Making connections between arithmetic sequences and linear function


Use the graph or the formula to write the equation of the line in a slope intercept form

Answer: An increase of a population doubling every year can be modeled using repeated multiplication. The expression for the increase after five years would be:

2 * 2 * 2 * 2 * 2 = 2^5 = 32

This can also be written as an exponential expression, with 2 as the base and 5 as the exponent:

2^5 = 32

This notation expresses that 2 is multiplied by itself 5 times to get 32, which represents the population size after 5 years with the rate of doubling every year.

It's also worth mentioning that exponential notation is used to represent the quantity of growth (doubling every year) , and using exponents makes it easy to see how much the population has grown over a certain period of time.

Step-by-step explanation:

Answer:

Step-by-step explanation:

2^5= 32

=2 x 2 x 2 x 2 x 2

= 2^5

= 32

The correct answer is option c as DE/AE=BC/AC

In Mathematics, a slope of a line is the change in y coordinate with respect to the change in x coordinate. The net change in y-coordinate is represented by Δy and the net change in x-coordinate is represented by Δx. Where “m” is the slope of a line. So, tan θ to be the slope of a line.

Given here ΔADE ≈ΔABC and thus  DE/AE=BC/AC

but tanθ = DE/AE=BC/AC where tan θ is the slope of the line thus both the lines have identical slopes.

Hence, The correct answer is option c)  ΔADE ≈ΔABC thus DE/AE=BC/AC Therefore the slope of AD is the same as the slope of AB

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Solving the modular arithmetic will result to

The term modular arithmetic is the arithmetic that works with whole numbers in a such that, following division by a modulus, all numbers are replaced by their remainders.

Solving the modular arithmetic: 2 + n = 0 (mod 6)

2 + n = 0 (mod 6)

n = -2

Solving the modular arithmetic: 2 + (3 + n) = 1 (mod 6):

2 + (3 + n) = 1 (mod 6)

5 + n = 1 (mod 6)

n = 1 - 5

n = -4

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By geometric formulas, the gray square has an area of 25 square units.

Squares are quadrilaterals with four right angles. In this question we find a geometric system formed by seven squares with distinct dimensions. By geometry, the area of a square (A), in square units, is equal to following formula:

A = l²

Where l is the length of the side of a square, in units.

All the required lengths are represented in the image attached below. First, determine the side length of the gray square:

7 + x = 12

x = 12 - 7

x = 5

The gray square has four sides with a length of 5 units.

Second, determine the area of the gray square by means of square formula:

A = x²

A = 5²

A = 25

According to area formula for squares, the area of the gray square is equal to 25 square units.

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The domain and scope of the identity function are the same. Equation for the identity function is  f(x) = x, or y = x.

An statement, rule, or law in mathematics that specifies the connection between an independent variable and a dependent variable (the dependent variable). A relationship between a group of inputs and one output each is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. Y=X2 is an illustration of this. You only receive one output for y if you enter anything for x. The fact that x is the input value leads us to argue that y is a function of x.

identity function are the same.

Equation for the identity function is

f(x) = x, or y = x.

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Solving provided question, we can say that the given linear equation is  y = -2x + 4 and -x -x +3(-2x + 4 ) + 9 = x = 7/3

A linear equation is one that has the form y=mx+b in algebra. B is the slope, and m is the y-intercept. It's usual to refer to the previous clause as a "linear equation with two variables" because y and x are variables. The two-variable linear equations known as bivariate linear equations. There are several instances of linear equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. It is referred to as being linear when an equation has the form y=mx+b, where m stands for the slope and b for the y-intercept. When an equation has the formula y=mx+b, with m denoting the slope and b the y-intercept, it is referred to as being linear.

the given linear equation is

y = -2x + 4 and -x +3y = -9

now,

-x +3(-2x + 4 ) + 9

-x - 6x - 12 +9 = 0

-7x = -3

x = 7/3

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The amount of money owned by JIm is $9

From the question, we have the following parameters that can be used in our computation:

The ratio of the amount owned by Tim tohe ratio of the amount owned by Jim is 6 : 1

This means that

Tim : Jim = 6 : 1

Also from the question, we have

Tim gets $54

This means that

Tim = 54

Substitute the known values in the above equation, so, we have the following representation

54 : Jim = 6 : 1

Multiply by 9

54 : Jim = 54 : 9

So, we have

Jim = 9

Hence, the amount is $9

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The predicted winning time in year 10 of the race is 2.83 minutes.

