Constant Rate of Change: 2/3
Initial Value:10

The eigenvalues of a matrix are roots of its characteristic polynomial using this we prove  that a square matrix and its transpose have the same characteristic polynomial.

A collection of numbers lined up in rows and columns to form a rectangular array is called a matrix. The elements, or entries, of the matrix are the integers. In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics. In computer graphics, where they have been used to describe picture rotations and other transformations, matrices have vital applications as well.

Recall that the eigenvalues of a matrix are roots of its characteristic polynomial.

Hence if the matrices A and A Transpose have the same characteristic polynomial, then they have the same eigenvalues.

So we show that the characteristic polynomial pA(t)=det(A−tI) of A is the same as the characteristic polynomial p[tex]A^T[/tex] (t)=det(A ^T−tI)of the transpose [tex]A^T[/tex].

We have,

p[tex]A^T[/tex](t)=det([tex]A^T[/tex] −tI) =det([tex]A^T[/tex] −tI T )

=det((A−tI) [tex]A^T[/tex] )

=det(A−tI)

=pA(t)

Since I ^T =I

Since det([tex]B^T[/tex] )=det(B)for any square matrix B

Therefore, we obtain p[tex]A^T[/tex] (t)=pA(t), we conclude that the eigenvalues of A and [tex]A^T[/tex] are the same.

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An equation of the best-fit line of data can be used to estimate untested values. Graphical data can be used to show relationships between variables, but those relationships do not have to be linear and do not always have a positive relationship.

Graphical data is a way of visually representing information in a way that can be easily understood. It is often used to determine relationships between variables, such as the relationship between prices and sales. Graphical data can be used to show correlations, even if the relationship is not linear. An equation of the best-fit line of data can be used to estimate untested values, which can be useful in predicting future values. The x-values and y-values of a graph can both be estimated based on the data presented, and do not necessarily need to start at the origin. Graphical correlations can have both positive and negative relationships, depending on the data presented. Graphical data can be a powerful tool for understanding relationships between data points.

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The size of angle BAC in the given circle is 50°

A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.

Since, no figure is given, we will draw a circle according the given information, [attached in the solution]

So, we get a circle with center O, having chords, BC, which is extended to F, DC and AB,

We have, Angle CBD = 28° and angle FCD = 78°

Therefore,

∠ DCB = 180° - ∠ DCF [Linear pair]

∠ DCB = 180° - 78°

∠ DCB = 102°

In Δ DCB,

∠ DCB + ∠ BCD + ∠ BDC = 180°

∠ BDC = 180° - (102°+28°)

∠ BDC = 50°

Since, the angles made by the same chord are equal in measure,

Therefore,

∠ BDC = ∠ BAC

Therefore,

∠ BAC = 50°

Hence, the size of angle BAC in the given circle is 50°

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Work Shown:

[tex](f+g)(\text{x}) = f(\text{x})+g(\text{x})\\\\(f+g)(\text{x}) = (\text{x}+3)+(\text{x}^2-2)\\\\(f+g)(\text{x}) = \text{x}^2+\text{x}+1\\\\(f+g)(2) = 2^2+2+1\\\\(f+g)(2) = 4+2+1\\\\(f+g)(2) = 7\\\\[/tex]

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

Another approach:

[tex]\begin{array}{lll}f(\text{x}) = \text{x}+3& & g(\text{x}) = \text{x}^2-2\\f(2) = 2+3 & \text{ and } & g(2) = 2^2-2\\f(2) = 5 & & g(2) = 2\\\end{array}\\\\\\\text{Therefore, } (f+g)(2) = f(2)+g(2) = 5+2 = 7[/tex]

The total surplus received by both Patricia and Cecilia is $55.

The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.

Given that;

Patricia decides to buy a pair of boots for $90.00. She was willing to pay $120.00.

And, Her friend Cecilia sees the boots, she loves them and thinks they are worth $170.00. So she offers Patricia $145.00 for the boots, and Patricia accepts.

Hence, Consumer surplus is the difference between the willingness to pay of a consumer and the price of the good.

