What is the proof of the non-existence of a solution of a Diophantine equation?

Answer:

Question 8: (-4,7)

Question 10: (5,2)

Step-by-step explanation:

What are equations?:

What is Substitution Method?:

y = -2x - 1

3x - 4y = -40

Question 8:

3x - 4(-2x-1) = -40

3x + 8x + 4 = -40

11x + 4 = -40

11x = -44

x = -4

To find y we can substitute x into the equation.

y = -2(-4) -1

y = 8 -1

y = 7

(-4,7)

Question 8 Solution - (-4,7)

Question 10:

2x - y = 8

x = 5

Since this equation already given x we can substitution to find y.

2(5) - y = 8

10 - y = 8

-y = -2

y = 2

(5,2)

Question 10 Solution - (5,2)

Lenvy~

Answer:

33.33%

Step-by-step explanation:

You start with 36 eggs, and 24 are removed and fried. This means that the remaining eggs are the difference of 36 and 24:

36 - 24 = 12

There are 12 eggs left! To find the percentage of remaining eggs, write a ratio comparing the number of eggs that remain to the original amount of eggs:

[tex]\frac{12}{36}[/tex]

Now, simplify the fraction into a decimal:

12 ÷ 36 = 0.3333...

Multiply a decimal by 100 to convert it to a percentage:

0.3333... ⋅ 100 = 33.33...%

Rounded to the nearest hundredth, the percent of remaining eggs is 33.33%

The required vertex form of the given equation is y = x² - 4x  + 7.

The equation is the relationship between variables and represented as y = ax + b is an example of a polynomial equation.

here,
y = (x - 2)² + 3

Simplifying the expression,
y = x² -4x + 4 + 3   {since (a - b)² = a² + b² -2ab}

y = x² - 4x  + 7

Thus, the required vertex form of the given equation is y = x² - 4x  + 7.

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Answer: To calculate the balance of the savings plan after 9 months with an APR of 3% and monthly payments of $200, we need to take into account the interest earned on the account.

First, we will calculate the interest earned by multiplying the principal (initial amount) by the interest rate (APR) and the number of months.

Interest = Principal * Interest Rate * Number of months

Interest = 200 * 0.03 * 9 = 54

Then, we will add the interest earned to the total amount of payments made.

Total = Principal + Interest + Payments

Total = 200 + 54 + (200 * 9) = 200 + 54 + 1800 = 2054

So the savings plan balance after 9 months with an APR of 3% and monthly payments of $200 is $2054.

Step-by-step explanation:

Answer:

To graph the equation y = 1/2x - 4, we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept.

In this equation, the slope is 1/2, and the y-intercept is -4. The slope represents the steepness of the line, and the y-intercept is the point at which the line crosses the y-axis.

To graph the equation:

Plot the y-intercept, which is (-0, -4)

Using the slope, we can pick a second point on the line. The slope of 1/2 tells us that for every 2 units in the x direction, we will move 1 unit in the y direction. So we can pick a point that is 2 units to the right of the y-intercept, such as (2,0).

Connect the two points to get a straight line

The resulting graph will be a straight line that passes through the point (-0, -4) and (2, 0) and continues in both directions.

Alternatively, you can use a table of values to plot the points that belongs to the line and connect them, for example:

x | y

-2 | -5

-1 | -4.5

0 | -4

1 | -3.5

2 | -3

These are few points on the graph and with more points you can connect them and see the shape of the graph.

Step-by-step explanation:

Answer:

x=91

Step-by-step explanation:

Follow the steps in the image.

add 16 to both sides

subtract 3 from both sides

divide by 1x

Plug answer back into equation

-Hope this helped

Answer:

Input = 5

Step-by-step explanation:

Output = Answer a.k.a. f(x)

Input = x

We start by replacing f(x) with the output, which is 11

11 = 4x-9. I like making it opposite. It's easier in my eyes.

