2x - 7y=-3
4x - 5y = 12

If (x, y) is the solution to the system of equations above, what is the
value of y²?

Let's find the magnitude of each component:

Let:

 [tex](x1,y1)=(4,4)[/tex]

[tex](x2,y2)=(-12,8)[/tex]

     [tex]u=ax+by[/tex]

  [tex]||u||=\sqrt{a^2+b^2}[/tex]

So, let's find a and b:

[tex]a=|x2-x1|=|-12-4|=|-16|=16[/tex]

       [tex]b=|y2-y1|=|8-4|=|4|=4[/tex]

so:

[tex]||u||=\sqrt{16^2+4^2}[/tex]

   [tex]||u||=\sqrt{272}[/tex]

 [tex]||u||\thickapprox16.492[/tex]

And the direction is:

         [tex]\theta=180-\text{tan}^{-1}\huge \text(\dfrac{b}{a}\huge \text)[/tex]

        [tex]\theta=180-\text{tan}^{-1}\huge \text(\dfrac{4}{16}\huge \text)[/tex]

[tex]\theta\thickapprox180-\text{tan}^{-1}\huge \text(\dfrac{4}{16}\huge \text)\thickapprox165.964[/tex]

It is not possible to have each of the 15 participants play against exactly 5 others in your chess tournament.

Here's why:
1. In a tournament where each person plays against 5 others, the total number of matches played can be calculated using the combination formula: C(n, k) = n! / (k!(n-k)!),

where n is the number of participants and k is the number of opponents each player faces.

In this case, n = 15 and k = 5.
2. Applying the formula, we get C(15, 5) = 15! / (5!(15-5)!) = 15! / (5!10!) = 3,003 matches in total.

However, since each match involves two players, the actual number of matches played is 3,003 / 2 = 1,501.5.

This is not a whole number, so it is impossible to have exactly 5 matches per player in this scenario.
3. To further confirm this, we can also analyze the problem using graph theory.

Each participant can be represented as a node in a graph, and a match between two participants can be represented as an edge connecting the corresponding nodes.
4. In this case, we want each node to have a degree of 5 (i.e., 5 edges connecting to it).

The Handshaking Lemma states that the sum of the degrees of all nodes in a graph is equal to twice the number of edges in the graph.

Thus, in our scenario, the sum of all degrees would be 15 * 5 = 75.

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In the absence of any specific story or context for "Secrets of Oak Park," However, based on the given prompt, here are five possible feelings that Zach might have if he believes his dad is behind the incidents.

Betrayed: Zach might feel a sense of betrayal if he believes that his dad, who he presumably trusts and loves, is responsible for the incidents in question.

Confused: Zach may feel confused about why his dad would do something like that or what his motives might be.

Angry: Zach could feel angry that his dad could cause harm or trouble, and be causing problems in their community or family.

Worried: Zach might feel worried about what will happen if his dad is found to be responsible for the incidents.

Determined: Zach might also feel determined to find out the truth, confront his father, and set things right.

Therefore, These are just a few possibilities, as the story's context would greatly influence Zach's feelings.

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Answer:

x = - 2

Step-by-step explanation:

using the rules of exponents

[tex]a^{m}[/tex] ×[tex]a^{n}[/tex] = [tex]a^{(m+n)}[/tex]

[tex]\frac{a^{m} }{a^{n} }[/tex] = [tex]a^{(m-n)}[/tex]

given

[tex]\frac{9^{3}(9^{1}) }{9^{6} }[/tex]

= [tex]\frac{9^{(3+1)} }{9^{6} }[/tex]

= [tex]\frac{9^{4} }{9^6} }[/tex]

= [tex]9^{(4-6)}[/tex]

= [tex]9^{-2}[/tex] = [tex]9^{x}[/tex]

then equating exponents

x = - 2

Answer:

$4.40

Step-by-step explanation:

The price per foot is $13.20/60 = 0.22 per foot. To find the cost of the extra 20 feet of cable, we can multiply the price per foot by the number of feet needed:

0.22 x 20 = $4.40

Therefore, Cora will pay $4.40 for the extra 20 feet of cable.

The air pressure at the peak of Mount Marcy is approximately 164.17 pascals.

Air pressure, also known as atmospheric pressure, is the force exerted by the weight of the Earth's atmosphere on any given surface. It is the weight of the column of air that extends from the Earth's surface up to the top of the atmosphere.

The Pascal (symbol: Pa) is the SI (International System of Units) unit of pressure, named after the French mathematician and physicist Blaise Pascal. One pascal is defined as the pressure exerted by the force of one Newton on an area of one square meter. In other words, 1 Pa = 1 N/m².