Consumer surplus = willingness to pay – price of the good

Consumer surplus received by Patricia = $120 - $90 = $30

Consumer surplus received by Cecilia = $170 - $145 = $25

Thus, Sum of the surpluses = $30 + $25 = $55

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Answer:

4^5 = 1024

Step-by-step explanation:

The unit rate for the games per loss are:

Jules’ Rules unit rate = 3 or 3/1

Pink sox unit rate = 4 or 4/1

Go - Girl unit rate = 4 or 4/1

High- 5s unit rate = 9 or 9/1

Unit rate = Games/Losses

Jules’ Rules unit rate = 36/12

Jules’ Rules unit rate = 3 or 3/1

Pink sox unit rate = 68/17

Pink sox unit rate = 4 or 4/1

Go- Girls unit rate = 52/13

Go - Girl unit rate = 4 or 4/1

High-5s unit rate = 72/8

High- 5s unit rate = 9 or 9/1

Therefore the Jules’ Rules unit rate 3, Pink sox unit rate 4, Go - Girl unit rate 4, High- 5s unit rate 9.

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Answer:

Cosine (cos)

Step-by-step explanation:

Sine, Cosine, and Tangent have their own properties.

Cosine is adjacent over hypotenuse. We have the angle of 59, the line adjacent is 17 and the hypotenuse is x. You could also do it for the right angle.

Based on the information in the table, a graph can be created to relate the number of Supply Dolls and Demand Dolls with respect to the sale value.

To create a graph with the information in the table, we must identify what information should go on the x-axis and the y-axis. In this case, we are being asked to locate the sale value on the x axis and on the y axis we must locate the number of units supplied and demanded.

Based on this information, we can infer that at a lower sale price, more dolls are demanded, but not as many are supplied. On the contrary, when the price is higher ($4.00) more dolls are supplied but not as many are demanded. This means that the relationship is inversely proportional.

Note: This question is incomplete because there is some misinformation. Here is the complete information:

Task.

Joan King is marketing director for the BurgeRama restaurant chain. BurgerRama has decided to have a cartoon-character doll made to sell at a premium price at participating BurgerRama locations. The company can choose from several different versions of the doll that sell different prices. King's problem is to decide which selling price will best suit the needs of BurgerRama's customers and store managers. King has data from previous similar promotions to help her make a decision.

1. Use the data from the table above to plot points representing selling price and supply price on a graph. (Selling Price of Each Doll should appear on the x-axis, and Number of Dolls Per Week per Store should appear on the y-axis). Draw the line through the data points and write the word "Supply on this line.

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The formula for the sum of cubes of the first n natural numbers is:

(1³+2³+3³+ ... + n³) = (1 + 2 + 3 + ... + n)²

Let us consider:

A(n)=1³+2³+⋯+n³--------(1)

B(n)=(1+2+⋯+n)²---------(2)

Now plug the numbers in the above equations(1)&(2) then,we get:

A(1)=B(1)=1

A(2)=B(2)=9

A(3)=B(3)=36

A(4)=B(4)=100

A(5)=B(5)=225

So both A and B are polynomials of degree at most 4 on n, and they coincide at 5 points.

Hence A(n) = B(n) for all n.

In general, p(n) is a polynomial of degree d on n,

∑kn=1 P(k)

is defined as a polynomial of the degree of d+1 on n. This trick will help solve every single issue involving the summation of polynomials, and then find the value on d+2 points, and we will determine the sum for all n.