Making it opposite it's 4x-9 = 11

Now we need to get x by itself. The opposite of -9 is +9. So if we add 9 to each side, it deletes the 9 from the left and puts it on the other side, making it;

4x = 11 + 9. 11 + 9 is 20, so....

4x = 20.

Now 4x is the same as 4 times x. The opposite of times is divide. So let's divide by 4 on both sides. That gets rid of the 4 on the left side and puts it on the other side, making it;

x = 20/4. 20/4 is 5, so...

x = 5

Step-by-step explanation:

f(x) = 4x - 9

4x - 9 = 11

4x = 11 + 9

4x = 20

x = 20 ÷ 4

x = 5

Answer:

≈51.55

Step-by-step explanation:

∆/360 πD = 7.2

x/360 *22/7*16= 7.2

x/360 *50/2/7= 7.2

x/360 = 63/440

x =51.54545455

≈51.55

f(h(x)) = -2x^2 + 12x + (-12) in terms of x.

What is a function?

A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and codomain or range.

To find f(h(x)), we first need to substitute h(x) into f(x).

f(h(x)) = f(4(x-3)²)

Now we can substitute 4(x-3)^2 into f(x)

f(h(x)) = 6 - 1/2(4(x-3)^2)

= 6 - 2(x-3)^2

= 6 - 2(x^2 - 6x + 9)

= 6 - 2x^2 + 12x - 18

Hence, f(h(x)) = -2x^2 + 12x + (-12) in terms of x.

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The P(n) represents the maximum no of regions into which n lines divide the plain then P(k) = P(k-1) + k. (b). The formula for P(n) is [tex]1+{ }^{n+1} c_2\end{aligned}$[/tex]. (c). The 5152 regions can divide the plane for 100 lines.

When no. lines are drawn, we get one plain region.

Let the number of regions when divided by lines be represented by

P(0), P(1), P(2), P(3), P(n)

Therefore P(0) = 1

P(1) = 2

P(1) = P(0) + 1

P(2) = P(1) + 2

P(3) = P(2) + 3

P(4) = P(3) + 4………………….

P(n) = P(n-1) + n

Therefore, if P(n) represents the maximum no. of regions into which n lines divide the plain

Then P(k) = P(k-1) + k

(b).

⇒ P(0) = 1

⇒P(n) = P(n-1) + n

= P(n-2) + (n-1)+n

= P(n-3) + (n-2) +(n-1)+n……..

= P(0) + 1 + 2+ 4=3 + 4 + 5 +……+n

= P(0) + (Sum of the first n natural numbers)

[tex]$$\begin{aligned}& =P(0)+\frac{n(n+1)}{2} \\& =1+{ }^{n+1} c_2\end{aligned}$$[/tex]

Therefore, the formula for P(n) is [tex]1+{ }^{n+1} c_2\end{aligned}$[/tex].

(c)

Compute the value for n=100

[tex]$$1+{ }^{n+1} c_2=1+{ }^{101} c_2=1+\frac{101 \times 102}{2}=5152$$[/tex]

Therefore, the 5152 regions can divide the plane for 100 lines.

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A single line divides the plane into two regions. Two lines (by crossing) can divide the plane into four regions; three lines can divide it into seven regions, etc. Let P(n) be the maximum number of regions into which n lines divide a plane, where n is a positive integer.

(a) Derive a recurrence relation for P(k) in terms of P(k-1) for all integers [tex]$k \geq 2$[/tex].

(b) Use iteration to find an explicit (closed) formula for P(n).

(c) Into how many regions can 100 lines divide the plane?

Answer:

P = 5x + 33 = 58

Q = R = 3x+46 = 61

Step-by-step explanation:

angles in a triangle add up to 180 degrees

the angles in an isosceles triangle that correspond to the two congruent sides (are opposite the sides) are also congruent

that means that angle R = angle Q because QP and RP are congruent

angles = 5x+33, 3x+46, 3x+46

5x+33 + 3x+ 46 + 3x+46 = 180

11x + 125 = 180

subtract 125 from both sides to isolate the x and its coefficient

11x = 55

divide both sides by 11 to find x

x = 5

P = 5x + 33 = 58

Q = R = 3x+46 = 61

34x56x79447886447567

Question 1

[tex]11+2x+29=x+28 \\ \\ 2x+40=x+28 \\ \\ x+40=28 \\ \\ x=-12[/tex]

Question 2

A, B, E

The final statement is: The system of equations have unique solution.

