A mathematical ________________ is a statement that contains _______________, _________________, and ___________________.

The final solution to the given simultaneous equation is (0, -6)

A finite set of equations for which common solutions are sought is referred to as a set of simultaneous equations, often known as a system of equations or an equation system.

Given the set of equations below:

y = -x-6

x + 3y = -18

Using the substitution method;

Substitute equation 1 into 2 to have:

x + 3(-x - 6) = -18
x - 3x - 18  =-18
-2x = 0

x = 0

Since y = -x- 6

y = -0 - 6

y = -6

Hence the solution to the system of equation is (0, -6)

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Los intervalos numéricos en la recta numérica para x=2, en donde x ∈R y x>2 y x<7, son: (2, 7).

En la recta numérica, los intervalos se definen utilizando dos números, uno inferior y uno superior. Por ejemplo, el intervalo (2, 7) representa todos los números entre 2 y 7, incluyéndolos. Esto significa que el intervalo (2, 7) incluiría 2, 3, 4, 5, 6 y 7. Además, los intervalos numéricos pueden incluir o excluir los números de los extremos. Por ejemplo, el intervalo (2, 7) representa todos los números entre 2 y 7, incluyéndolos, mientras que el intervalo (2, 7] sólo incluiría 3, 4, 5, 6 y 7.

The number intervals on the number line for x=2, where x ∈R and x>2 and x<7, are: (2, 7).

On the number line, intervals are defined using two numbers, one lower and one upper. For example, the interval (2, 7) represents all numbers between 2 and 7, including them. This means that the interval (2, 7) would include 2, 3, 4, 5, 6 and 7. In addition, numerical intervals can include or exclude the numbers at the ends. For example, the interval (2, 7) represents all numbers between 2 and 7, including them, while the interval (2, 7] would only include 3, 4, 5, 5, 6 and 7.

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Answer:

density of apple juice

1.034-1.051g.cm‐³

Note that where the figure below contains two semi-circles that are tangent to each Other at point B and ray DE is tangent to both semicircles at points E and F, it is correct to state that ED = 6√2 (Option A).

A semicircle is a one-dimensional point collection that forms half of a circle in mathematics. It is a circular arc with a circumference of 180°. It just has one symmetry line.

Solving the ED, we have:

ΔDEM ~ ΔDFP

This means that MD/PD = EM/FP

MD/(MD + 9) = 3/6
6MD = 3MD + 27 (Collect like terms over the equation)
3MD = 27
MD = 9 (If both sides are divided by 9)

Note that

ED = √[MD²-EM²]
ED  = √9²-3²]
ED = 6√2

Thus, it is correct to state that ED =  6√2

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Answer: (x = 6, y = -0.5)

Step-by-step explanation:

Substitute into one of the equations: 3(5 - 2y) + 2y = 17

Apply Multiplicative Distribution Law: 15 - 6y + 2y = 17

Rearrange unknown terms to the left side of the equation: -6y + 2y = 17 - 15

Combine like terms: -4y = 2

Divide both sides of the equation by the coefficient of variable: y = 2 / (-4)

Remove the parentheses: y = -2 / 4

Calculate the product or quotient:  y = -0.5

Substitute into one of the equations: x = 5 - 2 * (-0.5)

Remove the parentheses: x = 5 - 2 * -0.5

Calculate the product or quotient: x = 5 + 1

Calculate the sum or difference: x = 6

The solution of the system is: {x = 6, y = -0.5}

Hope this helps!

The correct matching of the given numbers is as follows:

Natural numbers, whole numbers, integers, rational numbers, and irrational numbers can all be classified as real numbers.

An imaginary number is created by multiplying a real number by i where i = (-1). To calculate the square root of negative values, we employ imaginary numbers. For instance, (√-7) Equals (-1) √7

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Answer:

x > 50

Step-by-step explanation:

[tex]\frac{x}{10}[/tex] > 5 ( multiply both sides by 10 to clear the fraction )

x > 50

Numbers that summarize a sample of data are called "sample statistics," and numbers that summarize populations are called "population parameters."

A sample statistic is a single value that describes a characteristic of a sample, such as the mean, median, or standard deviation. Sample statistics are used to make inferences about the population from which the sample was drawn.

A population parameter, on the other hand, is a single value that describes a characteristic of an entire population, such as the population mean, population standard deviation, or population proportion. Population parameters are usually not known and must be estimated from a sample.

For example, if you have a sample of 100 students' heights, the mean height of the sample (a sample statistic) can be calculated and used to make inferences about the mean height of the population of all students.

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The diagonals of the parallelogram intersect at the center. Then the value of the variable 'x' is 38.

It is a polygon with four sides. The total interior angle is 360 degrees. A parallelogram's opposite sides are parallel and equal. Its diagonals intersect in the center.

