number:

Every duck has a number, so the probability of choosing a duck with a number is certain (1/1 or 100%).

A duck with a number greater than 10:

is 2/12 or 1/6, since there are 2 ducks with numbers greater than 10 out of a total of 12 ducks.

Duck number 4:

There is only one duck with the number 4, so the probability of choosing this duck is 1/12.

A duck with sunglasses:

There are no ducks with sunglasses so the probability is impossible.

A duck with a hat:

We don't know how many ducks wear hats, so we cannot determine the probability.

An even-numbered duck:

There are 6 even-numbered ducks (2, 4, 6, 8, 10, 12) out of 12 ducks in total, so the probability of choosing an even-numbered duck is 6/12, which simplifies to 1/2.

A duck with a number less than 7:

There are 6 ducks with a number less than 7 (1, 2, 3, 4, 5, 6) out of 12 ducks in total, so the probability of choosing a duck with a number less than 7 is 6/12, which simplifies to 1/2.

A duck with a mustache:

There are no ducks with mustaches, so the probability of getting a duck with a mustache is impossible, 0.

A duck with a bow tie:

Ducks with bowties are 8, so the probability of choosing a duck with bowtie is likely 8/12, 2/3

A duck with a number lower than 25:

All ducks have a number lower than 25, so the probability of choosing a duck with a number lower than 25 is certain (1).

A duck with a number greater than 5:

There are 7 ducks with a number greater than 5 (6, 7, 8, 9, 10, 11, 12) out of 12 ducks in total, so the probability of choosing a duck with a number greater than 5 is 7/12.

A duck with number that's a multiple of 3:

There are 4 ducks with a number that's a multiple of 3 (3, 6, 9, 12) out of 12 ducks in total, so the probability of choosing a duck with a number that's a multiple of 3 is 4/12, which simplifies to 1/3.

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The solution of the function f(h.g)(x) will be 6x³ -3x² + 5. The correct option is C.

In mathematics, expression is defined as the relationship of numbers, variables, and functions using mathematical signs such as addition, subtraction, multiplication, and division.

The given functions are,

h(x) = 3x +5

g(x) = 2x³ - x ²

The value of the function f(h.g)(x) will be calculated as,

f(h.g)(x) = 3 ( 2x³ - x² ) + 5

f(h.g)(x) = 6x³ - 3x² + 5

Therefore, the solution of the function f(h.g)(x) will be 6x³ -3x² + 5. The correct option is C.

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The expression that is equivalent to [tex](\frac{x^2}{x^-4})^5[/tex] is x³⁰

From the question, we have the following parameters that can be used in our computation:

[tex](\frac{x^2}{x^-4})^5[/tex]

An equivalent expression is a different way of writing the same mathematical statement using equivalent mathematical operations or properties.

Evaluate the quotients using the law of indices

So, we have the following representation

(x⁶)⁵

Remove the bracket using the law of indices

So, we have the following representation

x³⁰

Hence, the solution to the expression is x³⁰

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To determine whether the two statement forms are equivalent, we need to create a truth table for each statement form and compare the results.

First, let's create a truth table for the statement form (p∧q)∨∼(p∨q):


 
   p
   q
   p∧q
   p∨q
   ∼(p∨q)
   (p∧q)∨∼(p∨q)
 
 
   T
   T
   T
   T
   F
   T
 
 
   T
   F
   F
   T
   F
   F
 
 
   F
   T
   F
   T
   F
   F
 
 
   F
   F
   F
   F
   T
   T
 

Next, let's create a truth table for the statement form (p∨∼q)∧(∼p∨q):


 
   p
   q
   ∼q
   ∼p
   p∨∼q
   ∼p∨q
   (p∨∼q)∧(∼p∨q)
 
 
   T
   T
   F
   F
   T
   T
   T
 
 
   T
   F
   T
   F
   T
   F
   F
 
 
   F
   T
   F
   T
   F
   T
   F
 
 
   F
   F
   T
   T
   T
   T
   T
 

Comparing the results of the two truth tables, we can see that the two statement forms are not equivalent. The statement form (p∧q)∨∼(p∨q) is true when both p and q are true or when both p and q are false. The statement form (p∨∼q)∧(∼p∨q) is true when both p and q are true or when both p and q are false, but it is also true when p is false and q is true. Therefore, the two statement forms are not equivalent.