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Answer:

the number is 78

Step-by-step explanation:

call x is the tens digit (0<x<9)

y is the unit digit (0[tex]\leq[/tex]y[tex]\leq[/tex]9)

we have:

10y+x=[tex]\oveline{yx}[/tex] (1)

10x+y= [tex]\overline{xy}[/tex] (2)

acording (1),(2) and the topic we have: (10y+x)-(10x+y)=9 ⇔9y-9x=9

⇔y-x=1 (3)

and x+y=15 (4)

from (3), (4) we have a system of equation

[tex]\left \{ {{y-x=1} \atop {x+y=15}} \right.[/tex]

⇔[tex]\left \{ {2y=16} \atop {x+y=15}} \right.[/tex]

⇔[tex]\left \{ {{y=8} \atop {x+8=15}} \right.[/tex]

⇔[tex]\left \{ {{y=8} \atop {x=7}} \right.[/tex] (conditions are satified)

the number is 78

The z score is -0.18 and 57 % of all Atlantic cod have more than 49.23 cm in length and

The relationship between a value and the mean of a set of values is expressed numerically by a Z-score. The Z-score is computed using the standard deviations from the mean. A Z-score of zero indicates that the data point's score and the mean score are identical.

The Z-score is calculated using the formula:

z = (x - μ)/σ

where z: standard score

x: observed value

μ: mean of the sample

σ: standard deviation of the sample

Given data ,

Let the probability that a randomly selected Atlantic cod has a length of 53.68 cm or less be represented as P

Now , the equation will be

The observed value of be x

The mean of the sample μ = 49.9 cm

The standard deviation of the sample σ = 3.74 cm

Now , z-score is calculated using the formula:

z = (x - μ)/σ

And , the length of 57% of all Atlantic cod have more than is calculated by

Now , the z score with a p value of 0.43 is

z = -0.18

Substituting the values in the equation , we get

-0.18 = ( x - 49.9 ) / 3.74

Multiply by 3.74 on both sides of the equation , we get

-0.6732 = x - 49.9

Adding 49.9 on both sides of the equation , we get

x = 49.2268 cm

Hence , 57 % of all Atlantic cod have more than 49.2268 cm in length and the z score is -0.18

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The area of the topper will be 24 square units.

A rectangle is a quadrilateral having four sides and the sum of the angles is 180 in the rectangle the opposite two sides are equal and parallel and the two sides are at 90-degree angles.

The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the rectangle in a two-dimensional plane is called the area of the rectangle.

The area of the glass topper will be:-

Area = Length x Width

Area = 6 x 4

Area = 24 square units

The image of the glass topper is attached with the answer below.

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Answer: 101.6

Since we are trying to find the sides of both the triangles, we will need to use the Sin, Cos, & Tan trigonometry functions. There are 3 equations:

[tex]Sin (Angle) = \frac{Opposite}{Hypotenuse} \\Cos (Angle) = \frac{Adjacent}{Hypotenuse} \\Tan (Angle) = \frac{Opposite}{Adjacent}[/tex]

For this triangle shown in the picture, we can infer that the 88 is OPPOSITE to the 60° and 30° angles. So we already have the opposite, and since we are trying to find the adjacent, we will use the formula that consists both opposite and adjacent, which is the following:

[tex]Tan(Angle) = \frac{Opposite}{Adjacent}[/tex]

Since [tex]x[/tex] is only a part of the long triangle, we will subtract the small triangle's side from the big one, which will give us [tex]x\\[/tex].

We can input our values to solve each triangle's side lengths.

(You have to use the calculator for these)

Small one:

[tex]Tan(60) = \frac{88}{Adjacent}\\Adjacent = \frac{88}{Tan(60)}\\Adjacent = \frac{88\sqrt{3} }{3} = 50.807 (3 d.p)[/tex]

Big one:

[tex]Tan(30) = \frac{88}{Adjacent} \\Adjacent = \frac{88}{Tan(30)} \\Adjacent = 88\sqrt{3} = 152.420 (3d.p)[/tex]

Now, to calculate [tex]x[/tex] , we will subtracat the small one from the big one like this: (it is better to input the values with the surds and not the real numbers for accurate marks and calculations)

[tex]88\sqrt{3} - \frac{88\sqrt{3} }{3} = 101.6 (1 d.p)[/tex]

That is the answer. If you did not understand it completely, please reach me back and I will help you :)

The length of a pendulum that makes one full swing in 1.9 seconds is 3 feet.

As per the given data, the given formula is:

[tex]$T=2 \pi \sqrt{\left(\frac{L}{32}\right)}[/tex] gives the time it takes in seconds.

