What postulate, if any, can be used to prove the triangles are congruent?

The values of a and b such that y∼n(0,1) are a = ± (1/σ) and b = ± (μ/σ)

of a gaussian distribution.

Gaussian distribution (also known as normal distribution) is a bell-shaped curve, and it is assumed that during any measurement values will follow a normal distribution with an equal number of measurements above and below the mean value.

E(X) = μ, Var(X) = σ^2

Y = aX + b

We want E(Y) = 0 and Var(Y) = 1

E(Y) = a E(X) + b

0 = a μ + b

b = -a μ  --- (1)

Var(Y) = (a^2) Var(X)

1 = (a^2) σ^2

a^2 = 1/σ^2

a = ± 1/σ  --- (2)

From (1) and (2)

b = (± 1/σ) μ

So, a = ± (1/σ) and b = ± (μ/σ)

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The complete question is :

A random variable X ∼ N (µ, σ2 ) is Gaussian distributed with mean µ and variance σ 2 . Given that for any a, b ∈ R, we have that Y = aX + b is also Gaussian, find a, b such that Y ∼ N (0, 1)

The triangle that will be formed using the sides 12 mm , 16 mm and 20 mm will be a right trinagle.

A right triangle, also known as a right-angled triangle, right-perpendicular triangle, orthogonal triangle, or previously rectangled triangle, is a triangle with one right angle, or two perpendicular sides. The foundation of trigonometry is the relationship between the sides and various angles of the right triangle.

The characteristics of a right-angled triangle.

Let is consider the triangle is a right triangle so the base and height will be 12 mm and 16 mm respectivly.

Now using Pythagoras Theorem :

√(height² + base²) = hypotenuse

⇒√(12²+16²) = √(144+256) = √(400) = 20 mm = hypotenuse.

The triangle that will be formed using the sides 12 mm , 16 mm and 20 mm will be a right trinagle.

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Answer:

D. (3.5, 2.25)

Step-by-step explanation:

A solution to the system is where the two equations intersect.

looking at the graph, we see that the intersection point is between x= 3 and 4, and Y = 2 and 3.

Answer:

(3.5, 2.25)

Step-by-step explanation:

To find the coordinate of the solution to the system of linear equations −3x + 2y = −6 and y = -1/2x + 4, we need to solve the system. One method to do this is substitution. We can solve one of the equations for y, then substitute the expression for y into the other equation. This will give us an equation in terms of x, which we can then solve for x. Then, we can substitute the value of x back into one of the equations to find the corresponding value of y.

Using substitution, we can solve the system as follows:

y = -1/2x + 4

-3x + 2y = -6

Substitute y = -1/2x + 4 into -3x + 2y = -6:

-3x + 2(-1/2x + 4) = -6

-3x + (-x + 8) = -6

-4x + 8 = -6

-4x = -14

x = 3.5

Substitute x = 3.5 into y = -1/2x + 4:

y = -1/2(3.5) + 4

y = 2.25

Therefore, the solution to the system is approximately (3.5, 2.25).

F(4) < F(5) means that the temperature at 7 A.M. is less than the temperature at 8 A.M.

An inequality is used to compare two or more expressions.

For example -

ax + b > c

ax + b < c

Given is that F(t) is the outside temperature (*F)t hours after 3 A.M.

It is given that -

F(4) < F(5)

This means that the temperature at 7 A.M. is less than the temperature at 8 A.M.

Therefore, F(4) < F(5) means that the temperature at 7 A.M. is less than the temperature at 8 A.M.

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On solving the provided question, we can say thatarea of the figure = area of triangle + area of semi-circle = 12 + 14.13 = 26.13 yd sq.

