Write an equation for each line in point-slope form and then convert it to standard form.
through (2,3) and (3,5)

The equation for the vertical translation of y=f(x). 2 units up is y=f(x)+2.

Using equations of the form y=f(x), you may be able to modify the graph in various ways. For example, you can move the chart up, down, left, or right, flip it around the x or y-axis, or stretch or shrink it vertically or horizontally. Understanding these transformations makes it easy to graph different functions. Start with a "basic model" and apply a series of transformations/modifications to change it to the desired function.

A point on the graph of y=f(x) has the shape (x,f(x)).

The points on the graph of y=f(x)+2 are of the form (x,f(x)+2).

So, the graph for y=f(x) +2 is the same as the graph for y=f(x) shifted up by 2 units.

The new equation is:

y=f(x)+2

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The distance between points F and G is 15 units.

The length of the line segment connecting two points on a plane is known as the distance between the points. The formula for distance between two points ([tex]x_{1}[/tex],[tex]x_{2}[/tex]) and ([tex]y_{1}[/tex],[tex]y_{2}[/tex]) is given by

distance between two points =[tex]\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2 } }[/tex]

The equation can be used to calculate the distance between any two locations on an x-y plane or coordinate plane.

The point F is (-9,14) and point G is (1,-19)

distance between F and G

=[tex]\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2 } }[/tex]

=[tex]\sqrt{(-9-1)^2+(-14-(-19))^2 } }[/tex]

=[tex]\sqrt{(-9-1)^2+(-14+19)^2 } }[/tex]

=[tex]\sqrt{(-10)^2+(5)^2 } }[/tex]

=[tex]\sqrt{100+25}[/tex]

=[tex]\sqrt{125}[/tex]

=15 units.

Distance between point F and G is 15 units.

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36 is 17th term of sequence a₁₈ = 32, d=-4

Given the 18th term and the common difference, we subtract the common difference from the 18th term to determine the 17th term:

a₁₇ = a₁₈ - d

     = 32 - (-4)

     =  36

To solve linear equations, utilize the steps that are provided below.

Lets take an example of solve z for 7z – (3z – 4) = 12

Here is no multiplication or division, so this will be easy to solve

⇒ 7z – (3z – 4) = 12

⇒ 7z – 3z + 4 = 12

⇒ 4z = 12 – 4

⇒ 4z = 8

⇒ z = 8/4

⇒ z = 2

See, that was so simple, using the mentioned methods, every linear equation can be solved easily.

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Answer:

2.5t

Step-by-step explanation:

Let the man be

x

feet away from the street light, and his shadow length be

y

, Let the tip of the shadow be

l

feet away from the light such that

l

=

x

+

y

and we are given that

d

x

d

t

=

2

By similar triangles:

y

6

=

x

+

y

30

∴

30

y

=

6

(

x

+

y

)

∴

30

y

=

6

x

+

6

y

∴

30

y

−

6

y

=

6

x

∴

24

y

=

6

x

∴

y

=

1

4

x

Now,

l

=

x

+

y

⇒

l

=

x

+

1

4

x

∴

l

=

5

4

x

Differentiating wrt

x

gives;

d

l

d

x

=

5

4

And by the chain rule we have:

d

l

d

t

=

d

l

d

x

d

x

d

t

∴

d

l

d

t

=

5

4

×

2

∴

d

l

d

t

=

2.5

feet/sec

The slope of the line is 150.

It is given the number of songs (in million) downloaded from the internet in 2006 and 2004.

We actually need to graph this data which will be a straight line. We try to find two points in this graph using the given data.

Represent the years as x coordinates and the number of songs as y coordinates. Also take 2006 as 6 only and 2004 as 4 only for convenience.

Now we get two points [tex](x_1,y_1)[/tex] = (4, 200) and [tex](x_2,y_2)[/tex] = (6, 500)

Slope of the line = [tex]\frac{y_2-y_1}{x_2-x_1} = \frac{500-200}{6-4}[/tex] = 300/2 = 150

Here the slope is positive, which means that the number of downloads are increasing with increase in years. The rate of change in number of downloaded songs from internet is 150 million in two years.