The solution of a linear equation is defined as the points, in which the lines represent the intersection of two linear equations. In other words, the solution set of the system of linear equations is the set of all possible values to the variables that satisfies the given linear equation.

Given here: x-2y = 2 & y=-2x+5

Solving these two equations by method of substitution we get

x-2(-2x+5)=2

x+4x-10=2

5x=12

x=2.4

∴ y=2×2.4+5

    =4.8+5

    =9.8

Hence, The system has one solution.

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Francis purchases a package of 450 plastic forks but only uses 315 of them, therefore he uses 70% of the available supply.

In math, a % is a number or ratio that is expressed as a fraction of 100. There are also sporadic uses of the abbreviations "pct.," "pct," and "pc." However, the percent sign "%" is frequently used to indicate it. The percentages' total quantity has no dimensions. Percentages are fundamentally fractions when the denominator is 100. Put a percent sign (%) next to a number to indicate that it is a percentage. For instance, if you successfully answer 75 out of 100 questions on a test (75/100), you get a 75%. Divide the money by the total to get percentages, then multiply the resulting number by 100. The formula (value/total) x 100% is used to calculate the percentage.

given

Francis purchases 450 plastic forks and

utilizes 315 of them, for

a percentage of 315/450*100

= 70%.

Francis purchases a package of 450 plastic forks but only uses 315 of them, therefore he uses 70% of the available supply.

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there is not enough information here to complete the problem.

Answer:

k = -12.6

Step-by-step explanation:

Given rational polynomial:

[tex]\dfrac{x^3-2x^2+kx-12}{x-5}[/tex]

When we divide a polynomial f(x) by (x − d) the remainder is f(d).

If the remainder is zero, then (x - d) must be a factor of f(x).

If a cubic polynomial is divided by (x - d) then:

[tex]\implies (x-d)(ax^2+bx+c)[/tex]

where:

If the constant of the cubic polynomial is -12 and d = 5:

[tex]\implies -cd=-12[/tex]

[tex]\implies -5c=-12[/tex]

[tex]\implies c=2.4[/tex]

The leading coefficient of the cubic polynomial is one.  Therefore:

[tex]\implies (x-5)(x^2+bx+2.4)[/tex]

Expand and collect like terms:

[tex]\implies x^3+bx^2+2.4x-5x^2-5bx-12[/tex]

[tex]\implies x^3+(b-5)x^2+(2.4-5b)x-12[/tex]

The find the value of b, compare the coefficients of the term in x²:

[tex]\implies b-5=-2[/tex]

[tex]\implies b=3[/tex]

To find k, substitute the found value of b into the coefficient of x:

[tex]\implies k=2.4-5b[/tex]

[tex]\implies k=2.4-5(3)[/tex]

[tex]\implies k=-12.6[/tex]

Answer:

–12.6

Step-by-step explanation:

[tex] \frac{ \\ {x }^{3} - {2x}^{2} + kx - 12}{x - 5} = 0[/tex]

that means (x–5) is a factor of x³ – 2x² +kx – 12

since it resulted to zero according to the rules of zeros of polynomial.

x–5 = 0

x = 5

by substituting

f(x) = 5³ – 2(5)² + k(5) – 12 = 0

125 – 50 + 5k – 12 = 0

75 + 5k – 12 = 0

5k + 75 – 12 = 0

5k + 63 = 0

5k = –63

[tex] \frac{ \\ 5k}{5} = \frac{ - 63}{5} [/tex]

[tex]k = - 12.6 \: or \: - 12 \frac{3}{5} [/tex]

I hope I helped

Answer:

52.5

Step-by-step explanation:

58.1 - 5.6= 52.5

The midpoint of the points A (1, 5) and B (-7, -5) is found to be (-3, 0)

The midpoint of a line segment is the centroid of the segment. It is equally distant from both of the ends and hence cuts the section in half.