We are given the formula that relates the altitude (a) and the atmospheric air pressure (P):

a = 15,500 (5 - log₁0 P)

We are also given that the altitude at the peak of Mount Marcy is 1629 meters. We can use this information to find the atmospheric air pressure at the peak of Mount Marcy by plugging in a = 1629 and solving for P:

1629 = 15,500 (5 - log₁0 P)

Dividing both sides by 15,500, we get:

(5 - log₁0 P) = 1629 / 15,500

Subtracting 5 from both sides, we get:

-log₁0 P = (1629 / 15,500) - 5

Simplifying the right-hand side, we get:

-log₁0 P = -1.5454

Taking the inverse logarithm (base 10) of both sides, we get:

P = 10⁻¹.⁵⁴⁵⁴

Using a calculator, we can evaluate this expression to get:

P ≈ 164.17 pascals

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The percentage error between the estimated age and the carbon-14 dating age of the fossil is 12.5% and an estimated age of approximately 2978 years would give the same percent error of 12.5%.

What is the definition of percentage?

Percentage is a way of expressing a fraction or proportion as a part of 100. It is denoted by the symbol "%". The term "percent" means "per hundred". For example, if 50% of a group of people are men, it means that 50 out of every 100 people in the group are men.

Now,

a). To find the percent error between the estimated age and the carbon-14 dating age of the fossil, we can use the formula:

percent error = |(estimated age - carbon-14 dating age) / estimated age| x 100%

Substituting the given values, we get:

percent error = |(3200 - 3600) / 3200| x 100%

= |-400 / 3200| x 100%

= 12.5%

Therefore, the percent error between the estimated age and the carbon-14 dating age of the fossil is 12.5%.

b). To find other estimates that give the same percent error, we can use the formula:

percent error = |(estimated age - carbon-14 dating age) / estimated age| x 100%

If we let x be the estimated age that gives the same percent error, then we can write:

12.5% = |(x - 3600) / x| x 100%

Simplifying, we get:

0.125 = |(x - 3600) / x|

Taking the positive square root of both sides (since percent error cannot be negative), we get:

√0.125 = (x - 3600) / x

Simplifying, we get:

x = 3600 / (1 - √0.125)

x ≈ 2978

Therefore, an estimated age of approximately 2978 years would give the same percent error of 12.5% as the original estimated age of 3200 years. This means that if the fossil were actually 2978 years old, the carbon-14 dating would have given an age estimate of approximately 3299 years, which is 12.5% higher than the true age.

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Answer:

np

Step-by-step explanation:

To solve the inequality 4x - 7 >= 13, we can follow these steps:Add 7 to both sides of the inequality:

4x - 7 + 7 >= 13 + 7Simplify the left side and right side:

4x >= 20Divide both sides by 4 (since we want to solve for x, not 4x):

4x/4 >= 20/4Simplify:

x >= 5Therefore, the solution to the inequality 4x - 7 >= 13 is x >= 5.

To find the volume of the solid generated by revolving about the y-axis the region under the curve y = 5e^-2x in the first quadrant, we can use the method of cylindrical shells.

The formula for the volume generated by a cylindrical shell is V = 2πrh Δx, where r is the radius of the shell, h is the height of the shell, and Δx is the thickness of the shell.

In this case, we are rotating about the y-axis, so our cylindrical shells will have a height of y and a thickness of Δy. The radius of each shell will be the distance from the y-axis to the curve, which is x.

We can solve for x in terms of y by rearranging the equation y = 5e^-2x:

y/5 = e^-2x

ln(y/5) = -2x

x = -ln(y/5)/2

So the volume of each cylindrical shell is:

V = 2πx * y * Δy

Substituting for x:

V = 2π(-ln(y/5)/2) * y * Δy

V = -πln(y/5) * y * Δy

To find the total volume, we need to integrate this expression from y = 0 to y = infinity:

V = ∫[0,∞] -πln(y/5) * y dy

Using integration by parts:

u = ln(y/5), dv = y dy

du = 1/y dy, v = 1/2 y^2

∫[0,∞] -πln(y/5) * y dy = [-π(1/2y^2 ln(y/5) - 1/4y^2)] [0,∞]

= [π/4]

Therefore, the volume of the solid generated by revolving about the y-axis in the region under the curve y = 5e^-2x in the first quadrant is π/4, which is a finite value.

Hence, the answer is 0.7854 (rounded to four decimal places).

Thus, the total time to reach the ball at ground after being kicked is 6 seconds.