Where,

T is the time it takes for a pendulum to make one full swing back and forth (in seconds).

L is the length of the pendulum, in feet.

Here we need to find the length of a pendulum that makes one full swing in 1.9 seconds to the nearest foot.

Substitute the given parameters into the formula will give;

[tex]$1.9=2 \pi \sqrt{\left(\frac{L}{32}\right)}[/tex]

Now divide both sides by 2π.

[tex]$ \frac{1.9}{2 \pi}=\sqrt{\left(\frac{L}{32}\right)} \\[/tex]

[tex]$ 0.3024=\sqrt{\left(\frac{L}{32}\right)}[/tex]

Then taking square on both sides.

[tex]$ (0.3024)^2=\frac{L}{32} \\[/tex]

0.09144576 = [tex]$\frac{L}{32}[/tex]

Later multiply both sides by 32.

2.9262 = L

The approximate value of the length of a pendulum:

L ≈ 3

Therefore, the length of a pendulum is about 3 feet.

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Answer:

[tex]y=x-2[/tex]

Step-by-step explanation:

The slant, or oblique asymptote, is found by taking the quotient of the two polynomials and disregarding the remainder:

-3 |  1  1  2

____  -3_6

      1 -2 | 8

Hence, our slant asymptote is [tex]y=x-2[/tex]

95% CI for the true posttreatment self-esteem score is:

(24.47, 28.73)

The formula for calculating a 95% confidence interval for the mean self-esteem score is as follows:

(standard deviation/square root of sample size) * (mean (critical value)

The standard deviation of a set of data is a measure of its spread or dispersion. It denotes the average deviation of individual values in a data set from the data set's mean.

A low standard deviation indicates that the data points are close to the mean, whereas a high standard deviation indicates that the data points are scattered across a wide range.

For a normal distribution and a 95% confidence level, the critical value is 1.96.

As a result, the 95% confidence interval for the mean posttreatment self-esteem score is as follows:

26.6 ± 1.96 * (7.4/sqrt(21)) = 26.6 ± 2.13

As a result, the 95% CI for the true posttreatment self-esteem score is:

(24.47, 28.73)

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The break-even point refers to the amount of revenue necessary to cover the total fixed and variable expenses incurred by a company.

In business accounting, the break-even point refers to the amount of revenue necessary to cover the total fixed and variable expenses incurred by a company within a specified time period.

To calculate the break-even point in units use the formula:

Break-Even point (units) = Fixed Costs ÷ (Sales price per unit – Variable costs per unit) or in sales dollars using the formula: Break-Even point (sales dollars) = Fixed Costs ÷ Contribution Margin.

Insurance Requirements the part of a commercial contract in which the types and minimum amounts of insurance the parties agree to provide in connection with their performance of the contract are specified.

A break even point is used in multiple areas of business and finance. In accounting terms, it refers to the production level at which total production revenue equals total production costs. In investing, the break even point is the point at which the original cost equals the market price.

Therefore, the break-even point refers to the amount of revenue necessary to cover the total fixed and variable expenses incurred by a company.

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The probability that the two tallest are in the same subgroup is given as follows:

2/n².

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes.

In this problem, we have that:

For each student, the probability of belonging to a group is given as follows:

1/n.

Hence the probability that the two tallest are in the same subgroup is obtained as follow:

2 x 1/n x 1/n = 2/n².

The multiplication by 2 is because there are two groups.

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The variance clearing account is used to accumulate individual variances until they can be allocated to their respective variance accounts, and the formula used to calculate the variance clearing account is the sum of all the individual variances, plus the beginning balance of the account.

A variance clearing account is a temporary ledger account used to accumulate individual variances such as direct labor cost, direct material cost, and overhead cost. The variance clearing account is used to collect all the individual variances in one account until it can be allocated to its respective variance accounts. This account has a debit side and a credit side. The debit side of the variance clearing account is used to accumulate all the unfavorable variances, while the credit side is used to accumulate all the favorable variances.

The formula used to calculate the variance clearing account is the sum of all the individual variances, plus the beginning balance of the account. This can be expressed as follows:

Variance Clearing Account = Beginning Balance + Sum of All Individual Variances

For example, if the beginning balance of the variance clearing account is $500, and the sum of all individual variances is $400, the balance in the variance clearing account would be $900 ($500 + $400). This balance can then be allocated to the respective variance accounts.

The variance clearing account is used to accumulate individual variances until they can be allocated to their respective variance accounts, and the formula used to calculate the variance clearing account is the sum of all the individual variances, plus the beginning balance of the account.

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Answer:

$400

Step-by-step explanation:

Here is my crack at it:

To find Heather's weekly salary, we can set up an equation using the information given. Let's call Heather's weekly salary "x".

According to the problem, $20 is 5% of her salary, so we can write this as an equation:

$20 = 0.05x

Now we can solve for x by dividing both sides by 0.05:

x = 400$

So Heather's weekly salary was $400.

Heather’s weekly salary would be 400$.  

If heather saved $20 a week. This was 5% of her salary. Heather’s weekly salary would be

Given, $20 = 5%

As we know, the total salary will be 100 %

If 5% = $20

then, 100% = $400 (multiplying both the sides by 20)

Thus, Heather’s weekly salary was 400$.

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This means that the average sale price for a home in this sample was "$773" and that half the sales were for less than the median price of "$577", while half were more than the median price of "$577".

(a) To calculate the sample mean, we add up all the values and divide by the number of values:

x = (592 + 817 + 577 + 608 + 353 + 1,281 + 412 + 535 + 550 + 679)/10 = 7,727/10 = $773

The sample median is the middle value when all the values are ordered from smallest to largest:

353, 412, 535, 550, 577, 608, 592, 679, 817, 1,281

The median is 577.

This means that the average sale price for a home in this sample was $773 and that half the sales were for less than the median price of $577, while half were more than the median price of $577.

(b) If the 6th observation had been 985 rather than 1,281, the new mean would be calculated as:

x = (592 + 817 + 577 + 608 + 353 + 985 + 412 + 535 + 550 + 679)/10 = 7,610/10 = $761

The new median would still be $577.

(c) To calculate a 20% trimmed mean, we first need to find the two smallest and two largest observations, then remove them and take the average of the remaining values:

Two smallest: 353, 412

Two largest: 1,281, 817

Remaining values: 535, 550, 577, 608, 592, 679

The trimmed mean is (535 + 550 + 577 + 608 + 592 + 679)/6 = $592

(d) To calculate a 15% trimmed mean, we first need to find the three smallest and three largest observations, then remove them and take the average of the remaining values:

Three smallest: 353, 412, 535

Three largest: 817, 1,281, 679

Remaining values: 550, 577, 608, 592

The trimmed mean is (550 + 577 + 608 + 592)/4 = $576.

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Answer:

5,4,2,0

Step-by-step explanation:

f(g(4))=5

g(f(3))=4

f(f(2))=2

g(g(5))=0

The value of composition function is:

1. f(g(4)) = 5

2. g(f(3)) = 4

3. f(f(2)) = 2

4. g(g(5)) = 0

To find the value of function focus on the graph of the function.

1. f(g(4))

From the graph of g(x), when x= 4 the value of y is 2.

So, g(4) = 2

Now, the value of f(g(4))

f(g(4))= f(2)

        = 5.

2. g(f(3))

From the graph of f(x), when x= 3 the value of y is 0.

So, f(3) = 0

Now, the value of g(f(3))

g(f(3))= g(0)

        = 4.

3. f(f(2))

From the graph of f(x), when x= 2 the value of y is 5.

so, f(2)= 5

Now, the value of f(f(2))

f(f(2)) = f(5)

        = 2

4. g(g(5))

From the graph of g(x), when x= 5 the value of y is 3.

so, g(5)= 3

Now, the value of g(g(5))

g(3) = 0

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Answer:

x * 2.54 = centimeters

Step-by-step explanation:

The equation to convert inches to centimeters is as follows:

x inches * 2.54 cm/inch = y centimeters

Where x is the number of inches and y is the equivalent length in centimeters.