The size of an area on a surface can be expressed as an area. The area of an open surface or the boundary of a three-dimensional object is referred to as the surface area, whereas the area of a planar region or planar region refers to the area of a form or planar layer. The total amount of space filled by a planar (2-D) surface or shape of an object is known as its area. Draw a square on a piece of paper using a pencil. a character with two dimensions. The area of a shape on paper is the space it takes up. Imagine that the square is composed of more compact unit squares.

area of semi-circle = 1/2 * [tex]\pi *r^2[/tex] = 0.5*3.14*3*3 = 14.13yd sq.

area of triangle = 1/2*b*h = 0.5*6*4 = 12sq. yd

area of the figure = area of triangle + area of semi-circle = 12 + 14.13 = 26.13 yd sq.

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The minimum degree of a Taylor polynomial required to calculate to 3 decimal places depends on the complexity of the function being approximated.

A Taylor polynomial is a way of approximating a function with a polynomial. The polynomial is built based on the function's value and its derivatives at a specific point.

To calculate a Taylor polynomial to 3 decimal places, the degree of the polynomial required depends on the complexity of the function and the desired accuracy.

If the function is smooth, with no sudden changes, a lower degree polynomial may be sufficient to approximate the function to 3 decimal places.

A general rule is to use a polynomial of degree at least two times the number of decimal places desired. So, for 3 decimal places, a polynomial of degree 6 or higher is required.

However, it is important to note that the minimum degree required can be determined by experimenting with different degrees of polynomials and checking the accuracy of the approximation.

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Homogeneous systems of linear equations can have either a unique solution or infinitely many solutions, depending on the rank of the matrix representing the system.

A homogeneous set of equations is a set of linear equations that has a trivial solution (that is, a solution consisting of only the zero vector). In other words, if the coefficients of the variables in a set of linear equations are all equal to zero, the set of equations is considered homogeneous.

The trivial solution, that is, the solution in which all variables are equal to zero, will always be a solution to a homogeneous set of equations. This is because, if all coefficients are equal to zero, then each equation in the set will reduce to 0 = 0, which is always true. Thus, the trivial solution is always a solution to a homogeneous set of equations.

In addition to the trivial solution, a homogeneous set of equations may also have non-trivial solutions, which are solutions that are not equal to only the zero vector. To find these non-trivial solutions, we typically use techniques such as Gaussian elimination, row reduction, or the use of matrices and determinants.

It is important to note that homogeneous systems of linear equations can have either a unique solution or infinitely many solutions, depending on the rank of the matrix representing the system.

Therefore, homogeneous systems of linear equations can have either a unique solution or infinitely many solutions, depending on the rank of the matrix representing the system.

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Adding and subtracting mixed numbers with unlike denominators requires converting them into equivalent fractions with a common denominator. for example 3 3/5 + 2 1/7 = 5 8/35.

3 3/5 + 2 1/7 = 5 8/35, 5 1/5 - 3 2/3 = 1 7/15, 4 1/4 + 2 2/7 = 6 11/28, 6 3/4 - 3 5/6 = 2 13/24, 1 1/2 + 3 4/7 = 4 11/14, 2 5/6 - 1 1/4 = 1 7/12, 5 1/2 + 3 3/4 = 9 1/4, 3 1/5 - 2 3/7 = 0 16/35, 4 2/3 + 2 4/5 = 6 17/15 , 2 3/4 - 1 5/6 = 1 1/12

To add or subtract mixed numbers with different denominators, you must convert them into identical parts with common factors. Similarities can be found by looking at the most unusual differences between the two denominators. Mixed numbers are converted to bad parts, added or subtracted, and returned to perfect mixed numbers. How to find the common factor, that is,

Switch completely to unwise parts, extend or subtract, then switch completely to mixed numbers. Consider exact calculations of mixed numbers with different denominators.

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Using the details given, the exponential growth at rate of 10% over 14 years is 26582.5

An exponential function is a mathematical function in which the variable appears as an exponent. It can be written in the form of y = ab^x, where a is the base, b is a constant, and x is the variable. This function has the property that y grows or decays at an exponential rate as x increases or decreases.

y = a(1 + r)^t

In the given problem, we have the data given as;

Substituting the values into the formula;

y = 7000(1 + 0.1)^14

y = 26582.5

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The mean/standard error of the sampling distribution of the proportion is 0.0111.

