For what value of k does the pair of equations 3x y − 8 and 6x KY 16 0 have infinitely many solutions?.

If the discriminant b² - 4ac = 0 for a quadratic equation ax² + bx + c = 0, it means that the nature of roots of the equation has one real and identical (repeated) root.

When the discriminant is equal to zero, the square root of the discriminant will be zero, and the equation will have one solution for x, regardless of the values of a, b and c.

This can be observed in the quadratic formula, where the square root of the discriminant is used to determine the solutions of x. In this case the equation will be reduced to:

x = -b/2a

It's also possible to observe this by factoring the equation, in this case the equation can be factored as (x - r)^2 = 0, where r is the common root. This means that the graph of the equation will touch the x-axis only once.

In summary, when the discriminant b² - 4ac = 0, the roots of the quadratic equation are real and identical (repeated).

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To have a unique solution for the given equations, k should not be equal to 3.

Given equations are:

kx - y = 2

and, 6x - 2y = 3

The second equation can be written as, 3x - y = 3/2

The system of equations kx-y=2 and 3x - y = 3/2 has a unique solution when the coefficients of x and y are not equal, meaning k is not equal to 3.

This is because when the coefficients are not equal, the equations are not dependent on each other and can be solved independently. If the coefficients were equal, then the equations would be parallel and have infinitely many solutions.

Additionally, the specific value of k does not affect the uniqueness of the solution, as long as it is not equal to 3.

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The formulas of closure property is

where a, b and R are real numbers.

In mathematics, the Closure property can be defined as when an arithmetic operation is performed on any two numbers from a set and the result obtained is a number from the set itself, it is considered closed.

Closure property of addition : The closure property of an integer says that when we add two real numbers, the result will always be a real number. The formula for this is written as

a + b = R; where a, b and R are real numbers.

For example, 6 + 13 = 19 ; (1/3) + (5/2) = 17/6

Closure property under multiplication: The product of two real numbers is always a real number, that is, real numbers are closed under multiplication. The formula, ab = R , where a, b, and R are real numbers. For example, 9 × 0 = 0

(3/4) × (-1/4) = -3/16, √3 ×√5 = √15.

The Closure Property of Subtraction states that if any two real numbers a and b are subtracted from each other, the difference or result will also be a real number. For example, 9 - 4 = 5. This property is only applicable to integers and rational numbers.

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Answer:

4.5x + 27.5

Step-by-step explanation:

first you must simplify the equation using distributive property:

2.5(3x +8) - 1.5(2x - 5) = 7.5x + 20 - 3x - 7.5

Now, you and or subtract the like terms

7.5x + 20 - 3x - 7.5 = 4.5x + 27.5

Answer:

-4.5x - 27.5

Step-by-step explanation:

   1.5(2x - 5 ) - 2.5(3x + 8) -

=  3x -  7.5 - 7.5x  - 20

=  -4.5x - 27.5

To find the direction and magnitude of the resultant of these three forces, we can use the law of cosines and the law of sines.

First, we need to resolve each force into its x and y components using trigonometry.

F1x = 80 * cos(30) = 69.28

F1y = 80 * sin(30) = 40

F2x = 95 * cos(45) = 67.08

F2y = 95 * sin(45) = 67.08

F3x = 125 * cos(120) = -62.5

F3y = 125 * sin(120) = -106.6

Then we can find the x and y components of the resultant force by adding the x and y components of each force respectively.

Rx = F1x + F2x + F3x = 69.28 + 67.08 - 62.5 = 74.86

Ry = F1y + F2y + F3y = 40 + 67.08 - 106.6 = 0.48

Finally, using these values we can use the law of cosines to find the magnitude of the resultant force:

R = sqrt(Rx^2 + Ry^2) = sqrt(74.86^2 + 0.48^2) = sqrt(5633.79) = 75.24

And we can use the law of sines to find the direction of the resultant force:

theta = atan2(Ry, Rx) = atan2(0.48, 74.86) = 0.01

Therefore, the magnitude of the resultant force is 75.24 pounds and the direction of the resultant force is 0.01 degrees.