The equation for the line of fit is obtained inserting the points of the data-set into a linear regression calculator.

Given:

From the scatter plot, the points are given as follows:

(1, 5.4), (2, 5), (3, 4.6), (4, 4.9), (5, 4.1), (6, 3.9), (7, 3.6), (8, 3.3).

Inserting these points into the linear regression calculator, we get

y = -0.29x + 5.67.

Now, for x= 10 year

y = -0.29(10) + 5.67 = 2.77 minutes.

which is closest to 2.83 minutes.

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a) After 8 hours, a person earned $148 in wages.

The function is, $x = 18.5 y

b) The linear function to represent the table is, y = 10x -4

A function with one or two variables and no exponents is referred to as a linear function. It is a function with a straight line as its graph.

Given:

a) After 8 hours, a person earned $148 in wages.

So, the unit rate change = 148/ 8

                                         = 18.5

So, if y represents the number of hour and $x the wages, then the function is

$x = 18.5 y

b) The unit rate change for the function is

= (56-36)/(6-4)

= 20/ 2

= 10

So, the linear function to represent the table is

y - 36 = 10(x - 4)

y- 36 = 10x - 40

y = 10x -4

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Answer:

A right triangular prism contains rectangular faces.

Step-by-step explanation:

Answer:

(C) A right triangular prism contains rectangular faces.

Step-by-step explanation:

hope this helps <3

The angle measures : m∠1 = 107° and m∠4= 73°

When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.

Given:

There are 2 parallel lines

A transversal cuts them

∠5 and ∠7 are corresponding angles

Corresponding angles are equal in measures

m∠5 = m∠7

m∠7 = 107°

m∠5 = 107°

∠1 and ∠7 formed a pair of alternate exterior angle.

And m∠1 = m∠7

∠1 and ∠4 formed a pair of linear angle.

The sum of the measures of the linear angles is 180°

m∠1 + m∠4 = 180°

Substitute ∠1 by 107°

m∠4 + 107° = 180°

m∠4 = 180 - 107

m∠4= 73°

Therefore, m∠1 = 107° and m∠4= 73°

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Therefore , the solution of the given question of triangle comes out to be y=2.

You can link any three points in a plane to form triangles, which have three sides and three angles. They were among of the first geometric forms to be examined. Since triangles may be divided into any polygon, regardless of whether it has four, five, six, or n sides, they are very significant.

Here,

Since it is an isosceles trapezoid the length of the two legs (i.e.  AB =  4y - 6 and CD = 3y - 4) are equal.  In other words,

4y - 6  = 3y - 4.

Subtract 2y from each side:

4y - 3y - 6= -4

Now add 10 to each side

=> 4y - 3y - 6 + 4  

=> y - 2 =0

=> y = 2

Therefore , the solution of the given question of triangle comes out to be y=2.

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Pat needs 16[tex]\sqrt{1200}[/tex]cm of trim for each placemat to braid trim around the edges.

To find out how much trim Pat needs for each placemat, we need to first find the perimeter of the rectangle.

The area of the rectangle is given as 1200 [tex]cm^2[/tex].

The perimeter of the rectangle is the sum of all four sides of the rectangle. If the length and breadth of the rectangle are L and B respectively, then the perimeter P can be calculated using the formula:

P = 2(L+B)

Now, let's find the length and breadth of the rectangle using the area formula,

Area = LB = 1200 [tex]cm^2[/tex]

so LB = 1200 [tex]cm^2[/tex]

Now we can substitute the value of L and B in the perimeter formula

P = 2(L+B) = 2(sqrt(1200)+sqrt(1200))

As the area of the rectangle is given as 1200[tex]cm^2[/tex] , the length and breadth can be calculated as [tex]\sqrt{1200}[/tex]cm.

So the perimeter of the rectangle is

P = 2*([tex]\sqrt{1200}[/tex] + [tex]\sqrt{1200}[/tex])

  = 2*2[tex]\sqrt{1200}[/tex]

  = 4[tex]\sqrt{1200}[/tex]cm

To find the total length of trim Pat needs for the placemat, we need to multiply the perimeter by the number of edges that need a trim. Since the edges are around the rectangular placemat, so it will be 4 edges.

So, the total length of trim Pat needs for the placemat is

P4 = 4[tex]\sqrt{1200}[/tex]*4 = 16[tex]\sqrt{1200}[/tex]cm

So Pat needs 16[tex]\sqrt{1200}[/tex]cm of trim for each placemat.