The formula to determine the center point of a straight line using the coordinates of its endpoints is known as the midpoint formula in coordinate geometry.

The formula is of the form

X= (x₂ + x₁)/2

Y = (y₂ + y₁)/2

Given points A (1, 5) and B (-7, -5)

X= (1 - 7)/2 = -3

Y = (5 - 5)/2 = 0

the midpoint is (-3, 0)

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47.925 can be written as a mixed number as 47(925/1000).

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

47.925

This can be written as,

= 47925 / 1000

Now,

47925 = 47000 + 925

1000 x 47 = 47000

So,

47925/1000

= 47(925/1000)

Thus,

The mixed number is 47(925/1000)

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C. Determine if the system containing u, v, and b is consistent. If the system is consistent, then b is in Span{u,v} is correct.

To show that a vector b is in Span{u,v}, one way is to determine if it can be written as a linear combination of the vectors in the set, i.e., if there exist scalars h and k such that b = hu + kv. If this is the case, the system containing the equations hu + kv = b is consistent, meaning that there exist values of h and k that make the equation true. Therefore, if the system is consistent, it can be concluded that the vector b is in Span{u,v}.

Therefore, option C is the correct answer.

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Answer:

[tex]\text{f)} \quad y=80(1.1)^x[/tex]

[tex]\text{g)} \quad y=225(0.8)^x[/tex]

[tex]\text{h)} \quad y=5(3)^x[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]

Given points:

Substitute the given (x, y) points into the exponential function formula to create two equations:

[tex]\implies 72.73=ab^{-1}[/tex]

[tex]\implies 106.48=ab^3[/tex]

Divide the second equation by the first equation to eliminate a:

[tex]\implies \dfrac{106.48}{72.73}=\dfrac{ab^3}{ab^{-1}}[/tex]

[tex]\implies \dfrac{106.48}{72.73}=\dfrac{b^3}{b^{-1}}[/tex]

Solve for b:

[tex]\implies \dfrac{106.48}{72.73}=b^3 \cdot b^{1}[/tex]

[tex]\implies \dfrac{106.48}{72.73}=b^{3+1}[/tex]

[tex]\implies \dfrac{106.48}{72.73}=b^4[/tex]

[tex]\implies b=1.09998968...[/tex]

[tex]\implies b=1.1[/tex]

Substitute the found value of b into one of the equations and solve for a:

[tex]\implies 72.73=a \cdot (1.09998968...)^{-1}[/tex]

[tex]\implies a=80.0022499...[/tex]

[tex]\implies a=80[/tex]

Substitute the found values of a and b into the exponential function formula:

[tex]\implies y=80(1.1)^x[/tex]

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

Given points:

Substitute the given (x, y) points into the exponential function formula to create two equations:

[tex]\implies 351.56225=ab^{-2}[/tex]

[tex]\implies 115.2=ab^3[/tex]

Divide the second equation by the first equation to eliminate a:

[tex]\implies \dfrac{115.2}{351.56225}=\dfrac{ab^3}{ab^{-2}}[/tex]

[tex]\implies \dfrac{115.2}{351.56225}=\dfrac{b^3}{b^{-2}}[/tex]

Solve for b:

[tex]\implies\dfrac{115.2}{351.56225}=b^3 \cdot b^{2}[/tex]

[tex]\implies \dfrac{115.2}{351.56225}=b^{3+2}[/tex]

[tex]\implies \dfrac{115.2}{351.56225}=b^5[/tex]

[tex]\implies b=0.800000113...[/tex]

[tex]\implies b=0.8[/tex]

Substitute the found value of b into one of the equations and solve for a:

[tex]\implies 115.2=a(0.800000113...)^3[/tex]