Given parallelogram A B C D below, DE = 31 If EB = x - 7.

By the definition, then the equation is given as,

EB = DE

x - 7 = 31

Simplify the equation, then we have

x - 7 = 31

x = 31 + 7

x = 38

The value of the variable 'x' is 38.

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Answer: The coordinates of a point in the plane are represented by a pair of values, (x, y), where x is the horizontal position and y is the vertical position.

When a point is reflected across the x-axis, the x-coordinate remains the same, but the sign of the y-coordinate changes. This is because the x-axis acts as a mirror, flipping the point over to the opposite side of the axis.

So, the coordinates of the reflected point would be (x, -y).

In summary, the x-coordinates of a point and its reflection across the x-axis are the same, while the y-coordinates are different (one is positive, and the other is negative).

Step-by-step explanation:

The minimum number of rotations required for a 3500 km race is 52,800 rotations.

This is calculated by dividing the total distance of the race, 3500 km, by the circumference of the wheel, which is 2πr, where r is the radius of the wheel. The radius of the wheel must be between 27.5 cm (half of 55 cm) and 37.5 cm (half of 75 cm). So, the minimum number of rotations is given by 3500/(23.14r), where r is the minimum radius of the wheel. Substituting in r = 27.5 cm, we get the minimum number of rotations required for a 3500 km race as 52,800.

To arrive at this result, we must first calculate the circumference of the wheel, which is given by 2πr, where r is the radius of the wheel. Since the diameter of the wheel must be between 55 cm and 75 cm, the radius of the wheel must be between 27.5 cm (half of 55 cm) and 37.5 cm (half of 75 cm).

We then divide the total distance of the race, 3500 km, by the circumference of the wheel, which is given by 2πr. Substituting in the minimum radius of the wheel, 27.5 cm, we get the minimum number of rotations required for a 3500 km race as 52,800.

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Answer:

201 eighth graders and 201 + 54 = 255 seventh graders at the middle school.

Step-by-step explanation:

The system of equations representing the situation is:

x + 54 = 456 - x

2x + 54 = 456

2x = 402

x = 201

where x represents the number of eighth graders. So, there are 201 eighth graders and 201 + 54 = 255 seventh graders at the middle school.

Answer: 7000 (4 s.f)

Step-by-step explanation:

[tex]x[/tex] x [tex]1.06^{11} = 13,288.09[/tex]

[tex]x = \frac{13,288.09}{1.06^{11} }[/tex]

[tex]x = 7000[/tex] (4 s.f)

This is a question where you make the equation by yourself using compound interest. If you have any further questions, feel free to ask me :)

Answer:

  4% and 8%

Step-by-step explanation:

You want to know the interest rates that yield $168 on investments of $2200 at one rate and $1000 at a rate 4% higher.

Let r represent the lower interest rate. Then x+0.04 is the higher rate, and the total interest earned is ...

  2200r +1000(r +0.04) = 168

Simplifying the equation, we have ...

  3200r +40 = 168

  3200r = 128 . . . . . . . . subtract 40

  r = 128/3200 = 0.04 = 4% . . . . . divide by the coefficient of r

The rates of interest on Lindays's investments were 4% and 8%.

the lunch cost, including tax and tip 67.31$.

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, the Patterson family went to eat lunch at a new restaurant. They spent $52.30 on their meal. The restaurant will add a 7 1/4% tax to the cost of the meal.

total cost after the tax = 52.30*(1 + 7 1/4)%

total cost after the tax = 52.30* 7.25/100 + 52.30

total cost after the tax = 3.79 + 52.30

total cost after the tax = 56.09

Since,

Mr Patterson was very happy with the food and service so he gave a 20% tip to the waitress.

Thus,

Total cost after the tip = 56.09(1 +20/100)

Total cost after the tip = 56.09 * 1.2

Total cost after the tip =  67.308

Therefore, the lunch cost, including tax and tip 67.31$.

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P should stand for the population in a certain year. P = 97*(0.875) is the function to model the population change (year-2021).

P should stand for the population in a certain year. To simulate how the animal population changes over time, we can utilise a function. The 12.5% annual reduction rate will be considered by the function. This can be stated mathematically as P = 97*(0.875), which is how it will appear in the equation (year-2021). In this equation, the number 97 stands in for the population in 2021, and the number 0.875 denotes the 12.5% annual decline rate. The exponent (year 2021) represents the number of years that have passed since 2021, allowing the equation to take the population decline over a number of years into consideration. As a result, the equation can be used to determine the animal species' population at the start of a particular year.

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The constant of proportionality is, 12

Yes, The graph of equation is a straight line.

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

Given that;

Mark is planning a birthday party at the local skating rink.