The maximum value or minimum value of a quadratic function are located at its vertex, which is shaped like a U. The quadratic function's axis of symmetry crosses the function (a parabola) at its vertex.

This graph's opening side plus vertex determine the quadratic function's range. To ascertain the quadratic function's range, locate the lowest and highest f(x) values on the function graph.

Standard form for quadratic function:

f(x) = ax² + bx + c,

In which, a, b, and c -  real numbers and  a ≠ 0.

Given function for the which explain the height of ball with time:

f(x) = -6x² + 36x + 0.23

f(x) is the height of ball taken from ground.

x is the time in seconds.

Draw the graph of the function using the graphing tool.

The diagram is attached.

The vertex of the graph lies at (3, 54.23).

x - 3 seconds

f(x) = 54.23 ft

So, time taken by ball to reach at height of 54.23 ft is 3 seconds.

Total time to reach the ball at ground = 2*3 =  6 seconds

Thus, the total time to reach the ball at ground after being kicked is 6 seconds.

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To find the minimum number of subsets that must be printed to ensure at least two identical subsets on the list, we can use the concept of Pigeonhole Principle. At least 65 sets must be printed to be sure of having at least 2 identical subsets on the list.

1. First, we need to find the total number of possible subsets in a 6-element set. We use the formula 2^n, where n is the number of elements. In this case, n = 6.
2. Calculate the total number of subsets: 2^6 = 64.
3. According to the Pigeonhole Principle, if there are N pigeonholes (subsets) and more than N pigeons (printed subsets), at least one pigeonhole will have more than one pigeon.
4. To ensure that we have at least 2 identical subsets, we need to print one more subset than the total number of possible subsets: 64 + 1 = 65.

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The ratio of the time taken to compute LU factorization vs. back/forward substitution depends on several factors such as the size and sparsity of the matrix, the specific implementation of the algorithms, and the hardware used to perform the computations.

However, in general, LU factorization takes more time than back/forward substitution. This is because LU factorization involves decomposing the original matrix into two matrices, L and U, which requires more computations than simply solving the system of equations using back/forward substitution once the L and U matrices are available.

As a rough estimate, the time taken for LU factorization can be several times larger than the time taken for back/forward substitution, especially for large and dense matrices.

However, the exact ratio can vary widely depending on the specific factors mentioned above.

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Answer:

for fog(x)

fog=f(g(x))

= 2(-4x +5)

=-8x+10

When two variables are related in such a way that people with high scores on one variable tend to have high scores on the other, and people with low scores on one variable tend to have low scores on the other, it is called a positive correlation. This type of relationship between variables is appropriate for a correlation analysis.

A correlation is a statistical technique used to measure the strength and direction of the association between two variables. The most common correlation coefficient is the Pearson's correlation coefficient, which ranges from -1 to 1. A positive correlation indicates a direct relationship between the variables, while a negative correlation indicates an inverse relationship. A correlation close to 0 suggests no relationship between the variables.

To conduct a correlation analysis, the variables should be continuous or ordinal, which means that they can be measured on a scale with meaningful intervals. Examples of continuous variables include age, height, and weight, while ordinal variables could include rankings, such as class rank or satisfaction levels.

Here's a step-by-step explanation for conducting a correlation analysis:

1. Identify the two variables of interest, ensuring they are continuous or ordinal.
2. Collect the data for each variable from a sample of individuals or observations.
3. Calculate the mean and standard deviation for each variable.
4. Compute the covariance between the two variables, which measures the degree to which they change together.
5. Normalize the covariance by dividing it by the product of the standard deviations of the two variables. This produces the correlation coefficient, which represents the strength and direction of the relationship between the variables.

In summary, a correlation analysis is appropriate when examining the relationship between two continuous or ordinal variables, such as in the case of a positive correlation where high scores on one variable are associated with high scores on the other, and low scores on one variable are associated with low scores on the other.

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Answer:

Below.

Step-by-step explanation:

Let the number of bananas I ate be x.

Number of apples [tex]= 2x[/tex].

Number of blueberries [tex]= 3x[/tex].

So we have the equation:

[tex]x + 2x +3x = 12[/tex]

[tex]6x = 12[/tex]

[tex]x = 2[/tex].

So I ate 2 bananas, 4 apples and 6 blueberries.

No, there cannot be a right triangle with sides of lengths 1, 2, and 3. This is because of the Pythagorean theorem.

The theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, this can be written as:

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

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

If we substitute the values of 1, 2, and 3 for a, b, and c, respectively, we get:

[tex]3^2 = 1^2 + 2^2[/tex]

which simplifies to:

9 = 1 + 4

This is not true, so the sides 1, 2, and 3 do not form a right triangle.