To elaborate, the conversion factor between inches and centimeters is 2.54 cm/inch, meaning that for every inch there are 2.54 centimeters. To convert from inches to centimeters, we simply multiply the number of inches by the conversion factor. This gives us the equivalent length in centimeters.

Answer:For part (a), the overall percentage of American voters who say the government shutdown is a "very serious problem for the country" can be calculated as follows:

P(A) = 54% (probability of being a Democrat)

P(B|A) = 80% (probability of saying the shutdown is a "very serious problem" given that the person is a Democrat)

P(A and B) = P(A) * P(B|A) = 54% * 80% = 43.2%

P(B) = 35% (probability of saying the shutdown is a "very serious problem" for Republicans)

P(A|B) = (P(A and B) / P(B)) = (43.2% / 35%) = 123.14%

P(A or B) = P(A) + P(B) - P(A and B) = 54% + 35% - 43.2% = 45.8%

Therefore, the overall percentage of American voters who say the government shutdown is a "very serious problem for the country" is 45.8%.

For part (b), the 2x2 probability table can be set up as follows:

Serious Not serious Total

Democrat 43.2% 6.8% 50%

Republican 10.5% 35.5% 46%

Total 53.7% 42.3% 100%

Step-by-step explanation:

As a result, the original discounted price is $1 while the weekly normal price is $9.

The fundamental formula for calculating a reduction is to increase the initial price by the provided percentage rate in decimal form. We must deduct the reduction from the initial price to determine the item's sale price.

From the given information, we have:

d + 6x = 55 (discounted price for 7 weeks)

d + 11x = 100 (discounted price for 12 weeks)

We can solve this system of equations by eliminating d. Subtracting the first equation from the second, we get:

5x = 45

Dividing both sides by 5, we get:

x = 9

Substituting x = 9 into one of the equations, we can solve for d. Using the first equation, we get:

d + 6(9) = 55

d + 54 = 55

d = 1

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Answer:

3a^2

Step-by-step explanation:

one way of answering this question is by finding the area of the whole shape, including the non-shaded shape, then taking away the area of the non-shaded shape from the whole thing, leaving you the area of the shaded shape

Step-by-step

find the area of the whole shape

2a x 2a

= 4a^2 (or 4a to the power of 2)

Now that we have the area of the whole shape, we must find the area of the non-shaded shape and take it away from 4a^2, we do this by

a x a

= a^2 (or a to the power of 2)

now we minus the non-shaded shape away from the total area

4a^2 - a^2

= 3a^2 (or 3a to the power of 2)

Answer: 3y + 2

Step-by-step explanation: First, you can organize the equation into -3y + 6y + 4 - 2. Then, you get 3y + 2. :)))))

Solve for x in one of the equations and substitute it into the other equation.

To solve the system of equations 3x + 5y = 25 and x - 2y = -6 using substitution, we first solve for x in one of the equations. Let's solve for x in the second equation: x = 2y + 6. Next, we substitute this expression for x into the first equation: 3(2y + 6) + 5y = 25. This gives us 6y + 18 + 5y = 25, which simplifies to 11y + 18 = 25. Finally, we subtract 18 from both sides to obtain 11y = 7, and divide both sides by 11 to find y = 7/11. Now that we have y, we can substitute it back into either equation to find x.

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Step-by-step explanation:

The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. If a line is parallel to another line, it means they have the same slope. To find the slope of the line 7x - y = 3, we can rearrange it into slope-intercept form: y = 7x - 3. So, the slope of the line 7x - y = 3 is 7.

Since the line we want to find is parallel to 7x - y = 3 and has a slope of 7, the equation of the line can be written as y = 7x + b, where b is the y-intercept. To find b, we use the point (6, 3) as a solution:

y = 7x + b

3 = 7 * 6 + b

3 = 42 + b

b = -39

So, the equation of the line that passes through (6, 3) and is parallel to 7x - y = 3 is y = 7x - 39.