Its root-mean The square root of the mean of the squares of all the values in a series calculated from the arithmetic mean is the square deviation, also referred to as the standard deviation and denoted by the symbol.

The standard deviation reveals the degree of data dispersion. It expresses how far away from the mean each observed value is.

Almost 95% of the values in any distribution will be within two standard deviations of the mean. The degree of data dispersion from the mean is described by the term "standard deviation" (or " σ").

The data tend to cluster around the mean when the standard deviation is low; when it is high, the data are more dispersed.

The formula to calculate the standard error of sampling distribution of sample proportion is:

[tex]$ SE(p) =\sqrt{\frac{p(1-p)}{n} }[/tex]

It is given that the bank believes that 8% of people who receive loans will not make payments on time. That is,

Population proportion, p =0.08

Sample size, n= 800

Using the formula defined :

[tex]$ SE(p) =\sqrt{\frac{0.08(1-0.08)}{600} }[/tex]

[tex]$ SE(p) =\sqrt{0.000123}[/tex]

SE(p) = 0.0111

Thus, the mean/standard error of sampling distribution of the proportion is 0.0111.

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The solution is given below.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

The perimeter of a rectangle is 40 cm. The length is 14 cm.

Let x = width of the rectangle.

Ravi says he can find the width using the equation 2(x + 14) = 40.

Fran says she can find the width using the equation 2x + 28 = 40.

now, we get,

1. Divide both sides by 2

 2(x+14) = 40

 x+14 = 20

2. Isolate the x term by subtracting 14 from both sides

3. x = 6. The width of the triangle is 6 cm.

4. Isolate the x term by subtracting 28 from both sides

 2x + 28 = 40

 2x = 12

5. Divide both sides by 2

6. x = 6

7. The two equations have the same solution, because by the distributive rule, 2(x+14) = 2x+28.

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The recursive formula for the arithmetic sequence in this problem is given as follows:

c(1) = -16.

c(n) = c(n - 1) - 17.

Arithmetic sequences are collections of numbers that share a consistent difference between them; that is, there is a constant amount of subtraction between any two entries in an arithmetic sequence.

The recursive formula for an arithmetic sequence is defined as follows:

c(n) = c(n - 1) + d.

In which d is the common difference of the arithmetic sequence.

The sequence in this problem is given as follows:

-16, -33, -50, -67,...

Then the common difference is obtained as follows:

d = -33 - (-16) = -33 + 16 = -17.

So, c(n) = c(n - 1) - 17.

c(1) = -16.

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x = 217.9927  = amount of mL of the 62% acid solution

y = 116.7873 = amount in mL of the 19% acid solution

This can be solved using a system of two equations and two variables.  

Let x = amount of mL of the 62% acid solution

Let y = amount in mL of the 19% acid solution

We know the total mL of the mixture of the two solutions is 600 mL,

so x + y = 334.78    ..... eq(1)

62% of solution x is acid which can be written as 0.62x and similarly acid from the second solution can be written as 0.19y. We know that 47% of the 334.78 mL solution will be acid or 0.47(334.78).

The acid in the mixture solution comes from the acid in solution x and solution y. So 0.62 x + 0.19 y = 0.47 (334.78)      ... eq(2)

from equation (1)

x = 334.78-y

So, x = 217.9927

  y = 116.7873.

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The surface area of ​​this rectangular prism 290 square inches.

A rectangular prism is a three-dimensional solid shape with six faces that including rectangular bases. A cuboid is also a rectangular prism. The cross-section of a cuboid and a rectangular prism is the same.

The surface area of a solid is defined as the sum of the areas of each of its faces (or surfaces), hence the name.

The surface area of rectangular prism formula is

Surface Area = 2(lw+lh+wh)

Here, length=5 inches, width= 5 inches and height=12 inches

Now, surface area = 2(5×5+5×12+5×12)

= 2(25+60+60)

= 2×145

= 290 square inches

Therefore, the surface area of ​​this rectangular prism 290 square inches.