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If Martina has 16 cars available to rent, and if c is the number of cars she would have left after renting r of them, then the equation relating c to r will be c = 16 - r, and the number of cars she would have been left with after renting 14 of them is calculated to be 2.

An algebraic expression is a type of expression consisting of both numerals and letters.

If Martina has a total of 16 cars available to rent, then the equation or algebraic expression relating c to r is,

c = total number of cars available to rent - r

c = 16 - r

After renting 14 of the total cars available, then the number of cars left can be calculated using this equation,

c = 16 - r

c = 16 - 14

c = 2

Thus, the number of cars she would have been left with is calculated to be 2.

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I bought 2 pens and 4 pencils from given amount.

Let us represent the number of pens and pencils as x and y respectively. Forming the equation as per the given information -

Equation representing total number of pens and pencils bought

x + y = 6 - Equation 1

Rearranging the equation 1 as per the variable x

x = 6 - y - Equation 2

Equation representing cost of bought pens and pencils

1x + 0.5y = 4 - Equation 3

Keep the value of x from Equation 2 in Equation 3

1 (6 - y) + 0.5y = 4

6 - y + 0.5y = 4

Rearranging the equation for positive sign on each side -

y - 0.5y = 6 - 4

Performing subtraction on Right Hand Side of the equation to find the value of y

0.5y = 2

Shifting 0.5 to Right Hand Side of the equation and performing division

y = 2 ÷ 0.5

y = 4

Calculating the value of x by keeping the value of y in Equation 2

x = 6 - 4

x = 2

Hence, I bought 2 pens and 4 pencils from given amount.

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Answer:

Choice 4

Step-by-step explanation:

A one to one function has every domain with only one range as their individual output. Choice 4 has the output 6 mentioned twice for both 2 and 4 so therefore it's not a one to one function because the range 6 has two outputs 2 and 4.

Answer:

-7

Step-by-step explanation:

-3x-6>15

-3x/-3>21/-3=-7

x<-1+3

x<2.

Answer:

1: P = -3

2: X = -60

3: N = 5

Step-by-step explanation:

Ok, for number one you are going to add 4 on the -4 canceling that out, and adding 4 on the -7 making that -3.

Number 2 you times 10 on the x/10 to cancel out the 10 (making it x) then you do -6 times 10, and then you get -60

Last one is a little tricky here's the equations I used to get the last one

-3 x 8n = -24x

New equation: 4-24x=-104

                        +4         +  4

___________________________

                        -24x = -100

                         ___     ___

                         -24      -24

                                                   _

so the answer would be x = 5

The geometric mean between pair of numbers √20 and √80 is 6.32

We know that for a sequence of numbers x1, x2,.., xn

The formula of geometric mean is,

[tex]\Pi=\sqrt[n]{x_1.x_2.x_3...x_n}[/tex]

In this question, we have been given two numbers √20 and √80

We need to find the geometric mean between pair of numbers, √20 and √80

Using the formula for the geometric mean,

[tex]\Pi=\sqrt[n]{x_1.x_2.x_3...x_n}[/tex]

In this question, we have two numbers.

So, the value of n is 2.

[tex]\Pi=\sqrt{\sqrt{20}\times \sqrt{80} }\\\\ \Pi=\sqrt{\sqrt{1600} }\\\\ \Pi=\sqrt{40}\\\\ \Pi=6.32[/tex]

Therefore, the geometric mean between pair of numbers √20 and √80 is 6.32

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According to the question, the model has linear relationship between the temperatures which are in degree Fahrenheit and in Celsius.

Now, to calculate the rate of change of the temperature which is defined in the ratio of y-axis and x-axis. Therefore, the expression can be written as:

Rate of change of the temperature = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

From the graph model, the coordinate values are (0,32) and (30,86)

[tex]x_{2} = 30; x_{1}=0; y_{2}=86; y_{1}=32[/tex]

Substituting the given values in the standard expression to calculate the rate of change of the temperature:

Rate of change of the temperature = [tex]\frac{86-32}{30-0}=\frac{54}{30} =\frac{9}{5}[/tex]

Hence, the rate of change of the temperature is 9/5.

What is rate of change in the graph?

In the graph, the rate of change of a quantity defines the relationship between the two quantities. The calculated values are either positive, negative or zero. It is a slope of a line in the form of ratio of vertical to horizontal axis.