And, The table shows the rates that are charged for parties at the skating rink.

Let two points on table are (5, 60) and (10, 120)

Now, The equation of line is,

⇒ y - 60 = (120 - 60) / (10 - 5) (x - 5)

⇒ y - 60 = 60/5 (x - 5)

⇒ y - 60 = 12(x - 5)

⇒ y - 60 = 12x - 60

⇒ y = 12x

Thus, The number of guests proportional to the cost.

And, Constant of proportionality = 12

And, The graph of y = 12x is a straight line.

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Complete question is shown in figure.

The definition of the f, h, and g functions, indicates the possible damains, range and composite functions are as follows;

(a) The domain of f is the set of real numbers except x = 5

The range is the set of all real numbers except f(x) = 0

(b) (f○g)(x) = 1/(3·x + 1)

(f○h)(x) = 1/(cos(x + 1) - 5)

(g○f)(x) = (6·x - 27)/(x - 5)

(c) g⁻¹(x) = (x - 6)/3

(d) The domain is; x ∈ R, and x ≠ 5

The domain of a function is the set of the possible input values of the function.

The specified functions are;

f(x) = 1/(x - 5), g(x) = 3·x + 6, h(x) = cos(x + 1)

(a) The domain of f is the set of numbers such that the rational polynomial, 1/(x - 5) is defined

The polynomial, f(x) = 1/(x - 5) is defined when x - 5 ≠ 0, therefore;

x - 5 + 5 ≠ 0 + 5 = 5

x - 0 ≠ 5

x ≠ 5

The polynomial, f(x) = 1/(x - 5) is difined where x ≠ 5

The domain is the set of all real numbers, such that x ≠ 5

The range of the function, is the set of the possible output values, such that as (x - 5) ≠ 0, f(x) ≠ 0

The range is therefore; -∞ < f(x) < ∞, f(x) ≠ 0

(b) The composite functions (f○g)(x) = f(g(x))

Therefore; (f○g)(x) = f(g(x)) = f(3·x + 6)

Plugging in x = (3·x + 6) in the function, f(x) = 1/(x - 5), we get;

f(g(x) = 1/((3·x + 6) - 5) = 1/(3·x + 1)

f(g(x) = 1/(3·x + 1)

(f○h)(x) = f(h(x))

h(x) = cos(x + 1), therefore; (f○h)(x) = f(cos(x + 1))

Plugging in x = cos(x + 1) in f(x) = 1/(x - 5), we get;

(f○h)(x) = f(cos(x + 1)) = 1/((cos(x + 1)) - 5)

(g○f)(x) = g(f(x))

g(f(x)) = g(1/(x - 5)) = 3/(x - 5) + 6 = (6·x - 27)/(x - 5)

(c) g⁻¹(x)

g⁻¹(x) is the inverse of the function, g(x), which can be foiund as follows;

Plugging in; y = g(x) = 3·x + 6

y = 3·x + 6

y - 6  = 3·x

x = (y - 6)/3

Therefore;

(d) (f + g)(x) = f(x) + g(x)

f(x) + g(x) = 1/(x - 5) + 3·x + 6

The use of a graphing calculator indicates that we get;

1/(x - 5) + 3·x + 6 = (3·x² - 9·x - 29)/(x - 5)

Therefore; (f + g)(x) = (3·x² - 9·x - 29)/(x - 5)

The domain is therefore, the possible values of x at which the denominator is not zero, that is x ≠ 5

The domain is the set real numbers except x = 5, which is; x ∈ R, x ≠ 5

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The locus of point P is a circle with center at the midpoint of points M and N, and radius equal to half the distance between the two points.

The midpoint of M and N is given by:

((-4+1)/2, (3+0)/2) = (-1.5, 1.5)

The distance between M and N is given by the distance formula:

√((1-(-4))^2 + (0-3)^2) = sqrt(25 + 9) = √(34)

So, the radius of the circle is √(34)/2.

Hence, the equation of the locus of point P is given by:

(x-(-1.5))^2 + (y-1.5)^2 = (√(34)/2)^2

or

(x+1.5)^2 + (y-1.5)^2 = (√(34)/2)^2

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You need to buy 80 cubic feet of sand to fill the sandbox.

a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.

The sandbox is a rectangular prism.

Length is 10 feet.

Width is 8 feet

Height is 1 feet.

Volume of prism=Length×Width×Height

=10×8×1

=80 cubic feet.

Hence, you need to buy 80 cubic feet of sand to fill the sandbox.

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The complete question is given in the attachment.

Based on the information provided, it can be concluded the sum is $11,000.

To know what is the sum, let's calculate the difference in the interest rate:

13% - 12% = 1%

This means the difference between the two interest rates presented is 1%. Based on this, let's calculate the 1% of the sums given:

Based on the above, the correct sum is $11,000 because only in this one 1%  is equivalent to $110.

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The evaluation of the expression is 1198.4.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

Given;

Expression;  52 × (7.2 + 16) − 8

Now,

=52*23.2-8

=1206.4-8

=1198.4

Therefore, the algebra will give 1198.4.

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Given that x, y, z are different and satisfy the equations:

x^3 + x^2 - x = 0

y^3 + y^2 - y = 0

z^3 + z^2 - z = 0

We can multiply the first equation by y, the second by z, and the third by x to obtain:

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

y^3z + y^2z - yz = 0

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

Adding these three equations together, we get:

x^3y + x^2y - xy + y^3z + y^2z - yz + z^3x + z^2x - zx = 0

Expanding and simplifying, we find that:

x^2y + xy^2 + y^2z + yz^2 + z^2x + zx^2 - xy - yz - zx = 0

Now, we use the fact that x, y, and z are roots of the equations given above to simplify the expression further:

x^2 - x + y^2 - y + z^2 - z = 0

We can also write this as:

(x - 1)(x + 1) + (y - 1)(y + 1) + (z - 1)(z + 1) = 0

This expression can be factorized as:

(x - 1)(x + 1)(y - 1)(y + 1)(z - 1)(z + 1) = 0

Since x, y, and z are all different, we have that:

(x - 1)(x + 1) ≠ 0

(y - 1)(y + 1) ≠ 0

(z - 1)(z + 1) ≠ 0

Therefore, we have:

(x - 1)(x + 1)(y - 1)(y + 1)(z - 1)(z + 1) = (x - 1)(x + 1)(y - 1)(y + 1)(z - 1)(z + 1) = 0

So, we can conclude that:

xyz = -1

And,

1 + xyz = 0

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The cost of one rose bush is $2 and the cost of one bunch of grass is $17.

A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

Given that, Norachai and Rain each improved their yards by planting rose bushes and grass.

Let the cost of rose bushes be x and the cost of bunches of grass be y.

Norachai spent $209 on 11 rose bushes and 11 bunches of grass.

Now, 11x+11y=209

x+y=19 --------(i)

Rain spent $40 on 3 rose bushes and 2 bunches of grass.

3x+2y=40 --------(ii)

Substitute x=19-y in equation (ii), we get

3(19-y)+2y=40

57-3y+2y=40

57-40=y

y=17

Substitute y=17 in equation (i), we get

x=2

Therefore, the cost of one rose bush is $2 and the cost of one bunch of grass is $17.

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The base of the ladder must be placed 3 ft from the base of the building. The total length of the ladder is 15 ft.

1. The ladder must reach 12 ft up the wall, so we need to calculate the distance from the base of the building to 12 ft.

2. The total length of the ladder is 15 ft.

3. Subtract the length of the ladder (15 ft) fromthe desired height (12 ft) to get the distance from the base of the building.

4. 12 ft - 15 ft = -3 ft

5. Since the distance cannot be negative, the base of the ladder must be placed 3 ft away from the base of the building.

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By AA similarity Triangle ABC and DEF are similar.

Two triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles. Similar figures are described as items with the same shape but varying sizes, such as two or more figures.

Given:

First in Triangle ABC

using Angle Sum Property

<C= 180 - <A - <B

<C = 180 - 77 - 48

<C = 55 degree

and, In Triangle DEF

using Angle Sum Property

ND= 180- <E- <F

<D = 180- 55 - 48

<D = 77

As, In both Triangle ABC and DEF all angles triangles are Equal.

<A = <D= 77

<B = <E = 48

<C = <F= 55

So, by AA similarity Triangle  ABC and DEF are similar.

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In linear equation, 3437.5  feet - lbs is the work done to haul the anchor .

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                          Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

Anchor  weighing = 100 lbs

   given 3  lb/ft

  the combined weight  = 3( 25 -y ) + 100

                                 = 175 - 3y

    the workdone on small solution is = (175 - 3y)Δy

                      w = ∫₀²⁵ (175 - 3y) dy

                        = 175[y]²⁵₀- 3/2[y²]²⁵₀

                       = 175 [ 25 - 0 ] - 3/2 [ 25²- 0²]

                       = 3437.5  feet - lbs

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Answer: NO

Step-by-step explanation:

Adult tickets cost $27

Child tickets cost $18

$18 x 4 child tickets = $72 + $27 adult ticket = $99

Answer:

2 or -1

Step-by-step explanation:

[tex]1. \: y + 1 = 2x \\ 2. \: \frac{y + 1}{2} = x \\ 3. \: x = \frac{y + 1}{2} [/tex]

[tex]1. \: y = 2 \times 0 - 1 \\ 2. \: y = - 1[/tex]