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The values that can be eliminated from the solution set of the inequality are given as follows:

Values of x ≤ 0.

The perimeter of a rectangle of length l and height h is given by the formula presented as follows:

P = 2(l + h).

The dimensions for this problem are given as follows:

The perimeter must be less than 120, hence the inequality is given as follows:

2(x + 4 + x) < 120

4x + 8 < 120

4x < 112

x < 28.

However, a side length must assume positive values, meaning that values of x ≤ 0, that is, zero and negative values, are eliminated from the solution.

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The value of x will be;

⇒ x = 35

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

Given that;

In Rectangle RSTW;

⇒ RZ = 3x + 6

⇒ SW = 7x - 23

Since, By the figure we conclude that;

⇒ 2 RZ = SW

Substitute the given values, we get;

⇒ 2 RZ = SW

⇒ 2 (3x + 6) = 7x - 23

⇒ 6x + 12 = 7x - 23

⇒ 12 + 23 = 7x - 6x

⇒ 35 = x

⇒  x= 35

Thus, The value of x = 35

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The probability the coin comes up all heads or all tails after 4 tosses is 12.5%.

Probability refers to the ratio, chance, likelihood, or odds that an expected outcome is achieved when there are many possible outcomes.

Probability is a fractional value, especially when the odds of the occurrence of the expected event are not 100% uncertain or certain.

In this case, probability lies between zero and one.

Odds of a coin coming up either heads or tails every time = 1:7

The sum of ratios = 8 (1 + 7)

Thus, the probability of the coin coming up with all heads or all tails = 12.5% (1/8 x 100)

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Solving for X in a Right Triangle

Subtract the sum of the two angles from 180 degrees. The sum of all the angles of a triangle always equals 180 degrees. Write down the difference you found when subtracting the sum of the two angles from 180 degrees. This is the value of X

A trapezoid will appear smaller than the original trapezoid if it is dilated with a scale factor between 0 and 1.

An open, flat shape with four straight sides and one pair of parallel sides is referred to as a trapezoid or trapezium. A trapezium's non-parallel sides are referred to as the legs, while its parallel sides are referred to as the bases. The legs of a trapezium can also be parallel.

What three characteristics do trapezoids have? An isosceles trapezoid has equal base angles and diagonals. The midpoints of the two opposing sides are parallel to the point where the diagonals intersect. An isosceles trapezoid has opposite sides that are congruent to one another or have the same length.

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The required maximum volume of the box is 2ft³.

Volume is a measurement of how much space an object takes up.

We can calculate the amount of something needed to fill it, like water for a bottle, tank, or aquarium.

Amount of space that stuff takes up - that's what we mean when we talk about volume.

Stuff is anything that has mass and takes up room.

When scientists are measuring volume, they usually use cubic meters.

So, calculate as:

V=x·y²

2x+y=3

y=3-2x

V = x·(3-2x)² = x·(9-12x+4x²) = 9x - 12x²+4x

V ` = 9-24x+12x² = 3(4x²-8x+3) = 3(4x²-6x-2x+3)

= 3[2x(2x-3)-(2x-3)] = 3(2x-3)(2x-1)

When the most amount is present: V ` = 0

Then,

2x-3 = 0  

2x = 3              

x = 1.5 (That's wrong, 'cause y = 0.)

or: 2x-1 = 0

2x = 1

x = 0.5,  y = 3 - 2 · 0.5 = 3 - 1 = 2

V max = 0.5 · 2² = 0.5 · 4 = 2 ft³

Therefore, the required maximum volume of the box is 2ft³.

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The total quantity of water that the container would have to hold is given as 8.

We know that the container holds 1 gallon of water, and that each bottle holds 1/8 of a gallon. To find out how many bottles the container can hold we need to divide the container size by the size of each bottle.

We can use the following formula to find the number of bottles:

Number of bottles = Container size / Size of each bottle

We can substitute the values into the formula:

Number of bottles = 1 / (1/8) = 8

So the container can hold 8 bottles of water.

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The measure of side KL in the given quadrilateral is 31.88 units.

A quadrilateral is a four-sided polygon with four edges and four corners that is used in geometry.

The Latin words Quadri, a variation of four, and latus, meaning "side," are the source of the name.

A two-dimensional shape with four sides, four vertices, and four angles is referred to as a quadrilateral.

Convex and concave are the two main forms.

Convex quadrilaterals can also be divided into a number of subgroups, including trapezoids, parallelograms, rectangles, rhombus, and squares.

