Use a truth table to determine whether the given statement is a tautology.

Answer:

x = 24.5

Step-by-step explanation:

27/(x + 7) = (27 - 6)/x

27x = (x + 7)(21)

27x = 21x + 147

6x = 147

x = 24.5

Answer:

x = 31.5

Step-by-step explanation:

given a line parallel to a side of a triangle and it intersects the other two sides then it divides those sides proportionally , that is

[tex]\frac{6}{27-6}[/tex] = [tex]\frac{7}{x}[/tex]

[tex]\frac{6}{21}[/tex] = [tex]\frac{7}{x}[/tex] ( cross- multiply )

6x = 21 × 7 = 147 ( divide both sides by 6 )

x = 24.5

The amount in the account after 4 years, rounded to the nearest cent, will be approximately $17,332.8.

Recall the compounding formula:

A = P * [tex]e^{rt}[/tex]

Where:

A = Final amount in the account

P = Initial principal (deposit)

e = Euler's number (approximately 2.71828)

r = Annual interest rate (as a decimal)

t = Time in years

Given:

Initial principal (P) = $16,000

Annual interest rate (r) = 2%  = 0.02

time (t) = 4 years.

Substitute these values into the formula, we get:

A = $16,000 * [tex]e^{0.02 * 4}[/tex]

Using a calculator, we can calculate:

A = $16,000 * [tex]e^{0.08}[/tex]

A = $16,000 * 1.0833

A = $17,332.8

Therefore, the amount in the account after 4 years, rounded to the nearest cent, will be approximately $17,332.8.

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For this problem, you need an on-hand calculator.

Problem:

sin (pi/8)

Step 1: divide pi by 8.

[tex]\pi[/tex]/8

=0.39269908169

Step 2: Find Cosine of (0.39269908169)

This is the part where you need a calculator.

Locate the SIN button on your graphing calculator.

Press the SIN button, then continue to type 0.39269908169 after the parenthesis.

The calculator gives a value of .0068538383

Final answer:

0.068538383

The first four terms of the sequence t/n = n(6n - 2) are: 4, 20, 48, 88.

Let's substitute n = 1:

t/1 = 1(6(1) - 2)

t = 1(6 - 2)

t = 4

The first term of the sequence (when n = 1) is 4.

Now, let's substitute n = 2:

t/2 = 2(6(2) - 2)

t = 2(12 - 2)

t = 2(10)

t = 20

The second term of the sequence (when n = 2) is 20.

Next, let's substitute n = 3:

t/3 = 3(6(3) - 2)

t = 3(18 - 2)

t = 3(16)

t = 48

The third term of the sequence (when n = 3) is 48.

Lastly, let's substitute n = 4:

t/4 = 4(6(4) - 2)

t = 4(24 - 2)

t = 4(22)

t = 88

The fourth term of the sequence (when n = 4) is 88.

Therefore, the first four terms of the sequence t/n = n(6n - 2) are: 4, 20, 48, 88.

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The solution for (P, Q, R) in the given equation (5400) + P x Ln30 + Q x Ln360 + R x Ln270, where P, Q, and R are real numbers, is (0, 0, 0). The correct solution is that P = Q = R = 0.

To find the values of P, Q, and R in the equation (5400) + P x Ln30 + Q x Ln360 + R x Ln270, use the properties of logarithms and solve the equation.

First, simplify the equation:

5400 + P x Ln30 + Q x Ln360 + R x Ln270

Next, express the logarithmic terms using their corresponding exponential forms:

5400 + P x Ln(30) + Q x Ln(360) + R x Ln(270)

= 5400 + P x Ln(2 x 3 x 5) + Q x Ln(2² x 3² x 5) + R x Ln(2 x 3³ x 5)

Using the properties of logarithms, we can rewrite the equation as follows:

5400 + P x (Ln(2) + Ln(3) + Ln(5)) + Q x (2 x Ln(2) + 2 x Ln(3) + Ln(5)) + R x (Ln(2) + 3 x Ln(3) + Ln(5))