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x = 73 is the value of x.

To find the value of x:

So,

Then rounding to the nearest tenths gives x = 73.

Therefore, x = 73 is the value of x.

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Step-by-step explanation:

i thank the answer is 10.5 thanks for your question

The translation from A to B is described as-

In mathematics, a translation is the movement of a shape to the left or right and/or up or down. The translated shapes appear to be the exact size as that of the original shape, therefore the shapes are congruent.

Now, according to the question.

Consider the equation for the shifting from A to B  as y = k x+b

The coordinates of the location A and B are-(see the diagram)

A(-3,-1) and  B(0,1)

Thus, A, B into y = k x+b

Substitute the value of A in the equation;

-1 = -3k + b

b = 1  

So, k = 2/ 3

Put value of k in 'y'.

Hence, AB ; y = (2/3)x + 1

Thus, the coin moves thought y = (2/3)x + 1 from x = -3 to x = 0.

In words; Shifting from A to B, the coin moves 3 spaces to the right after that moves 2 space up.

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The toy rocket will take a time of 8.999 seconds to hit the ground.

The height of the rocket (h), in meters, in terms of time (t), in seconds, is defined by a quadratic equation. In this question we must determine the time needed for the rocket to hit the ground, that is, a time t greater than zero such that the height is zero. Now we proceed to solve the polynomial:

- 4.9 · t² + 43.4 · t + 6.3 = 0

- 4.9 · (t² - 8.857 · t - 1.286) = 0

- 4.9 · (t - 8.999) · (t + 0.143) = 0

Only the first root offers a realistic value as time is a whole number. Hence, the toy rocket will take a time of 8.999 seconds to hit the ground.

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In linear equation, a = (n+p)/m is equation for the indicated variable.

What in mathematics is a linear equation?

In terms of algebra, writing "am" is the same as writing "a*m" or "a times m"

To undo the multiplication, we use division. We will divide both sides by m to get 'a' all by itself on the left side

So...

am = n+p

a*m = n+p

(a*m)/m = (n+p)/m .... divide both sides by m

a = (n+p)/m ... notice how the 'm's cancel out when we divide on the left side

a = (n+p)/m

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Anna , Bob , Chris weigh 108 pounds , 118 pounds  , 92 pounds respectively .

Let the Anna weigh = x pounds

Let the Bob weigh = y pounds

Let the Chris weigh = z pounds

According to the question Anna and Bob together weigh 226 pounds which can be written as  [tex]x+y=226[/tex]     ..(1)

According to the question Bob and Christ together weigh 210 pounds. which can be written as   [tex]y+z=210[/tex]      ...(2)

According to the question Anna and Chris together weigh 200 pounds. which can be written as  [tex]x+z=200[/tex]      ..(3)

From equation (1) [tex]y=226-x[/tex]  putting this value in equation (2) , we get

      [tex]226-x+z=210\\16=x-z\\16+z=x[/tex]       ..(4)

Putting above value in equation (3) , we get

     [tex]16+z+z=200\\2z=184\\z=92[/tex]

Putting z = 92 in equation (4) , we get

    [tex]x=16+92\\x=108[/tex]

Putting x = 108 in equation (1) , we get

   [tex]y=226-108\\y=118[/tex]

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The seventh term of the given geometric progression is: 729.

A geometric progression, sometimes referred to as a geometric sequence in mathematics, is a series of non-zero numbers where each term following the first is obtained by multiplying the preceding one by a constant, non-zero value known as the common ratio.

Now,

Given GP: 1, -3, 9, .....

Here, a₁ = 1

a₂ = a₁ (r)

=> 3 = 1 (r)

=> r = 3

Thus, a₇ = a₁ (r⁷⁻¹) = 1 (3⁶) = 729.

Hence, The seventh term of the given geometric progression is: 729.

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The leg length is 9/ root2

Here we are given a right angle triangle , and in a right triangle , sum of squares of sides is equal to the square of hypotenuse.

[tex]a^{2}+b^{2} =c^{2}[/tex]

This is right angle scalene triangle as the two angles other than right angle are 45° each.

