Exercises 4.4 1. For Propositions 1.5 through 1.8 consider the following: - Does the proposition hold on a sphere? - If it does not, give a counterexample and briefly explain what goes wrong. - If it does hold, does Euclid's proof work? If it does not, briefly explain what goes wrong. Note: you do not need to provide a valid proof.

Answer: Your welcome!

Step-by-step explanation:

The ratio of 24kg is 3:5, which means 3 parts of 24kg is 3 times the 5th part of 24kg.

Therefore, 3 parts of the 24kg is 18kg and 5 parts of the 24kg is 6kg.

The end behavior is as x approaches negative infinity, f(x) approaches negative infinity as x approaches positive infinity, f(x) approaches negative infinity The only x-intercept of f(x) is x = 3.

The given function is f(x) = -x³ + 2x² + 3

1. End behavior:

As the leading coefficient of f(x) is negative, -x^3 dominates the behavior of the function as x approaches negative or positive infinity. Therefore, the end behavior of the function is:

2. X-intercepts:

To find the x-intercepts of f(x), we need to solve the equation f(x) = -x^3 + 2x^2 + 3 = 0. We can factor out a common factor of -1 to simplify the equation:

-x^3 + 2x^2 + 3 = 0

x^3 - 2x^2 - 3 = 0

Using synthetic division or long division, we can find that one of the roots of this equation is x = 3. We can then factor out (x - 3) using polynomial division to find the other two roots:

x^3 - 2x^2 - 3 = (x - 3)(x^2 + x + 1)

The quadratic factor x^2 + x + 1 has no real roots, so the only x-intercept of f(x) is x = 3.

3. Positive/negative:

To determine the sign of f(x) for different values of x, we can look at the sign of each factor in the polynomial:

Putting these together, we can make the following sign chart for f(x):

x                         f(x)

x < 0                   -

0 < x < 3                  +

x > 3                  -

4. Multiplicity:

The only root of f(x) is x = 3, which has multiplicity 1. This means that the graph of f(x) crosses the x-axis at x = 3 and changes sign at that point.

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Answer:

0.65

Step-by-step explanation:

0.13 * 5 = 0.65

The overall percentage increase is approximately 35.7%.

Given that, a company sales increased by 15% in year one and 18% in the following year.

To calculate the overall percentage increase, we need to find the combined effect of the two increases.

One way to do this is to use the formula for compound interest, which is:

A = P(1 + r₁)(1 + r₂)

where:

In this case, let's assume that the initial sales amount is 100 (you could use any amount as long as you keep it consistent throughout the calculation).

Then, the first year's increase of 15% would result in a new amount of:

100(1 + 0.15) = 115

In the second year, the 18% increase would be applied to this new amount of 115, resulting in:

115(1 + 0.18) = 135.7

Hence, we have an overall increase of:

(135.7 - 100)/100

= 0.357 × 100%

= 35.7%

Therefore, the increase is approximately 35.7 percent.

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The volume of the triangular prism is 42 in³.

In geometry, a triangular prism is a three-sided polyhedron with a triangle base, a translated copy, and three faces joining equal sides. A right triangular prism is oblique if its sides are not rectangular.

We know that

Volume of Prism (V) = Area of triangle * Height of Prism

Area of triangle = (Base * Height) / 2

Area of triangle = (4 * 3) / 2

Area of triangle = 12 / 2

Area of triangle = 6 in²

Using this, we get

⇒V = 6 * 7

⇒V = 42 in³

Hence, the volume of the triangular prism is 42 in³.

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Since there are multiple questions so the question answered above is attached below

The triangles are similar by SSS and the scale factor is 0.625

If two triangles' corresponding angles are congruent and their corresponding sides are proportional, they are said to be similar triangles. In other words, similar triangles have the same shape but may or may not be the same size. The triangles are congruent if their corresponding sides are also of identical length.

Corresponding sides of similar triangles are in the same ratio. The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides

Given data ,

Let the first triangle be represented as ΔJKT

Let the second triangle be represented as ΔKLS

Now , the measure of side JT = 20

The measure of side JK = 14

The measure of side KS = 32

The measure of side KL = 22.4

The corresponding sides of similar triangles are in the same ratio

So , on simplifying , we get

JT / KS = JK / KL

20 / 32 = 14 / 22.4

On further simplification , we get

0.625 = 0.625

Hence , the triangles are similar

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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 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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Answer:13

Step-by-step explanation:

because when you

Answer: 9

Step-by-step explanation:

The Pythagorean Theorem is proven.

The reason that we can make the statement BC/AC = CD/AC is because corresponding sides of similar triangles are proportional. This is represented by option D. Corresponding sides of similar triangles are proportional.