However, there exist right triangles with consecutive natural numbers as their side lengths. One such example is the Pythagorean triple (3, 4, 5), where:

[tex]3^2 + 4^2 = 9 + 16 = 25 = 5^2[/tex]

This is a right triangle where the lengths of the sides are consecutive natural numbers. There are infinitely many such Pythagorean triples, and they can be generated using the formula:

[tex]a = 2mn, b = m^2 - n^2, c = m^2 + n^2[/tex]

where m and n are any two positive integers with m > n.

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In the above mentioned similar triangles the length of DE is 5.25 cm.

A triangle is a three-sided polygon with three vertices. The triangle's internal angle, which is 180 degrees, is constructed.

Since triangles ABC and ADE are similar, their corresponding sides are proportional. We can use this fact to set up a proportion:

AB / AD = BC / DE

Substituting the given values, we get:

8 / AD = 6 / DE

Multiplying both sides by DE, we get:

8DE / AD = 6

Dividing both sides by 8/AD, we get:

DE = (6/8)AD

We need to find AD to solve for DE. To do this, we can use the fact that BD is an altitude of triangle ABC and triangle ADE is similar to triangle ABC. Therefore, triangle ABD is also similar to triangle ABC, and we can set up another proportion:

BD / AB = AD / AC

Substituting the given values, we get:

4 / 8 = AD / (8 + 6)

Simplifying, we get:

1/2 = AD / 14

Multiplying both sides by 14, we get:

AD = 7

Substituting this value back into the first equation, we get:

DE = (6/8)AD = (6/8)7 = 21/4 = 5.25

Therefore, the length of DE is 5.25 cm.

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The expression that represents the maximum weight in tons of all the whales in the oceans is 150y tons.

A variable in mathematics is a symbol or letter that designates a value that is subject to variation or change. In algebraic equations and expressions, variables are used to represent unknown numbers or quantities with a range of possible values. Typically, variables are represented by letters like x, y, and z or a, b, and c. The variable is frequently placed on one side of the equal sign in an equation, and the value that the variable equals is denoted by the expression on the other side of the equal sign.

Given that, the weight of one single whale = 150 tons.

Now,

Maximum weight = 150y tons

Hence, the expression that represents the maximum weight in tons of all the whales in the oceans is 150y tons.

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Answer:

Step-by-step explanation:

The change Melody receives:

=

$

41.58

Explanation:

Cost of skateboard

=

$

79.81

Cost of helmet

=

$

26.41

Total

=

$

106.22

She uses a

45

%

off coupon, this means she pays

55

%

(

100

−

45

)

of the total cost:

She needs to pay only:

55

100

×

106.22

=

$

58.421

Since she uses a

$

100

bill the change she receives is:

100

−

58.42

=

$

41.58

Answer: She got back $42.67

Step-by-step explanation:

$79.81 + $24.41 = $104.22

%45 off of $104.22 = $57.32

$100 - $57.32 = $42.67

Answer:

The numerator is x²+6x+5.

We need to first factorise the expression

The coefficient of x² is 1, and the constant term of the quadratic expression is 5.

the product of coefficient of x² and constant term=5

∴ we break the middle term, 6x, such that,  the product of the coeffcients of  the two terms is 5, while their sum is 6

∴ 6x=5x+x

Now, rewriting the expression,

x²+5x+x+5

Taking x as common factor in x² and 5x,

x(x+5)+x+5

Again, taking x+5 as the common factor,

(x+5)(x+1)

Now, replacing x²+6x+5 with (x+5)(x+6) in the expression,

(x+5)(x+1)/(x+2)

Hence the value of the expression is (x+5)(x+1)/(x+2)

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Angle B and angle C must have a total measure of 180 - 48 = 132 degrees.

What is Linear Equation ?

A linear equation is an equation that represents a straight line on a graph. It is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.

Both Kwasi and Lola wrote true equations.

Kwasi's equation, b+132=180, is true because in a triangle, the sum of the interior angles is always 180 degrees. The angle measures opposite to sides b and c are a and f, respectively, so b+a+f=180. Since a+f=48+132=180- b, we can substitute and get b+132=180, which is true.

Similarly, Lola's equation, 132+f=180, is also true because the sum of the angles in a triangle is always 180 degrees, and angle A has a measure of 48 degrees,

Therefore,  angle B and angle C must have a total measure of 180 - 48 = 132 degrees.