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The surface area of a rectangular prism is given by the expression 2(lh+bh+lb) where, l = length, h = height and b = width

A rectangular prism is a three-dimensional solid shape with six faces that including rectangular bases.

Given that, what is the surface area of ​​this rectangular prism.

Let us consider a rectangular prism with l = length, h = height and b = width

Now, we will find its surface area,

Since, we know that, there are 6 rectangular faces in a rectangular prism

Therefore, for surface area of the rectangular prism will add the areas of all 6 faces,

Area of the faces = lb + bh + lh + lb + bh + lh (area of rectangle = length x width)

Area of the faces = 2(lb+bh+lh)

Hence, the surface area of a rectangular prism is given by the expression 2(lh+bh+lb) where, l = length, h = height and b = width

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the components of A in this system will be: Ah = Axi1 + Ayj1 + Azk1, A2 = Axi2 + Ayj2 + Azk2, and A3 = Axi3 + Ayj3 + Azk3. To find the components of A in a skewed coordinate system, we first need to express the vector A in terms of the unit vectors of the system.

i = i1i2i3

j = j1j2j3

k = k1k2k3

Ah = Axi1 + Ayj1 + Azk1

A2 = Axi2 + Ayj2 + Azk2

A3 = Axi3 + Ayj3 + Azk3

To find the components of A in a skewed coordinate system, we first need to express the vector A in terms of the unit vectors of the system. The unit vectors of the skewed coordinate system are given by i = i1i2i3, j = j1j2j3, and k = k1k2k3.

Therefore, the components of A in this system will be: Ah = Axi1 + Ayj1 + Azk1, A2 = Axi2 + Ayj2 + Azk2, and A3 = Axi3 + Ayj3 + Azk3.

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The slope of the line containing the points ( 3, 4 ) and ( 3, - 6 ) is -8.

The equation of the straight line has its slope and given point.

If we have a non-vertical line that passes through any point(x1, y1) has gradient m. then general point (x, y) must satisfy the equation

y-y₁ = m(x-x₁)

Its slope would be:

[tex]m = \dfrac{y-q}{x-p}[/tex]

Slope of parallel lines are same. Slopes of perpendicular lines are negative reciprocal of each other.

Given;

The two points ( 3, 4 ) and ( 3, - 6 ).

Now, to find the slope

m=-6-4/4-3

m=-8/1

m=-8

Therefore, the slope will be -8.

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[tex](\stackrel{x_1}{-3}~,~\stackrel{y_1}{-1})\qquad (\stackrel{x_2}{-5}~,~\stackrel{y_2}{4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{(-1)}}}{\underset{\textit{\large run}} {\underset{x_2}{-5}-\underset{x_1}{(-3)}}} \implies \cfrac{4 +1}{-5 +3} \implies \cfrac{ 5 }{ -2 } \implies - \cfrac{ 5 }{ 2 }[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-1)}=\stackrel{m}{- \cfrac{ 5 }{ 2 }}(x-\stackrel{x_1}{(-3)}) \implies y +1 = - \cfrac{ 5 }{ 2 } ( x +3) \\\\\\ y+1=- \cfrac{ 5 }{ 2 }x-\cfrac{15}{2}\implies y=- \cfrac{ 5 }{ 2 }x-\cfrac{15}{2}-1\implies {\Large \begin{array}{llll} y=- \cfrac{ 5 }{ 2 }x-\cfrac{17}{2} \end{array}}[/tex]

Answer: 40 pages/hour

Step-by-step explanation:

12pages / 18min  *   60min/1hr

= 40pages / 1hr

OR

12 pages / 18min    SIMPLIFY

2pages / 3 min

Multiply both by 20 to get how many pages per 60 minutes.

she has to pay $736 per month, next year in rent.

A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.

Given, lian currently pays 640 per month to rent her apartment she has been told by her landlord that her rent will be increasing by 15% next year.

Her rent is now = 640 per month

Since, After a year her rent will increase by 15 %

Thus,

Her rent next year per month = 640(1 + 15%)

Her rent next year per month = 640(1 + 0.15)

Her rent next year per month = 640* 1.15

Her rent next year per month = 736

Therefore, she has to pay $736 per month, next year in rent.