To complete the proof of the Pythagorean Theorem, a² + b² = c², we can use the following steps:

As similar triangles have proportional sides, we can write the proportions AB/AC = AC/BC and AB/BC = BC/AC.
By the cross-product property, AB * BC = AC * AC and AB * AC = BC * BC.
By the addition property of equality, AB² + BC² = AC² + BC².
By the segment addition postulate, AC + BC = c.
By factoring, (AC + BC)² = c².
By the distributive property, AC² + 2*AC*BC + BC² = c².
By rearranging the terms, AC² + BC² = c² - 2*AC*BC.
By substituting the values of AC and BC from the proportions in step 1, we get a² + b² = c² - 2*a*b.
By simplifying, we get a² + b² = c².

Therefore, the Pythagorean Theorem is proven.

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In answering the question above, the solution is As a result, (x,y) = is the system of equations' answer (-3,8).

A mathematical equation links two statements and utilises the equals sign (=) to indicate equality. In algebra, an equation is a mathematical assertion that proves the equality of two mathematical expressions. For instance, in the equation 3x + 5 = 14, the equal sign separates the numbers by a gap. A mathematical formula may be used to determine how the two sentences on either side of a letter relate to one another. The logo and the particular piece of software are usually identical. like, for instance, 2x - 4 = 2.

The equations in the system are:

[tex]y = -3x - 13 \sy = 2x + 2[/tex]

We may put the two equations equal to one another and get x to solve this system:

-3x - 13 = 2x + 2

3x added to both sides results in:

-13 = 5x + 2

By taking 2 away from both sides, we arrive at:

-15 = 5x

When we multiply both sides by 5, we get:

x = -3

We may use either of the original equations to calculate y now that we know x:

y = -3(-3) - 13 = 8

As a result, (x,y) = is the system of equations' answer (-3,8).

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The equation of the line passing through the point (8,3) that is parallel to the line y=-3/4+4 is y = (-3/4)x + 9.

The slope of the given line is -3/4. The desired line will likewise have a slope of -3/4 because it is parallel to this line.

The equation of the line travelling through the point (8,3) with a slope of -3/4 may be found using the point-slope form of a line's equation:

y - y1 = m(x - x1)

where the provided point is (x1, y1), and m is the slope.

When we change the values, we obtain:

y - 3 = (-3/4)(x - 8)

Making the right side simpler:

y - 3 = (-3/4)x + 6

To each sides, add 3:

y = (-3/4)x + 9

Hence, y = (-3/4)x + 9 is the equation of the line going through the point (8,3) that is parallel to the line y=-3/4+4.

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Answer:

The third one

Step-by-step explanation:

The only one which gives the last number of the sequence is the third one as subbing in 16 you get 3×16 + 3 = 48 + 3 = 51

a) A(t) = 1100(1 + 0.08/4)^(4t)
b) After 8 years, there will be $2203.99 in the account.

c) It will take approximately 8.03 years for there to be $2200 in the account.

d) With continuous compounding interest, there would be $2219.54 in the account after 8 years.

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

Where:
A(t) = the amount of money in the account after t years
P = the initial deposit
r = the annual interest rate
n = the number of times interest is compounded per year
t = the number of years

Find an equation that gives the amount of money in the account after t years:

A(t) = 1100(1 + 0.08/4)^(4t)

Find the amount of money in the account after 8 years:

A(8) = 1100(1 + 0.08/4)^(4*8)
A(8) = 1100(1.02)^32
A(8) = 2203.99

After 8 years, there will be $2203.99 in the account.

How many years will it take for the account to contain $2200?

2200 = 1100(1 + 0.08/4)^(4t)
2 = (1.02)^4t
log(2) = 4t*log(1.02)
t = log(2)/(4*log(1.02))
t = 8.03

It will take approximately 8.03 years for there to be $2200 in the account.

If the same account and interest were compounded continuously, how much money would the account contain after 8 years?

A(t) = Pe^(rt)

A(8) = 1100*e^(0.08*8)
A(8) = 1100*e^0.64
A(8) = 2219.54

With continuous compounding interest, there would be $2219.54 in the account after 8 years.

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Answer:

6(x)-5=90

Step-by-step explanation:

Move all terms that don't contain x to the right side and solve.

Exact Form:

x=95/6

Decimal Form:

x=15.83

Mixed Number Form:

x= 15 5/6

tbh i don't know if this is right

The range of allowable values 24; can vary by 3.5% is 0.9.

The difference between the highest and lowest values for a given data set is the range in statistics. For instance, the range will be 10 - 2 = 8 if the given data set is 2, 5, 8, 10, and 3.

As a result, the range can also be thought of as the distance between the highest and lowest observation. The range of observation is the name given to the outcome. Statistics' range reflects the variety of observations.

In statistics, we must put the given values, set of data, or set of observations in ascending order in order to determine the range. That indicates that you should start by writing the observations from lowest to highest value. The formula must now be used to determine the range of observations.

24 × 3.5% =0.84

so round to nearest tenth is 0.9.

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Complete question:

Find the range of allowable values based on the given information. Round to the nearest tenth.

24; can vary by 3.5%

Let's call the height of the building "h". We can use trigonometry to solve for "h" using the angle of elevation and the horizontal distance from the foot of the building to the point where the angle of elevation is measured.

In this case, we have a right triangle with the height of the building as one leg, the horizontal distance as the adjacent leg, and the angle of elevation as the angle opposite the height. So we can use the tangent function:

tan(60°) = h/55

Solving for "h", we get:

h = 55 tan(60°)

h ≈ 95.1

Rounded to the nearest tenth of a meter, the height of the building is approximately 95.1 meters.

The function [tex]g(x) = \frac{x}{x^2+3}[/tex] is neither even nor odd.

To determine if a function is even, we can use the following test:

f(x) = f(-x)

If this is true, then the function is even.

For the given function, [tex]g(x) = \frac{x}{x^2+3}[/tex], let's plug in -x for x and see if the function is equal to itself:

[tex]g(-x) = \frac{-x}{(-x)^2+3} = \frac{-x}{x^2+3}[/tex]

As we can see, g(x) does not equal g(-x), so the function is not even.

To determine if a function is odd, we can use the following test:

f(x) = -f(-x)

If this is true, then the function is odd.

For the given function, let's plug in -x for x and see if the function is equal to the negative of itself:

[tex]g(x) = \frac{-x}{(-x)^2+3} = \frac{-x}{x^2+3}\\\\ -g(-x) = \frac{x}{-x^2+3}[/tex]

As we can see, g(x) does not equal -g(-x), so the function is not odd.

After the algebraic analysis, we can conclude that the function g(x) is neither, i.e., it is neither even nor odd.

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The measures of arcs and segments are

NK = 12, MLK = 41.5 degrees, JK = 24 and JPK = 277 degrees

Length NK

Given the triangle as the parameter.

Using the Pythagoras theorem, we have

NK = √(KL² - LN²)

So, we have

NK = √(15² - 9²)

Evaluate

NK = 12

The arc measure MK

Here, we start by calculating the central angle MLK

This is done using the following cosine ratio

cos(MLK) = NK/KL

cos(MLK) = 9/12

cos(MLK) = 0.75

Evaluate

MLK = 41.5 degrees

This means that

Arc MK = 41.5 degrees

Length JK

This is calculated as

JK = NK * 2

So, we have

JK = 12 * 2

JK = 24

The arc measure JPK

This is calculated as

JPK = 360 - 2 * MK

So, we have

JPK = 360 - 2 * 41.5 degrees

Evaluate

JPK = 277 degrees

Hence, the arc measure is 277 degrees

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The number that was excluded from the 10 consecutive numbers would be = 228.

Let's take the smallest number of the 10 consecutive number to be = X

Then, the 10 consecutive numbers are x, x+1, x+2, x+3, x+4, x+5, x+6, x+7, x+8, and x+9.

If we exclude one of these numbers, then the sum of the remaining 9 numbers would be:

(x) + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) + (x+7) + (x+8)

Collect the like terms and add up.

= 9x + 36

Recall that the sum of the remaining 9 numbers = 2023.

That is:

= 9X + 36 = 2023

=9x = 1987

x = 221

Therefore, the smallest number in the consecutive sequence is 221, and the 10 numbers are 221, 222, 223, 224, 225, 226, 227, 228, 229, and 230.

If we exclude the number 228, then the sum of the remaining 9 numbers is 2023.

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Answer:

Step-by-step explanation:

sum of all angles in quadrilateral = 360

So, x+48+48+132 =360

x =132

When A U B = C & A ∩ B = φ, then A = C - B. The solution has been obtained by using properties of sets.

What is a set?

In mathematics, a set is a logically arranged group of items that can be represented in either set-builder or roster form. Curly brackets are typically used to represent sets.

We know that

A - B = (A ∪ B) - B

On substituting A U B = C in this, we get

A - B = C - B       ...(1)

Similarly,

A - B = A - (A ∩ B)

On substituting  A ∩ B = φ in this, we get

A - B = A        ...(2)

From (1) and (2), we get

A = C - B

Hence, when A U B = C & A ∩ B = φ, then A = C - B.

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The simplified form is 3/(x+7) + 14/(x)(x+7).

Simplified form of a fraction is the form of a fraction in which the numerator and denominator are in their lowest form, or the greatest common factor has been taken out of both the numerator and denominator. For example, the fraction 24/36 can be simplified to 2/3.