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Answer: Let's assume Ms. Wells bought x pounds of bananas and (11.75 - x) pounds of oranges.

The cost of bananas is $0.40 per pound, so the cost of x pounds of bananas is 0.4x.

The cost of oranges is $0.80 per pound, so the cost of (11.75 - x) pounds of oranges is 0.8(11.75 - x).

The total cost of the fruit purchase is $8.00, so we can write:

0.4x + 0.8(11.75 - x) = 8

Simplifying this equation:

0.4x + 9.4 - 0.8x = 8

-0.4x = -1.4

x = 3.5

Therefore, Ms. Wells bought 3.5 pounds of bananas.

So the answer is (B) 3.50 pounds.

Step-by-step explanation:

9 1/2% converted to decimal form is 0.095.

Answer:

k=198

Step-by-step explanation:

2+4=6

6+6=12

210-12=198

so K=198

The cost of admission for one child is $5.60.

To find the cost of cost of admisson for one child, we can set up a system of linear equations using the given information.
Let A represent the cost of admission for one adult and C represent the cost of admission for one child.

We can write two equations based on the information given:
3A + 5C = $32.50 (for the first group)---------(1)
4A + 4C = $34.00 (for the second group)---------(2)
We can solve these equations step-by-step to find the value of C:

Multiply equation (2) by -1 so that we can eliminate A when adding both equations:
-4A - 4C = -$34.00

Add equations (1) and the modified equation (2) to eliminate A:
(3A + 5C) + (-4A - 4C) = $32.50 + (-$34.00)
-A + C = -$1.50

Multiply the whole equation by -1 to get rid of the negative sign:
A - C = $1.50 ---------(3)

Add equation (2) to the equation we got in step 3 to eliminate C:
(4A + 4C) + (A - C) = $34.00 + $1.50
5A + 3C = $35.50

Divide the equation by 5 to find the value of A:
A = $7.10

Substitute the value of A back into the equation 3:
$7.10 - C = $1.50

Solve for C:
C = $7.10 - $1.50
C = $5.60

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The ratio for sin(a) is 8/17.

To find the ratio for sin(a),

we can use the Pythagorean identity for trigonometric functions in a right triangle:

sin²(a) + cos²(a) = 1. Given that cos(a) = 15/17, we can proceed with the following steps:
1. Square the given cosine ratio:

(15/17)² = 225/289.
2. Subtract the squared cosine ratio from 1:

1 - 225/289 = 64/289.
3. Take the square root of the result to find sin(a):

√(64/289) = 8/17.

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Answer: 120 %

Step-by-step explanation:

part/whole = percent/100

Let x be the percentage we are trying to find, then we can write:

5.40/4.50 = x/100

To solve for x, we can cross-multiply:

4.50x = 5.40 * 100

Dividing both sides by 4.50, we get:

x = (5.40 * 100) / 4.50 = 120

Therefore, $5.40 is 120% of $4.50.

Answer:

7.3km

Step-by-step explanation:

Using Pythagoras Theorem

D² = 7² + 2²

D = 7.280

Rounding off D = 7.3 km

Answer:

  7.3 km

Step-by-step explanation:

You want to know the straight-line distance between Ryan's house and Emmy's house if Ryan's is 7 km north of Shannon's and Emmy's is 2 km east of Shannon's.

The problem can be modeled by a right triangle with legs equal to the distances from Shannon's house to Ryan's and Emmy's houses. The hypotenuse of the triangle is the straight line between Ryan's and Emmy's.

The length of the hypotenuse can be found using the Pythagorean theorem.

  c² = a² +b²

  c² = 7² +2² = 49 +4 = 53

  c = √53 ≈ 7.3

The straight-line distance between Ryan's and Emmy's houses is about 7.3 km.

In the given word problem, they all start to mow their lawns at the same time, Beth will finish first.

Let's assume that Beth's lawn has an area of x, so Andy's lawn has an area of 2x, and Carlos' lawn has an area of (1/3) × 2x = (2/3)x. Let's also assume that Beth's lawn mower can mow at a rate of 1 unit of area per hour, so Carlos' mower can mow at a rate of 1/2 = 0.5 units of area per hour, and Andy's mower can mow at a rate of 1/3 units of area per hour.  We can calculate the time it will take each person to mow their lawn by using the formula:

Time = Area / Rate.

Beth's time = x / 1

                   = x hours

Carlos' time = (2/3)x / 0.5

                    = (4/3)x hours

Andy's time = 2x / (1/3)

                    = 6x hours

According to these facts, it is quite clear that Beth would finish first followed by Carlos and Andy would finish last.

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