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Her sales are at least $1000 during that week.

Let x stand in for her sales.

The equation to determine her sales is as follows, based on the data in the question:

Hillary works at a video store. She earns $8. 50 per hour plus a 12% commission on all the weekly sales she makes.

he works 24 hours a week if she earns at least $324 during that week

(8.50 × 24) + (12% × x) ≥ 324

204 + (0.12 × x) ≥ 324

204 + 0.12x ≥ 324

0.12x ≥ 324 - 204

0.12x ≥ 120

x ≥ 120/0.12

x ≥ 1000

Her sales are at least $1000

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The greatest common factor of 5x⁷, 30x⁴, 10x³ is 5x³

The Greatest Common Factor is defined as the largest number that is a factor of two or more numbers. For example, the GCF of 24 and 36 is 12, because the largest factor that is shared by 24 and 36 is 12. 24 and 36 have other factors in common, but 12 is the largest.

The greatest common factor between 5, 10 and 30 is 5. This means out all of the common factors between 5,10 and 30 , 5 is the greatest

The greatest common factor between x⁷ and x⁴ and x³ is x³.

Therefore the greatest common factor between 5x⁷, 30x⁴, 10x³ is 5× x³ = 5x³

This means 5x³ is the highest factor that can go through 5x⁷, 30x⁴, 10x³

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Answer:

20 pounds

Step-by-step explanation:

250% = 2.5

8 times 2.5 = 20 pounds

So, a one-year-old baby weighs 20 pounds.

"The function is increasing in the interval (-2.5,∞)".

To determine whether the function is increasing or decreasing we are going to use the derivative test.

f(x) = x² + 5x + 6

f'(x) = 0

2x + 5 = 0

x = -5/2

Intervals associated with this point are  (-∞,-2.5) and (-2.5,∞)

Now by checking in the interval  (-∞,-2.5)  at a point, let's say -3

Then,

f'(-3) = -1 so function is decreasing over this interval.

Now by checking in the interval  (-2.5,∞)  at a point, let's say 2

Then,

f'(2) = 1 so function is increasing over this interval.

Hence "The function is increasing in the interval (-2.5,∞)".

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The sign of the second derivative of a function can determine whether the function is increasing or decreasing. If the second derivative is positive, the function is concave up and therefore increasing.

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The function f(x,y) is defined for all (x,y) pairs that lie within the region d, which is bounded by the parabolas y= 2x2 and y= 1 x2. It returns the product of x and 2 to the power of y.

The function f(x,y) is a two-variable function defined on the region d, which is bounded by the parabolas y= 2x2 and y= 1 x2. In order to calculate the output of this function, we must first identify the values of x and y that lie within the region d. We can do this by graphing both parabolas and finding the intersection point, which is when x=1 and y=2. This tells us that any (x,y) pairs with x<1 and y<2 will be within the region d. Now that we have identified the region, we can calculate the output of the function by multiplying x and 2 to the power of y. For example, if x=0.5 and y=1, then f(x,y)= 0.52 = 0.25.

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Answer:

because the line has a slope of 0, you know that it is horizontal

therefore, the equation is y=-8

Step-by-step explanation:

The volume of the solid generated is [tex]\pi (2/3) (44)^{3/2}.[/tex]

We have,

To find the volume of the solid generated by revolving the region in the first quadrant bounded by the graph of y = √(44 - x²) and the x- and y-axes about the y-axis, we can use the method of cylindrical shells.

The volume can be calculated using the following integral:

V = ∫[a,b] 2πx f(x) dx

where [a, b] is the interval of x-values that defines the region in the first quadrant.