Hence ,

In this case we have

[tex]a^{2}+a^{2} =c^{2}[/tex]

[tex]2a^{2}=c^{2}[/tex]

[tex]a^{2}=\frac{c^{2} }{2}[/tex]

Taking square roots on both sides

[tex]a=\frac{c}{\sqrt{2} }[/tex]

Here we have hypotenuse c given as 9

Hence, the leg length is [tex]a =\frac{9}{\sqrt{2} }[/tex]

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The comparison and contrast function notation and vector notation for translations are-

Function notation is a method for writing functions that are simple to comprehend and read.

Whenever a vector is simply a list of numbers, it can be represented by an arrow in space.

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2 and 18 are the two pairs of whole numbers with a geometric mean that is also a whole number.

The geometric mean is defined as the nth root of the product of n numbers, which means for a set of numbers x1, x2, ..., xn. The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.

For example, we multiply them: 2 × 18 = 36

Then (as there are two numbers) take the square root: √36 = 6

Conditions: The square roots of the product of the pair of numbers must be whole-number. These two conditions must be met to have an exact geometric mean

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The value of x in equation is 8

To find the value of x, we will solve the given mathematical expression based on the sequence of signs and placement of bracket. Firstly bracket will be solved followed by subtraction or addition.

Rewriting the equation -

6(x - 4) = 3x

Performing multiplication on Left Hand Side of the equation to find the value of x

6x - 24 = 3x

Shifting 3x and 24 to other side of equation

6x - 3x = 24

Performing subtraction on Left Hand Side of the equation

3x = 24

Shifting 3 to other side of equation as denominator

x = 24 ÷ 3

Performing division on Right Hand Side of the equation

x = 8

Hence, the value of x is 8.

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The salary after 6 years is $83,000.

The salary after tt years is 53,000 + 5000tt.

The equation that can be used to determine Aisha's salary after a period of time can be represented by a linear equation. This is because that her salary increases by a constant rate each year.

A linear function represents a straight line when drawn on a coordinate plane. A linear function is a function that has a single variable raised to the power of 1. An example is y + 2. The variable y is raised to the power of 1. 2 is the constant term.

Total salary = beginning salary + (rate of increase x number of years)

Salary after 6 years: 53,000 + (5,000 x 6)

= 53,000 + 30,000 = $83,000

Salary after tt years: 53,000 + (5,000 x tt)

53,000 + 5000tt

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The three interior angles of a triangle have a sum of 180°. If two of the three angles are complementary, then the triangle is right. The correct option is B

A triangle is a polygon that has three angles and three sides. The sum of angles in a triangle is 180 degrees.

Types of triangles are isosceles, scalene and equilateral triangles. Two angles are said to be complementary if their sum is 90 degrees.

The three interior angles of a triangle have a sum of 180°. If two of the three angles are complementary, then the triangle is right.

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The factored form of the expression 16x² + 9y² over the complex numbers is; (4x + 3iy)(4x − 3iy)

We want to find the factored form of the expression 16x² + 9y² over the complex numbers.

We can rewrite the given expression 16x² + 9y² as perfect squares;

= (4x)² + (3y)²

Now, from using the difference of squares formula, we know that;

a² - b² = (a + b)(a - b)

Thus;

(4x)² + (3y)² = (4x)² - (√-9)²y²) can be expressed as;

(4x + 3iy)(4x − 3iy)

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Answer:

Step-by-step explanation:

18. Supplementary angles add up to 180°therefore

∠W+∠V=180°

∠W+49°=180°

∠W=180°-49°

∠W=131°

19.Complementary angles are angles that add up to 360°

∠2+∠3=90°

∠2+56°=90°

∠2=90°-56°

∠2=34°

Vertical angles are equal therefore this tells us that ∠1 and ∠2 are equal

∠1=∠2

∴∠1=34°

20. Angles which are in a linear pair add up to 180°(supplementary)

Mathematically what the statement is sayin is :

∠1+∠2=180°

and ∠1=5(∠2)-18°

therefore we can plug in ∠1=5(∠2)-18° in ∠1+∠2=180°

(5(∠2)-18°)+∠2=180°

5(∠2)+∠2=180°+18°

6(∠2)=192°

[tex]\frac{6 < 2}{6} =\frac{198}{6} \\ < 2=33[/tex]

∠1=5(33°)-18°

∠1=165°-18°

∠1=147°