Now, let's group the logarithmic terms together:

5400 + P x Ln(2) + P x Ln(3) + P x Ln(5) + Q x (2 x Ln(2) + 2 x Ln(3)) + Q x Ln(5) + R x Ln(2) + R x 3 x Ln(3) + R x Ln(5)

Simplifying further:

5400 + (P + 2Q + R) x Ln(2) + (P + 2Q + 3R) x Ln(3) + (P + Q + R) x Ln(5)

Then, the equation:

5400 + (P + 2Q + R) x Ln(2) + (P + 2Q + 3R) x Ln(3) + (P + Q + R) x Ln(5) = 0

Since this equation holds for all real numbers P, Q, and R, the coefficients of Ln(2), Ln(3), and Ln(5) must be equal to zero:

P + 2Q + R = 0 (1)

P + 2Q + 3R = 0 (2)

P + Q + R = 0 (3)

There is a system of three equations (equations 1, 2, and 3) with three unknowns (P, Q, and R). To solve this system, use any method such as substitution or elimination.

Subtracting equation (3) from equation (2), we get:

(P + 2Q + 3R) - (P + Q + R) = 0 - 0

2Q + 2R = 0

2Q = -2R

Q = -R/2 (4)

Substituting equation (4) into equations (1) and (3), we can solve for P and R:

P + 2Q + R = 0 (1)

P + (-R) + R = 0 (3)

P - R = 0

From equation (3), P = R.

Substituting P = R into equation (4):

Q = -R/2

Since P = R and Q = -R/2. Let's substitute these values back into the equations to find the final solution.

Substituting P = R and Q = -R/2 into equation (1):

P + 2Q + R = 0

R + 2(-R/2) + R = 0

R - R + R = 0

0 = 0

Equation (1) holds true for any real value of R.

Substituting P = R and Q = -R/2 into equation (2):

P + 2Q + 3R = 0

R + 2(-R/2) + 3R = 0

R - R + 3R = 0

3R = 0

R = 0

So, R = 0.

Substituting R = 0 into Q = -R/2:

Q = -R/2

Q = 0/2

Q = 0

Finally, substituting R = 0 into P = R:

P = R

P = 0

Therefore, the solution for (P, Q, R) in the given equation (5400) + P x Ln30 + Q x Ln360 + R x Ln270, where P, Q, and R are real numbers, is (0, 0, 0). The correct solution is that P = Q = R = 0.

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Answer:

B) 16.47

Step-by-step explanation:

In order to find the mean of grouped data with intervals, we use the formula

(∑f * m) / (∑f)

Step 1:  First, we need to find the sum of the frequencies: ∑f = 2 + 3 + 8 + 4 = 17

Step 2:  Next, we need to find the midpoint (m) of each interval.  We do this by averaging the end points of each interval

m of first interval:  (9.5 + 12.5) / 2 = 11

m of second interval:  (12,5 + 15.5) / 2 = 14

m of third interval:  (15.5 + 18.5) / 2 = 17

m of fourth interval:  (18.5 + 21.5) / 2 = 20

Step 3:  Now, we multiply the frequency for each interval by its corresponding midpoint and add them together to find the sum

f * m for first interval: (2 * 11) = 22

f * m for second interval:  (3 * 14) = 42

f * m for third interval:  (8 * 17) = 136

f * m for fourth interval:  (4 * 20) = 80

Sum of f * m for each interval:  22 + 42 + 136 + 80 = 280

Step 4:  Finally, we divide the sum of f * m for each interval by the sum of f to find the mean:

280 / 17 = 16.47058824 = 16.47

The length of the hypotenuse of the given right triangle is 117 millimeters because it is the longest side and it satisfies the conditions of the Pythagorean theorem.

The length of the hypotenuse of a right triangle with side lengths 45 millimeters, 108 millimeters, and 117 millimeters is 117 millimeters.

The hypotenuse of a right triangle is always the side opposite the right angle, and it is the longest side in a right triangle.