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1. 4(f+k)(x)=4(2x^2 - 5x - 8 + 2x + 4) = 8x^2 - 20x - 32.

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The perimeter of triangle QRS 60. Therefore, the correct answer is C: 60.

The perimeter of a triangle is the sum of the lengths of its sides. Therefore, the perimeter of triangle QRS is 14 + 19 + t.

Since t is the cube of an integer, we can write t as x³, where x is an integer. The perimeter of the triangle is then 14 + 19 + x³.

We also know that t is the longest side of the triangle, so it must be greater than both 14 and 19. This means that x³ must be greater than 19, so x must be greater than or equal to 3.

If x is 3, then t is 3³, or 27. The perimeter of the triangle is then 14 + 19 + 27, or 60.

Therefore, the correct answer is C: 60.

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Answer: Your welcome!

Step-by-step explanation:

This linear model tells us that, on average, for every additional square foot of area in a home, the selling price increases by 122.2 dollars. Therefore, this model suggests that the larger the area of a home, the higher the selling price of the home will be.

Answer:

Step-by-step explanation:

Answer: 5x + 2y = 6

Step-by-step explanation:

The standard form is Ax + By = C

Subtract 5x from each side:

- 5x - 2y = - 6

The first term in the standard form must be positive, so multiply by -1 to change the sign.

5x + 2y = 6

Hope this helps!

Answer:

0.65

Step-by-step explanation:

0.13 * 5 = 0.65

Answer:

10

Step-by-step explanation:

Answer:

10

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

In the word problem, the time (in seconds) the ball was in the air is D)0.50 seconds.

What is word problem?

Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.

Here expression which describes height is,

h(t)= [tex]-18t^2+3t+2[/tex]

Now h(t)=0 then,

=> [tex]-18t^2+3t+2=0[/tex]

=> [tex]18t^2-3t-2=0[/tex]

=> [tex]18t^2-3t=2[/tex]

=> 3t(6t-1)=2

=> 3t=2 , 6t-1=2

=> t=2/3 , 6t=3

=> t=2/3 , t=1/2

=> t= 0.5 seconds.

Hence  the time (in seconds) the ball was in the air is D)0.50 seconds.

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Answer:

16

Step-by-step explanation:

g(x)=-5x+1

g(-3)

x=-3

-5(-3) +1

15+1

16

Answer:

See below.

Step-by-step explanation:

For this problem, we are asked to find the value of g(-3).

We are given a Linear Function.

A Linear Function is a Polynomial Function that is commonly graphed. This function most of the time will simply be represented as a straight line when graphed.

For this problem;

[tex]g(x)=-5x+1 \ Find \ g(-3).[/tex]

We simply need to substitute -3 in for x.

[tex]g(-3)=-5(-3)+1[/tex]

Simplify:

[tex]g(-3)=16.[/tex]

Our final answer is g(-3) = 16.

The area of the hexagon is given by A = 585 units²

Six equilateral triangles make create a regular hexagon, which has six congruent sides and angles. The following is the formula to get the area of a regular hexagon.

Area of a hexagon A = 3 (√3)/ 2 ) a²

where, a represent the length of a side of the regular hexagon

Given data ,

Let the area of the hexagon be represented as A

Now , the value of A is

Let the side length of the hexagon be a = 15

Now , area of a hexagon A = 3 (√3)/ 2 ) a²

On simplifying , we get

The area of hexagon A = 3 (√3)/ 2 ) ( 15 )²

The area of hexagon A = 584.56715 units²

The area of hexagon A = 585 units²

Therefore , the value of A is 585 units²

Hence , the area of the hexagon is A = 585 units²

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