[tex]\implies a=224.999904[/tex]

[tex]\implies a=225[/tex]

Substitute the found values of a and b into the exponential function formula:

[tex]\implies y=225(0.8)^x[/tex]

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

Given points:

Substitute the given (x, y) points into the exponential function formula to create two equations:

[tex]\implies 405=ab^4[/tex]

[tex]\implies 98415=ab^9[/tex]

Divide the second equation by the first equation to eliminate a:

[tex]\implies \dfrac{98415}{405}=\dfrac{ab^9}{ab^4}[/tex]

[tex]\implies 243=\dfrac{b^9}{b^4}[/tex]

Solve for b:

[tex]\implies 243=b^9 \cdot b^{-4}[/tex]

[tex]\implies 243=b^{9-4}[/tex]

[tex]\implies 243=b^{5}[/tex]

[tex]\implies b=3[/tex]

Substitute the found value of b into one of the equations and solve for a:

[tex]\implies 405=a \cdot 3^4[/tex]

[tex]\implies 81a=405[/tex]

[tex]\implies a=5[/tex]

Substitute the found values of a and b into the exponential function formula:

[tex]\implies y=5(3)^x[/tex]

On solving the question, we can say that second function has vertical asymptote at x=2

The subject of mathematics includes quantities and their variations, equations and related structures, shapes and their locations, and places where they can be found.

The function that represents the graph is f(x) = 1/x-2 .

The graph of the function

From the given graph , the vertical asymptote is at x=2

Lets find out the vertical asymptote of each function

To find out vertical asymptote we set the denominator =0

f(x) = 1/2x

2x = 0

x = 0

f(x) = 1/x-2

x-2 = 0

x = 2

The second function has vertical asymptote at x=2

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Answer: 20

Step-by-step explanation: Since it is 1/5 you multiply the 5 by 4 and get 20.

Answer:

20

Step-by-step explanation:

It is 20 because you have to multiply the denominator by the whole number.

f (–3) = -1 , B) f (–1) = 2 , C) f (3) = 5, Mathematical equations that are described in the plane of cartesian coordinates are known as straight-line equations.

The general equation for every straight line is y = mx + c, where m is the gradient (or degree of steepness) of the line and c is the y-intercept (the point in which the line crosses the y-axis).

The variables x and y are related to coordinates on the line in the linear equation y = mx + c.

The formula y = mx + c yields a result for y when we enter a value for x.

As y depends on the value of x, it follows that x is an independent variable and y is a dependent variable.

According to our question-

m (x-x1) = y-y1 or

y = mx + c

Where,

Straight-line gradient, or m, is the line's slope.

The Cartesian coordinate that the line crosses is x1, y1.

the constant c

The calculation of the gradient (m) between any two locations on a line

m = Δy / Δx

f (-1) = Because x = -1, the function used is

y = 2, so f (-1) = 2 C) f (-3) = Because -3 -1,

the function used is y = -x-4, so f (-3) = - (- 3) -4 f (-3) = 3-4 f (-3) = -1 f (3) Since the function employed is

= -2x + 11 and x = 3, f (3) = -2 (3) +11, f (3) = -6 + 11, and f (3) = 5.

Hence, f (–3) = -1 , B) f (–1) = 2 , C) f (3) = 5, Mathematical equations that are described in the plane of cartesian coordinates are known as straight-line equations.

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Answer:-1
2
5

Step-by-step explanation:

simple answer

The domain and the range of this function shown in the graph include the following;

B. Domain: all real numbers.

   Range = f(x) ≥ -11

In Mathematics, the horizontal extent of a graph represents all domain values and they are read and written from smaller to larger numerical values, and from the left of a graph to the right.

Furthermore, the vertical extent of a graph represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph of this function (see attachment), we can reasonably and logically deduce the following domain and range:

Domain = -∞ ≤ x ≤ ∞ or all real numbers.

Range = f(x) ≥ -11 or [-11, ∞].

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