In this case, the curve y = √(44 - x²) intersects the x-axis when y = 0, which gives us:

0 = √(44 - x²)

Squaring both sides and solving for x, we get:

x² = 44

x = ±√44

Since we're considering the region in the first quadrant, we take the positive square root:

x = √44 = 2√11

Now we can set up the integral:

V = ∫[0, 2√11] 2πx (√(44 - x²)) dx

Simplifying the integrand:

V = 2π ∫[0, 2√11] x (√(44 - x²)) dx

To evaluate this integral, we can make the substitution u = 44 - x²,

du = -2x dx:

V = 2π ∫[0, 44] (√u) (-du/2)

V = -π ∫[0, 44] (√u) du

Integrating:

V = -π [2/3 u^(3/2)] [0, 44]

V = -π [2/3 (44)^(3/2) - 0^(3/2)]

V = -π (2/3) (44)^(3/2)

Finally, taking the absolute value and simplifying:

V = π (2/3) (44)^(3/2)

So, the volume of the solid generated is π (2/3) (44)^(3/2).

Thus,

The volume of the solid generated is [tex]\pi (2/3) (44)^{3/2}.[/tex]

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The given equation e(x)=f(x)f(-x) is an even function since e(x)=e(-x) and f(x)^2=f(-x)^2.

let f(x) be a function with domain double-struck r. (a) show that e(x) = f(x) f(−x) is an even function. (simplify your answers completely.) e(x) = f(x) f(−x) = f(x) f(x) = f(x)^2 e(x) = f(x)^2

e(x) = e(-x) Therefore, e(x) is an even function.

(-x) is an even function, we need to show that e(x)=e(-x).

We start by replacing -x with x in the equation:

e(x)=f(x)f(-x)

e(x)=f(x)f(x)

e(x)=f(x)^2

Now, we replace x with -x in the equation:

e(-x)=f(-x)f(-x)

e(-x)=f(-x)^2

Since the left-hand side of both equations are the same (e(x)=e(-x)), and the right-hand sides of both equations are the same (f(x)^2=f(-x)^2), we can conclude that e(x) is an even function.

The given equation e(x)=f(x)f(-x) is an even function since e(x)=e(-x) and f(x)^2=f(-x)^2.

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Answer:

600,000 is greater than 11,733.83. Or

600,000>11,733.83

Step-by-step explanation:

To compare the values of 600,000 and 11,733.83, we can simply look at the magnitude of the numbers.

600,000 is much larger than 11,733.83. This means that 600,000 is greater than 11,733.83

For the given coordinates, the equation of line is y = 1, The coordinates of foot of perpendicular will be (0, 5)

Assume that ax + by + c = 0. is a continuous line. If a perpendicular line is drawn from any point on the plane to the provided straight line, the intersection of the two lines is referred to as the foot of the matching perpendicular.

Because a perpendicular line is created from point P to the y-axis, the distance between its intersection with the x-z plane remains constant.

Given,

the points of line are - (1, 1) and (5, 5).

equation of line = (y - yo) = m (x - xo)

⇒ y - 1 = [tex]\frac{5 - 1}{5 - 1}[/tex] (x - 1)

⇒ y = 1. is the equation of the line,

Let us consider the point (a,b) such that it is the foot of the perpendicular,

Now, as the line line joining the points (1,5) and (a,b) will be perpendicular to each other,

⇒ y - 1 = b - 5 / a - 1 (x - 1)

The slope of one line is the negative inverse of other line,

⇒ b - 5 / a -1 = 0

⇒ b = 5

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The sum or difference matched with its simplified form;

(7x³ -12x + 4) - (6x²- 8x - 3)

open parenthesis

= 7x³ -12x + 4 - 6x² + 8x + 3

= 7x³ - 6x² - 4x + 7

(5x³ - 3x + 7) - (2x³ + 6x² - x)

= 5x³ - 3x + 7 - 2x³ - 6x² + x

= 3x³ - 6x² - 2x + 7

(3x³ - 10x² + 2x) - (10x² + 7x - 10)

= 3x³ + 10x² + 2x - 10x² - 7x + 10

= 3x³ - 5x + 10

(5x³- 3x + 7) + (2x³ + 6x² - x)

= 5x³- 3x + 7 + 2x³ + 6x² - x

= 3x³ + 6x² - 4x + 7

Therefore, a given expression can be written Ina simplified form.

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