Answer:

Step-by-step explanation:

You want the end behavior of the graphed line.

The end behavior of a graphed function is found by looking at the arrows on the ends of the graph. If the arrow points in a direction with positive slope, the sign of the function value (y) will match the sign of the independent variable (x).

If the arrow has negative slope, then the signs of the end behaviors will be opposite.

If the arrow points horizontally or vertically, then the end behavior will be an asymptote. The function value will approach the value of a horizontal asymptote. The function value will tend to infinity (up: +∞; down: -∞) as the independent variable approaches the location of the vertical asymptote.

Here, the line has positive slope everywhere, so the sign of y will match the sign of x as both go to ±∞.

__

Additional comment

Slope is positive if the sign of the "rise" matches the sign of the "run". A line that goes down to the left has positive slope, just as does a line that goes up to the right. (Right and up are positive directions.)

Using Trigonometric ratios, Tan(20 ) = -24/7 and cos(20) = 7/25

The trigonometric functions in mathematics are real functions that connect the right-angled triangle's angle to the ratios of its two side lengths. They are extensively employed in all fields of geometry-related study, including geodesy, solid mechanics, celestial mechanics, and many others.

It can be simpler to use a calculator when necessary. (See under)

The reference angle will be less than 45° since the magnitude of the tangent is less than 1. If the angle is less than 90 degrees when doubled, it will remain in the fourth quadrant, where the tangent is negative and the cosine is positive.

Additionally, you can use the identities.

cos(2) is equal to (1 - tan()2)/(1 + tan()2).

cos(2θ) = (1 -(-3/4)²)/(1 +(-3/4)²) = ((16-9)/16)/((16+9)/16)

cos(2θ) = 7/25

Tan(2)=2tan()/(1 -tan(2)2) = 2(-3/4)/((16-9/16) = (-6/4)(16/7)

tan(2θ) = -24/7

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Answer:

{-1, 2, 47}

Choice B

Step-by-step explanation:

Function is g(x) = x² - 2

Domain is the possible set of x values and range is the set of function values for that domain

Here Domain = {- 2, - 1, 7}. This means these are values for which the function value g(x) exists

(Note this is not the entire domain, only a subset of the domain)

At x = - 2, g(x) = (-2)² - 2 = 4 - 2 = 2

At x = - 1, g(x) = (-1)² -2 = 1 -2 = -1

At x = 7, g(x) = (7)² -2 = 49 - 2 = 47

These are the corresponding range values for the given domain

Both domain and range sets are expressed from lowest to highest so we get range of g(x) as

{-1, 2, 47}

Choice B

Answer:

the picture is blurry I can not see the question to help you!

Answer:

-2x - 1

Step-by-step explanation:

3x - ( -2x - 1 )

=3x + 2x + 1

= 5x + 1

If Phil received £235 in the ratio 3:5, we can use this information to find out how much money was shared in total.

We know that the ratio of Samantha's share to Phil's share is 3:5.

This means that for every 3 parts of the total money that Samantha gets, Phil gets 5 parts.

We can use this information to set up a proportion:

(Samantha's share) / (Phil's share) = (3) / (5)

We know that Phil's share is £235, so we can substitute that into the proportion:

(Samantha's share) / (235) = (3) / (5)

To find the total amount of money shared, we can cross-multiply and divide:

Samantha's share = (3/5) * 235

Samantha's share = 141

To find the total amount of money shared, we can add the amount Samantha received to the amount Phil received:

141 + 235 = £376

Therefore, £376 was shared in total.

Answer:

Hmmm, I do not know, but thanks for asking!

Step-by-step explanation:

...

Answer:

11.11 hours

Step-by-step explanation:

Average Bike Speed on Flat Terrain: 18 mph

So Colton can travel 200 miles in 200 / 18 = 11.11 hours

A polynomial function of least degree with a leading coefficient of 1 and zeroes at -2, 3, 6 is given by:  "f(x) = (x + 2)(x - 3)(x - 6) = x^3 - x^2 - 9x - 12)".

According to the Factor Theorem, k is a zero of a given polynomial function f (x), if and only if (x - k) is a factor of the given polynomial function f(x).

Since -2,3 and 6 are zeros to be given of f (x), so

you need to put -2,3 and 6 in (x - k)

such as

f (x) =  { (x - k) (x - k) (x - k) }

     =  { (x - (-2) (x - 3) (x - 6) }

      =  {(x+2) (x-3)(x-6) }

      =  {( x^3 - x^2 - 9x - 12) }

'f(x) = { (x^3 - x^2 - 9x - 12) }' is the required polynomial function.

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The Product of two mixed fractions is always greater than 1.

Now, According to the question;

1. Convert the mixed number into an improper fraction.

2: Multiply the numerators of the fraction and multiply the denominators of the fraction.

3: Convert it into simplified form if required.

A fraction represented with its quotient and remainder is a mixed fraction. For example, 2 1/3 is a mixed fraction, where 2 is the quotient, 1 is the remainder. So, a mixed fraction is a combination of a whole number and a proper fraction.

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The values that represent the inverse of the function which is represented in the table above include the following:

x  4    2    1    3   4

y -2   -3   6   7    5

In Mathematics, an inverse function simply refers to a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).

In order to determine the inverse of any function, you should interchange (swap) both the input value (x-value) and output value (y-value). This ultimately implies that, the inverse relation of any function has its input value (x-value) and output value (y-value) interchanged (swapped) as follows;

x           y

4          -2

2          -3

1           6

3          7

4          5

In conclusion, we can logically deduce that the above inverse relation is not a function because the same input value (4) appears more than once with different output values (-2 and 5).

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30 1/2 cups soup divided by 3: 10 1/6 cups or 10 1/2 cups per container.

What is interpretation ?

Interpretation is the process of understanding and explaining the meaning of something, such as a text, a piece of artwork, or a scientific data. In the context of mathematics and problem solving, interpretation refers to understanding the meaning of a mathematical expression or equation in the context of the problem or situation being described.

In this context, "30 and 1 half cups of oup" refers to the amount of soup that Avery has in the large pot, and "divided by 3" refers to the number of smaller containers Avery plans to divide the soup into. There are two different ways to interpret 30 and 1 half divided by 3 in this context ,

Avery will divide the 30 and 1 half cups of soup equally among 3 smaller containers. This means that each container will hold (30 + 1/2) / 3 = 10 and 1/6 cups of soup.

Avery will pour out 10 and 1/2 cups of soup from the large pot into each of the 3 smaller containers. This means that after Avery has poured out all the soup, the large pot will be empty and the 3 smaller containers will each contain 10 and 1/2 cups of soup.

30 1/2 cups soup divided by 3: 10 1/6 cups or 10 1/2 cups per container.

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Avery will divide the 30 and 1 half cups of soup equally among 3 smaller containers. This means that each container will hold (30 + 1/2) / 3 = 10 and 1/6 cups of soup.

Avery will pour out 10 and 1/2 cups of soup from the large pot into each of the 3 smaller containers. This means that after Avery has poured out all the soup, the large pot will be empty and the 3 smaller containers will each contain 10 and 1/2 cups of soup.

3/5 is greater than 1/3

What is the fraction?

A fraction is a way of representing a number that is not a whole number. It consists of two parts: a numerator and a denominator. The numerator represents the number of equal parts of a whole, and the denominator represents the total number of parts in the whole.

One way to compare the fractions 1/3 and 3/5 is to find a common denominator between them. In this case, the least common multiple of 3 and 5 is 15. So, you can convert 1/3 to 5/15 and 3/5 to 9/15. Now, you can compare the two fractions by looking at their numerators: 5 and 9. Since 9 is greater than 5, you can say that 3/5 is greater than 1/3.

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Answer:

The time t, in seconds, it takes a runner to finish a race depends on the distance in miles m of the race.

Step-by-step explanation:

Hope this helps !

5. 70

6. not sure sorry :/

The number 120 symbolizes  the restaurant that has been open for 120 months.

Use Slope-intercept form

The equation of the straight line is provided by:

y =mx+ b

m stands the slope of the line

b stands for the initial value or y-intercept.

As per the information:

The scatter plot displays the association between the average number of nightly consumers and the number of months since a restaurant opened.

The equation describes the linear standard for these details

y = 5x + 120

Where,

y denotes the number of months since a restaurant opened and,

x denotes the average number of nightly customers.

We have to discover the number 120 in the equation indicate in this context.

On analogizing provided equation with  we have;

m=5

b= 120

means y(0) = 120

and

Thus, The number 120 symbolizes  the restaurant that has been open for 120 months.

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The equation that relates the number of coins, y, in Janice's collection x months since she started adding to her collection is y = 26 + 5/3 x In the context of this problem, the y-intercept is the initial number of coins that Janice had in her collection while the slope is the rate at which Janice is adding to her collection which is 5/3.

Part A

The equation for the number of coins, y, in Janice's collection x months since she started adding to her collection is the total amount of coins she has and can be expressed as

y = 26 + 5/3 x

where y is the total number of coins, and x is the number of months.

Part B

The y-intercept of the equation is the point where the line of the equation crosses the y-axis. In this case, the y-intercept is the point (0,26) which means that when x = 0, the number of coins in Janice's collection is 26. This is the initial number of coins that Janice had in her collection before she started adding to it.

The slope of the equation is the rate of change of the dependent variable (y) with respect to the independent variable (x). In this case, the slope of the equation is 5/3, which means that for every 3 months that pass, the number of coins in Janice's collection increases by 5. So the slope is the rate at which Janice is adding to her collection.

The problem seems incomplete. It must have been

"Janice has a coin collection that began with 26 coins. Since then, she has been adding to her collection at a rate of 5 coins every 3 months. Part A Write an equation that relates the number of coins, y, in Janice's collection x months since she started adding to her collection. ​ ​ ​Part B ​ ​In the context of this problem, interpret and explain the y-intercept and slope of the graph of the equation."

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The inverse of y = 3x is f⁻¹(x) = 1/3x.

The inverse is a function or an operation which reverses the order of operation of another function or an operation”.

There are a few steps to find the inverse of a number:

We will be using the concept of inverse function to solve this.

Given that, y=3x

Let f(x) =y=3x

Switch x and y

x=3y

Express y in terms of x.

y = 1/3x

From the definition of the inverse function. f(x)=y⇒x=f⁻¹(y)

⇒ f⁻¹(y)=1/3x

Thus, the inverse of y = 3x is f⁻¹(x)=1/3x.

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Using addition, the total amount of sugar sold equals  13 16/20 Kg .

The numbers being added are known as addends, while the outcome of the addition process, or the final response, is known as the sum.

We add numbers in many different contexts.

The combination of at least two numbers constitutes basic addition. Once a student has mastered counting, they can begin to study basic addition.

By locating the addend at the top of the chart and tracing down to the row of the second addend, they can utilize an addition chart as a learning tool.

Given,

shopkeeper sold 7 3/4 kg, 2 1/2 and 3 3/5 of sugar to three customers in a day

Using addition,

⇒ 7 3/4 kg + 2 1/2 + 3 3/5

⇒ 31/4 + 5/2 + 18/5

⇒ 155 + 50 + 72 / 20

⇒ 277/20

⇒ 13 13/20 Kg

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The border line is so much shaded below the inequality symbol (shown in purple).

A relationship among two expresses or values that is neither equal in mathematics is referred to by the term inequality. Thus, unequal results from imbalance. In arithmetic, an inequality explains the links between two non-equal numbers. Egality nor disparities are not the same. Were using the not equal symbol most usually when two values are just not equal (). Values of any size can be contrast using a variety of discrepancies. By changing the two sides till only the variables are left, many straightforward inequalities can be solved. However, a variety of factors support injustice: Both sides' negative values are split or added. Change the left and the right.

given

Gold costs $25 a gram, therefore y grams would cost $25. Because silver costs $0.40 per gram, x grams will cost you $0.0040. If the entire cost of the ring must be less than $90, then 25y+0.40x90, making the cost of the ring (25y+0.40x) dollars. For y, resolve this inequality: 25+0.40<90/25−0.40+90

Take 0.40 off of both sides.