To subtract the two fractions, we need to find a common denominator. The common denominator of (x+7) and (x) is (x)(x+7). We can then rewrite the two fractions with this common denominator and subtract the numerators:

(x+8)/(x+7) - (x-2)/(x) = (x+8)(x)/(x)(x+7) - (x-2)(x+7)/(x)(x+7)

= (x^2+8x)/(x^2+7x) - (x^2-2x+7x-14)/(x^2+7x)

= (x^2+8x - x^2+2x-7x+14)/(x^2+7x)

= (3x+14)/(x^2+7x)

= 3x/(x^2+7x) + 14/(x^2+7x)

= 3/(x+7) + 14/(x)(x+7)

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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.

160 student tickets were sold.

Equations are used to model a wide range of phenomena, from physical laws in science to economic models in finance. They are also used in solving problems and making predictions in various fields.

Let's start by setting up a system of equations to represent the given information:

a + s = 200 (equation 1: total number of tickets sold)

7a + 5s - 210 = 870 (equation 2: profit after paying back Mr. Shum's investment)

We can simplify equation 2 by adding 210 to both sides:

7a + 5s = 1080 (equation 3)

Now we can use equation 1 to solve for one variable in terms of the other:

s = 200 - a (equation 4)

Substitute equation 4 into equation 3 and solve for a:

7a + 5(200 - a) = 1080

7a + 1000 - 5a = 1080

2a = 80

a = 40

So 40 adult tickets were sold. We can use equation 4 to find the number of student tickets sold:

s = 200 - a = 200 - 40 = 160

Therefore, 160 student tickets were sold.

To check our answer, we can verify that the total revenue from ticket sales is equal to the initial investment plus the profit:

7(40) + 5(160) = 210 + 870

280 + 800 = 1080

This checks out, so our answer is correct.

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Marcy spent 2.125 minutes of total time watching the millipede crawl.

To find the total time Marcy spent watching the millipede crawl, we need to add up the time it crawled during the first period and the time it crawled during the second period.

1 4/8 minutes is the same as 1 2/4 minutes, or 1.5 minutes.

2 1/8 minutes is the same as 2.125 minutes.

So the total time the millipede crawled is:

1.5 minutes + 2.125 minutes = 3.625 minutes

Therefore, Marcy spent a total of 3.625 minutes watching the millipede crawl.

We know that Marcy watched the millipede crawl for 1 4/8 minutes, which can be simplified to 1 2/4 minutes, or 1.5 minutes.

Next, the millipede stayed still for a while, and then crawled for 2 1/8 minutes. We can convert 2 1/8 minutes to a decimal by adding the whole number (2) to the fraction (1/8) converted to a decimal (0.125). So:

2 + 0.125 = 2.125 minutes

Marcy spent 2.125 minutes of total time watching the millipede crawl.

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The exponential growth equation is x ( t ) = 37,000,000 ( 1 + 1.3567% )⁶ , where x ( t ) is the population in the year 2015 = 40,115,896

The exponential growth or decay formula is given by

x ( t ) = x₀ × ( 1 + r )ⁿ

x ( t ) is the value at time t

x₀ is the initial value at time t = 0.

r is the growth rate when r>0 or decay rate when r<0, in percent

t is the time in discrete intervals and selected time units'

Given data ,

Let the exponential growth equation be represented as x ( t )

Now , the average percentage change from the year 2000 - 2009 is calculated by

The total percentage = ( 1.97+1.71+1.65+1.42+1.22+1.02+1.07+1.22+0.93 )

The total number of years = 9 years

So , the average percentage change = total percentage / number of years

On simplifying , we get

The average percentage change = 12.21 / 9 = 1.35667 %

b)

The population in 2009 was 37,000,000

So , the population growth rate r = 1.35667 %

And , the population in the years 2015 is given by

The number of years n = 6 years

x ( t ) = 37,000,000 ( 1 + 1.3567% )⁶

On simplifying , we get

x ( t ) = 37,000,000 ( 1.013567 )⁶

x ( t ) = 40,115,896

Hence , the population in the year 2015 is 40,115,896

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To write an equation in standard form using integers, we need to eliminate any fractions by multiplying both sides of the equation by the least common multiple of the denominators.

In this case, the denominator is 5. So we can multiply both sides of the equation by 5 to get:-

5Y = X

Now, we can rearrange the equation so that the variables are on the left-hand side and the constants are on the right-hand side, in the form of Ax + By = C.

X - 5Y = 0 (subtracted X from both sides)

Therefore, the equation in standard form using integers is -X - 5Y = 0.

To know more about integers -

brainly.com/question/490943

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