In this case, the side lengths are given as 45 millimeters, 108 millimeters, and 117 millimeters.

By comparing the lengths, we can see that 117 millimeters is the longest side, which makes it the hypotenuse.

To further understand why the hypotenuse is the longest side, we can refer to the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Mathematically, it can be written as c² = a² + b²,

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

In this case, if we square the lengths of the sides, we get 45² = 2025, 108² = 11664, and 117² = 13689.

By comparing these values, we can see that 13689 is the largest, which corresponds to the length of the hypotenuse, 117 millimeters.

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Darlene's net proceeds from the investment into shares would be $ 8, 279 .

First, find the initial broker fee:

=  Initial investment × Broker fee rate

= 20, 000 x 1. 5 %

= 20, 000 x 0. 15

= $ 300

The total fees incurred to buy and sell are:

= 300 initial broker + 21 later broker

= $ 321

The net proceeds would then be:

= Final value of stock - Initial investment - Total fees

= 28, 600 - 20, 000 - 321

= 8, 600 - 321

= $ 8, 279

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Full question is:

Darlene purchases $20,000 worth of stock on her brokers advice and pays her broker a 1.5% broker fee. She sells her stock when it increases to $ 28,600  two years later, and uses a discount broker who charges $21 per trade. Compute Darlene's net proceeds after the broker fees are taken out.

Answer:

(3xy - 1)/3x = b

Step-by-step explanation:

make b the subject of formula

then simply it

did you get it

Answer:

b=Y-x/3

Step-by-step explanation:

Solve for b by simplifying both sides of the equation, then isolating the variable.

have a great day and thx for your inquiry :)

The area of the rectangular shaped kitchen floor is A = 96 feet²

Given data ,

The side length of each tile is 2ft, then the area of each tile is:

Area of a tile = side x side = 2ft x 2ft = 4 sq ft

Since there are 6 rows of tiles, each row has 4 tiles, therefore the total number of tiles is:

Total number of tiles = 6 x 4 = 24 tiles

So, the total area covered by the tiles is:

Total area covered by tiles = Area of a tile x Total number of tiles

= 4 sq ft/tile x 24 tiles

= 96 sq ft

Hence , the area of the kitchen floor is 96 feet²

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The sales tax on the computer is $69.93.

To calculate the sales tax on the computer, we need to multiply the price of the computer by the sales tax rate. Here's how you can do it:

Price of the computer: $999.00

Sales tax rate: 7% (which can be expressed as 0.07)

Sales tax = Price of the computer * Sales tax rate

Sales tax = $999.00 * 0.07

To find the sales tax in dollars and cents, we can perform the calculation:

Sales tax = $999.00 * 0.07

Sales tax = $69.93

Therefore, the sales tax on the computer is $69.93.

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The total cost of materials needed to build the house is $4374.75.

Calculate the area of the front and back of the house.

Area = Length x Width

Area = 3.5 m x 2.5 m = 8.75 m²

Calculate the area of the sides of the house.

Area = 2 x Length x Height

Area = 2 x 3.5 m x 5 m = 35 m²

Calculate the total cost of glass.

Cost of glass = Area x Cost per square meter

Cost of glass = 8.75 m² x $37 / m² = $323.75

Calculate the total cost of roof tiles.

Cost of roof tiles = Area x Cost per square meter

Cost of roof tiles = 35 m² x $95 / m² = $3325

Calculate the total cost of reinforcement metal.

Cost of reinforcement metal = Perimeter x Cost per meter

Cost of reinforcement metal = 2 x (Length + Width) x $54 / m = $726

Add the costs of glass, roof tiles, and reinforcement metal to get the total cost of materials.

Total cost = $323.75 + $3325 + $726 = $4374.75

Therefore, the total cost of materials needed to build the house is $4374.75.

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The value of distance of two points shown in figure is,

⇒ d = √20

We have to given that;

Two points on graph are,

(0, 1) and (2, - 3)

Since, The distance between two points (x₁ , y₁) and (x₂, y₂) is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

Hence, The distance of two points shown in figure is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

⇒ d = √ (2 - 0)² + (-3 - 1)²

⇒ d = √4 + 16

⇒ d = √20

Thus, The value of distance of two points shown in figure is,

⇒ d = √20

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The properties of equality that shows justification of how the equation is solved is explained below.

Given the equation, 17/3 - 3/4x = 1/2x + 5, we have the following steps and justification which explains the property of equality that was used:

17/3 - 3/4x = 1/2x + 5 [given]

17/3 - 3/4x - 17/3 = 1/2x + 5 - 17/3 [subtraction property of equality]

3/4x = 1/2x - 2/3 [simplification]

3/4x - 1/2x = 1/2x - 2/3 - 1/2x [subtraction property of equality]

-5/4x = -2/3 [simplification]

-5/4x * -4/5 = -2/3 * -4/5 [multiplication property of equality]

x = 8/15 [simplification]

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The exponential value equation is A = 1/√x = 1/x^(-1/2)

Given data ,

Let the exponential equation be represented as A

Now , the value of A is

A = x ^ ( 1/6 ) / x ^ ( 2/3 )

On simplifying , we get

The different Laws of exponents are:

mᵃ×mᵇ = mᵃ⁺ᵇ

mᵃ / mᵇ = mᵃ⁻ᵇ

( mᵃ )ᵇ = mᵃᵇ

mᵃ / nᵃ = ( m / n )ᵃ

m⁰ = 1

m⁻ᵃ = ( 1 / mᵃ )

So , A = x ^ ( 1/6 - 2/3 )

On further simplification , we get

A = x ^ ( -3/6 )

A = x ^ ( -1/2 )

So , the value of A is

A = 1/√x

Hence , the exponential equation is solved

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The unknown length of the similar figure is:

x = 15 units

Two figures are similar if they have the same shape but different sizes. The corresponding angles are equal and the ratios of their corresponding sides are also equal.

Using the above concept, we can equate the ratio of the corresponding sides and solve for the unknown lengths. That is:

TV/AC = TU/AB

x/10 = 6/4

x/10 = 1.5

x = 10 * 1.5

x = 15 units

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Answer:

25

Step-by-step explanation:

There are only additions, so do each addition, in the order it appears, from left to right.

-11 + 3 + 10 + (-25) + 48 =

= -8 + 10 + (-25) + 48

= 2 + (-25) + 48

= -23 + 48

= 25

Answer:

471.1 square units

Step-by-step explanation:

A regular nonagon is a 9-sided polygon with sides of equal length.

The apothem of a regular polygon is the distance from the center of the polygon to the midpoint of one of its sides.

Therefore, the given diagram shows a regular nonagon with an apothem of 12 units.

The side length (s) of a regular polygon can be calculated using the apothem formula:

[tex]\boxed{\begin{minipage}{5.5cm}\underline{Apothem of a regular polygon}\\\\$a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\where:\\\phantom{ww}$\bullet$ $s$ is the side length.\\ \phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]

Given values:

Substitute the given values into the formula to create an expression for the side length (s):

[tex]12=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{9}\right)}[/tex]

[tex]12=\dfrac{s}{2 \tan\left(20^{\circ}\right)}[/tex]

[tex]s=24 \tan\left(20^{\circ}\right)[/tex]

The standard formula for an area of a regular polygon is:

[tex]\boxed{\begin{minipage}{6cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{n\cdot s\cdot a}{2}$\\\\where:\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\ \phantom{ww}$\bullet$ $s$ is the length of one side.\\ \phantom{ww}$\bullet$ $a$ is the apothem.\\\end{minipage}}[/tex]

Substitute the found expression for s together with n = 9 and a = 12 into the formula and solve for A:

[tex]A=\dfrac{9 \cdot 24 \tan(20^{\circ}) \cdot 12}{2}[/tex]

[tex]A=1296\tan(20^{\circ})[/tex]

[tex]A=471.705423...[/tex]

[tex]A=471.7\; \sf square\;units\;(nearest\;tenth)[/tex]

Therefore, the area of a regular nonagon with an apothem of 12 units is 471.1 square units, rounded to the nearest tenth.

Answer:

150

Step-by-step explanation:

To determine the number of possible combinations for the briefcase, we need to multiply the number of options for each wheel.

The first wheel has 10 digits, so there are 10 possibilities.

The second wheel has 5 digits, so there are 5 possibilities.

The third wheel has 3 digits, so there are 3 possibilities.

To calculate the total number of combinations, we multiply the possibilities for each wheel:

10 × 5 × 3 = 150

Therefore, there are 150 possible combinations for the briefcase.

Answer: 150

Step-by-step explanation:

I’m not completely sure, but this is going by my logic. The first one has ten digits and any of those ten digits could go with 1 digit from the 5 wheel, and 1 digit from the 3 wheel, so (I think) you have to multiply 10x5x3 to get the amount of possible combinations.

But, if this is wrong, please correct me! I love puzzles and just though this would be fun to try and figure out, I don’t want to spout out any mis or disinfo to anyone

But, in the case my answer’s right, hope this helped :D

The volume of the solid is given as follows:

55 ft³.

The volume of a rectangular prism, with dimensions length, width and height, is given by the multiplication of these dimensions, according to the equation presented as follows:

Volume = length x width x height.

The total volume would be given as follows:

V = 9 x 6 x 1.25

V = 67.5 ft³.

The volume of the removed part is given as follows:

V = 1.25 x 2.5 x (9 - 3 - 2)

V = 12.5 ft³.

Hence the volume of the solid is given as follows:

67.5 - 12.5 = 55 ft³.

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There are 4 variables in the expression 8d² + 2bc + 10ab

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

8d²+2bc+ 10ab

Express properly

So, we have

8d² + 2bc + 10ab

Consider an expression ax + by where the variable is a and b are constants

The variables in the expression are x and y

Using the above as a guide, we have the following:

The variables in the expression 8d² + 2bc + 10ab are a, b, c and d

This means that there are 4 variables in the expression

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Salina Grant's gross pay for assembling 47 chairs in one week at Chairs and More is $567.50.

To calculate Salina Grant's gross pay, we need to determine the pay for each range of chairs assembled and sum them up.

Let's break down the calculation based on the given differential piece rate scale:

1-25 chairs: $10 each

Salina assembled 25 chairs in this range, so the pay for this range is

25 * $10 = $250.

26-40 chairs: $13 each

Salina assembled 15 chairs in this range, so the pay for this range is

15 * $13 = $195.

Chairs over 40: $17.50 each

Salina assembled 47 chairs in total, and since she already assembled 25 chairs in the first range and 15 chairs in the second range, the number of chairs in this range is

47 - 25 - 15 = 7 chairs.

The pay for this range is

7 * $17.50 = $122.50.

Now, let's sum up the pay for each range to find Salina Grant's gross pay:

$250 + $195 + $122.50 = $567.50

Therefore, Salina Grant's gross pay for assembling 47 chairs in one week at Chairs and More is $567.50.

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The measure of angle ABC is given as follows:

m < ABC = 55º.

First we look to the exterior angle of angle C in the bottom parallel line, as a ray is formed, hence the missing angle measure is given as follows:

50 + 75 + x = 180

125 + x = 180

x = 55º.

The exterior angle relative to angle B is supplementary with the angle of 55º, as they are consecutive angles, hence it’s measure is obtained as follows:

y + 55º = 180

y = 125º.

By the exterior angle theorem, an interior angle is supplementary with it’s respective exterior angle, hence the measure of angle ABC is given as follows:

m < ABC + 125 = 180

m < ABC = 55º.

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Answer:  The jar contains a total of 18 coins (8 pennies + 3 nickels + 5 dimes + 2 quarters). To get more than 4 cents, we need to choose a nickel, dime, or quarter. There are 3 nickels, 5 dimes, and 2 quarters, for a total of 10 coins that satisfy this condition. Therefore, there are 10 ways to choose a coin from the jar that is more than 4 cents.

Based on the analysis, the correct transformation that maps triangle △UVW to triangle △EFG is option C: (x, y) ⟶ (2x, 2y), which represents an enlargement or dilation of the figure.

To determine which transformation maps triangle △UVW to triangle △EFG, we need to analyze the given options and understand the effects of each transformation.

A. (x, y) ⟶ (0.5x, 0.5y):

This transformation scales the coordinates of a point by a factor of 0.5 in both the x and y directions.

It represents a reduction or contraction of the figure.

Triangle △UVW would be scaled down to a smaller size, so option A is not the correct transformation.

B. (x, y) ⟶ (–0.5x, 0.5y):  

This transformation reflects the coordinates of a point across the y-axis and scales the x-coordinate by a factor of -0.5 and the y-coordinate by a factor of 0.5.

It represents a reflection and scaling.

Triangle △UVW would be reflected across the y-axis and scaled, which does not match the transformation to △EFG.

Therefore, option B is not the correct transformation.

C. (x, y) ⟶ (2x, 2y): This transformation scales the coordinates of a point by a factor of 2 in both the x and y directions.

It represents an enlargement or dilation of the figure.

Triangle △UVW would be scaled up to a larger size, which aligns with the transformation to △EFG.

Therefore, option C could be the correct transformation.

D. (x, y) ⟶ (–2x, 2y):

This transformation reflects the coordinates of a point across the x-axis and scales the x-coordinate by a factor of -2 and the y-coordinate by a factor of 2.

It represents a reflection and scaling.

Triangle △UVW would be reflected across the x-axis and scaled, which does not match the transformation to △EFG.

Thus, option D is not the correct transformation.

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The amount of money that will produce Rs. 15000 at the end of 6 month for 10years at 16% is Rs. 3750

Compound interest is the interest you earn on interest.

For example, if you have $100 and it earns 5% interest each year, you'll have $105 at the end of the first year. At the end of the second year, you'll have $110.25.

Compound interest is expressed as;

A = P( 1+r/n) ^{nt}

where A is the amount

n is the number of time its compounded

t is time

A = 15000

r = 16/100 = 0.16

n = 6/12 = 0.5

t = 10 years

15000 = P( 1 + 0.16/0.5) ^{ 0.5× 10}

15000 = P( 1 + 0.32)⁵

15000 = P( 1.32)⁵

15000 = 4P

P = 15000/4

P = Rs. 3750

Therefore the amount that will be deposited to give Rs. 15000 is Rs. 3750.

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The value of x, considering the Pythagorean Theorem, is given as follows:

x = 9.

The Pythagorean Theorem states that in the case of a right triangle, the square of the length of the hypotenuse, which is the longest side,  is equals to the sum of the squares of the lengths of the other two sides.

Hence the equation for the theorem is given as follows:

c² = a² + b².

In which:

The sides for the problem are given as follows:

15 and 8 (segment of 8 is the radius, which is the distance from the center to a point on the circumference of the circle).

Hence the hypotenuse is obtained as follows:

h² = 15² + 8²

h² = 289

h = 17.

Hence the value of x is obtained as follows:

x = 17 - 8

x = 9.

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The system of equations has infinitely many solutions, which corresponds to option (OB).

We can use the graphing function on a calculator to find the solution to the system of equations:

[tex]3y - 12x = 18\\\\2y - 8x = 12[/tex]

To do this, we can rearrange each equation to solve for y in terms of x:

[tex]3y = 12x + 18\\\\y =4x + 6[/tex]

For the second equation.

[tex]2y = 8x + 12\\\\y = 4x + 6[/tex]

We can see that the two equations have the same slope (4) and y-intercept (6). Therefore, the two equations represent the same line, and any point on that line will satisfy both equations.

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