<−0.016+3.6

25y+0.40x y900.40x+900.016x+3.6 = 25y+0.40x y900.40x+903.6

Remove 0.40x from both sides.

Add 25 to both sides.

The boundary line is therefore defined as =0.016+3.6 y=0.016x+3.6, which has a y-intercept of 3.6 and a slope of 0.016 0.016.

The border line is so much shaded below the inequality symbol (shown in purple).

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On solving the linear equations 2x + y = 5 and 3x - 2y = 4 we get x = 2 and y = 1.

Linear equations are the equations of degree 1. It is the equation for the straight line. The standard form of linear equation is ax+by+c =0, where a ≠ 0 and b ≠ 0.

The set of equations given are linear equations that can be solved simultaneously to find the values of variables x and y.

2x + y = 5------->(1)

3x - 2y = 4------->(2)

Multiply equation (1) by 2 and add to equation (2)

4x + 2y = 10

3x -  2y =   4

7x + 0  = 14

7x = 14

x = 2

Substituting x = 2 in (1) we get

2(2) + y = 5

y = 5 - 4 = 1

y = 1

Therefore on solving the system of linear equations simultaneously the values of x and y obtained are 2 and 1 respectively.

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Answer:

milo can put 5 framed photos on each shelf

Step-by-step explanation:

4÷20=5

A collection of values for a variable that simultaneously fulfil each equation is the solution to a system of equations.

A set of two linear equations with two variables is a system of linear equations.

1. Examine the two algebraic methods of substitution and elimination for solving systems of linear equations.

2. Visualize systems of linear equations to better grasp when they have a single solution, none at all, or an unlimited number of solutions.

3. Examine algebraic techniques for determining the quantity of solutions for systems involving two linear equations.

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Mixture A is more redder than mixutre B.

Percentage concentration is a way to express the amount of a substance in a solution in terms of the percentage of the total solution that the substance represents.

The mixture have more concentration of red dye is more redder .

In order to calculate concentration of red dye  we will check percentage concentration of red color in each mixture.

In mixute A concentration of red dye is given by:

Volume of (red dye)*100 / (total volume of solution)

=> (3*100) / (20+3)

=> 13.043%

In mixute B concentration of red dye is given by:

Volume of (red dye)*100 / (total volume of solution)

=> (4*100) / (30+4)

=> 11.764%

This indicate that mixure A have more reddish color.

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The expression C(g(m)) represents the total cost of filling the car's gas tank based on the number of miles driven.

What is an expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.

It is given that the car's gas mileage is 25 miles per gallon, then the number of gallons of gas needed to fill the car's tank can be found by dividing the number of miles driven by the gas mileage.

The expression for this is g(m) = m / 25 where m is the number of miles driven.

The total cost of filling the car's gas tank is represented by the expression C(x) where C is the cost in dollars and x is the number of gallons purchased.

So when g(m) is substituted into C(x), C(g(m)) is obtained which represents the total cost of filling the car's gas tank based on the number of miles driven.

Therefore, this expression is the cost of filling the gas tank given the number of miles driven.

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9 types of solution are gas in gas, liquid in liquid, solid in liquid, solid in solid, gas in liquid, liquid in gas, solid in gas, gas in solid and supercritical fluid.

There are many different types of solutions, but here are 9 common types with examples:

Gas in gas: air (a mixture of gases)

Liquid in liquid: oil and vinegar salad dressing

Solid in liquid: sugar in water

Solid in solid: alloy of gold and silver

Gas in liquid: carbonated water

Liquid in gas: fog (water droplets in air)

Solid in gas: smoke (solid particles in air)

Gas in solid: a gas bubble in a solid piece of plastic

Supercritical fluid: carbon dioxide in a supercritical state (above its critical temperature and pressure)

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