So, according to the given figure of FGHI and JKLM:

FG/JK = GH/KL = HI/LM = FI/JM

GH/KL = HI/LM

7/KL = 9/41

KL = 7*41/9 = 31.888

Therefore, the measure of side KL in the given quadrilateral is 31.88 units.

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Correct question with a diagram:

Quadrilateral FGHI is similar to quadrilateral JKLM. Find the measure of side KL. Round your answer to the nearest tenth.

standard form for a linear equation means

• all coefficients must be integers, no fractions

• only the constant on the right-hand-side

• all variables on the left-hand-side, sorted

• "x" must not have a negative coefficient

to get the equation of any straight line, we simply need two points off of it, let's use (-6,10), (-4,18)

[tex](\stackrel{x_1}{-6}~,~\stackrel{y_1}{10})\qquad (\stackrel{x_2}{-4}~,~\stackrel{y_2}{18}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{18}-\stackrel{y1}{10}}}{\underset{\textit{\large run}} {\underset{x_2}{-4}-\underset{x_1}{(-6)}}} \implies \cfrac{8}{-4 +6} \implies \cfrac{ 8 }{ 2 } \implies 4[/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}{10}=\stackrel{m}{ 4}(x-\stackrel{x_1}{(-6)}) \implies {\large \begin{array}{llll} y -10 = 4 ( x +6) \end{array}} \\\\\\ y-10=4x+24\implies -4x+y=34\implies \stackrel{standard~form}{{\Large \begin{array}{llll} 4x-y=-34 \end{array}}}[/tex]

Answer: Commutative property.

Step-by-step explanation: Basically, the commutative property can be expressed as a + b = b + a, which matches this scenario pretty well. You are correct.

Let me know if this is right! :)

Answer:

  (a)  (5, 9)

Step-by-step explanation:

You want the solution to the system ...

The value of x is already given, so no work is required to find that.

The value of y is found by using that value of x in the equation for y:

  y = 2x -1

  y = 2·5 -1 = 10 -1 = 9

The solution is (x, y) = (5, 9).

Answer:

it takes 110 minutes

Step-by-step explanation:

Answer:

it takes approximately 110 minutes

Exponential graph is identified by its curve.

An exponential graph is a graph that shows a rapid increase or decrease of a value over time, where the rate of change is proportional to the current value. It is usually represented by a curve that bends upward or downward.

The key characteristic of an exponential graph is that the Y-axis is on a logarithmic scale, and the curve of the graph is a steep curve going either up or down. The equation of an exponential graph is usually of the form y = ab^x, where a is the initial value, b is the growth or decay factor, and x is the independent variable.

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Answer:

A.   2w² + 8w

Step-by-step explanation:

rectangle:  A = w(w+4) = w² + 4w

triangle:  A = 1/2(2(w+4)(w)) = 1/2(2w+8)(w) = 1/2(2w²+8w) = w² + 4w

Total Area = w² + 4w + w² + 4w = 2w² + 8w

As per the decimal number system, the cost of 25 tickets is $32.5

The term decimal in math is defined as the numbers, which consist of two parts namely, a whole number part and a fractional part separated by a decimal point.

Here we have given that the product of 3 student tickets at $4.

Here we have know that the ticket cost for 3 students is $4.

Then the cost of one person is calculated by dividing the cost by the number of students then we get,

=> 4/3

=> 1.3

So, if the cost for one person or ticket is $1.3 then the cost of 25 tickets is calculated as,

=> 25 x 1.3

=> 32.5

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The collection of values that can be plugged into a function's domain are called its parameters.

A specific area of thinking, action, or interest is referred to as a domain, particularly one over which someone has authority. [formal]...the great artists who experimented with several mediums. Alternative words: region, globe, field, department [informal] More Domain synonyms.

The domain is used to facilitate user navigation and increase accessibility. Slugs are links that consist of the domain plus the text "/text."

Let y = f(x) be a function where x and y are the independent and dependent variables. The chosen x-value is said to fall within the domain of f if a function f offers a means of effectively producing a single value y while using a value for x to that end.

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Answer:

Mark the sides BC and DF as congruent---------------------------------------------------We are given a side and an adjacent angle and need to identify the missing side or angle for proof by SAS .SAS is side-angle-side and requires two sides and the included angle to be congruent on both triangles.The missing bit is the other side of the marked angle B and D:sides BC or DF.

Step-by-step explanation: