a) Positive correlation. b) Slope = 0.02, Intercept = 3.17; rate increases ~2%/yr, starting at 3.17%. c) Predicts interest rate of 5.17% in 2000.
a) The correlation between rate and year can be determined by constructing a scatter plot. From the given plot, we can see that there is a positive correlation between the two variables, meaning that as the year increases, the rate increases as well.
b) The regression model fit to the data provides the slope and intercept of the line of best fit. The slope is 0.02, which means that the rate is increasing by 0.02% per year. The intercept is 3.17, which means that the interest rate starts at 3.17% in 1950.
c) The model predicts that the interest rate in the year 2000 will be 5.17%. This can be calculated by plugging in 2000 for the year in the model equation (Rate = 0.02*Year + 3.17).
d) In summary, the regression model fit to the data shows that on average, interest rates during this period increased by about 0.02% per year, starting from an interest rate of about 3.17% in 1950. The model predicts that the interest rate in the year 2000 will be 5.17%.
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Boden has 45 CDs in his collection, and Camden has 61. If Boden buys 4 new CDs each month and Camden buys 2
new CDs each month, after how many months will Boden and Camden have the same number of CDs? Write an
equation and solve.
After 8 months both Boden and Camden will have the same number of CDs.
Define the term equation with one variable?Equations with a maximum from one variable of order 1 are referred to as linear equations in one variable equations. It has the formula axe + b = 0, with x serving as the variable. There is just one answer to this equation.The given data:
Current CD's of Boden : 45.Current CD's of Camden : 61.Let 'n; be the total number of months, when both have the same amount of CDs.
Thus,
For 4 new CDs each month bought by Boden: 45 + 4n.
For 2 new CDs each month bought by Camden : 61 + 2n.
Then, the equation becomes:
45+ 4n = 61 + 2n
45 + 2n = 61
2n = 16
n = 8
Thus, after 8 months both Boden and Camden will have the same number of CDs.
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Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning
The pair of figures are similar because:• Both figures are parallelograms
• All sides are congruents and • Scale factor 1: 2.5
A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).
The pair of figures are similar because:
• Both figures are parallelograms
• All sides are congruents
• Angles W and Y are congruent with angles P and R
• Angles X and Z are congruent with angles S and Q
• Scale factor 1: 2.5
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Need Help ASAP
I still don't understand this
Read carefully and do as it says
By using divisibility rule, the possible factors of number are-
495 : 3, 5 and 9924 : 3 and 4210 : 2, 5 and 10Explain the term divisibility rule?A simple technique to determine if a given number may be divided by such a fixed divisor before carrying out the division operation is to run a divisibility test. When a number is fully divided by yet another number, the remainder should equal zero and the quotient ought to be a whole number. A factor is really a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, or else the numbers we are adding are factors of both the product since the product is divisible by them.For the stated number:
495 : 3, 5 and 9 as 495 is completely divisible by 3, 5 and 9.924 : 3 and 4, as 924 is completely divisible by them.210 : 2, 5 and 10 as 210 is completely divisible by 2 and 5.To know more about the divisibility rule, here
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Two gentlemen are taking their morning walks, beginning from opposite ends of a park. They meet 720 meters away from one of the ends, after which they continue walking. After reaching the end of the park, each turns back. On the way back, they meet at a distance of 400 meters from other end of the park. What is the length of the park? Their speeds are constant, but not identical.
The length of the park will be 598 meters.
What is the Pythagorean theorem?Pythagorean theorem states that in the right angle triangle, the hypotenuse square is equal to the square of the sum of the other two sides. According to this statement, the areas of the squares on the other two sides add up to the size of the square whose side is the hypotenuse.
Given that two gentlemen are taking their morning walks, beginning from opposite ends of a park. They meet 720 meters away from one of the ends, after which they continue walking. After reaching the end of the park, each turns back. On the way back, they meet at a distance of 400 meters from another end of the park.
The length of the park will be calculated as,
H² = P ² + B²
B² = (720)² - ( 400 )²
B = √ ( 358400 )
B = 598 meters
The length of the park would be 598 meters.
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FILL IN THE BLANK when a study uses the__________method of data collection, participants complete a questionnaire about their thoughts, feelings, and behavior at the end of each day.
The correct answer is Diary.
We acquire a different understanding of participants' experiences than we might from other techniques when participants immediately record their observations or perceptions. Participants may feel more comfortable expressing themselves in a diary than in an interview since sometimes personal reflections are more open. The participant's commitment to following through is, of course, a requirement of diary systems. It can be difficult to anticipate someone's availability to write within the time range set by the researcher. To get over these challenges, contemporary diary researchers employ a range of strategies, including as utilizing apps and technology or combining textual and artistic techniques
Hence, The correct answer is Diary Method.
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One cubic foot of a chemical weighs 60 pounds. How many pounds of the chemical can a container with the dimensions shown hold?
The number of pounds of the chemical container can hold with given dimension is, 3600 pounds
What is Volume?In mathematics, Space occupied by three dimensional objects is called Volume. Basically it is a space within a closed region of every three dimensional object.
Given that,
1 ft³ = 60 pounds
A container which has dimensions,
length = 6 ft
height = 4 ft
width = 5/2 ft
Volume of container = length × height × width
= 6 × 4 × 5/2
= 60 ft³
1 ft³ = 60 pounds
60 ft³ = 60 × 60 pounds
= 3600 pounds
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What is the result of isolating y^2 in the equation below? 4x^2 + 25y^2= 100
Answer:
In factored form: [tex]y=\sqrt{-\frac{1}{25}(2x-10)(2x+10)}[/tex]
In standard form: [tex]y=\sqrt{-\frac{4}{25}x^2+4}[/tex]
Step-by-step explanation:
Generally for these type of equations we want to get the term with "y" by it self (ignoring coefficients and any operations such as square root, etc...) so we want to get rid of the [tex]4x^2[/tex] from the left side which we can do by subtracting this from both sides to maintain equality and get rid of it on the left side.
[tex]25y^2=-4x^2+100[/tex]
One thing I want to note here is we can actually factor the right side which we generally want to do, we can rewrite it as: [tex]-(4x^2-100)[/tex] and from here we have a difference of squares: [tex]a^2-b^2=(a-b)(a+b)[/tex] so we can rewrite this as: [tex]-(2x-10)(2x+10)[/tex]
[tex]25y^2=-(2x-10)(2x+10)[/tex]
From here we divide both sides by 25:
[tex]y^2=-\frac{1}{25}(2x-10)(2x+10)[/tex]
Now from here to get rid of that square we do the inverse which is taking the square root:
[tex]y=\sqrt{-\frac{1}{25}(2x-10)(2x+10)}[/tex]
The reason factoring is useful in this case is we can now easily determine the domain of the function since inside the square root we have a quadratic which is opening down meaning it's only not negative between the zeroes (including the zeroes since they're... zero not negative) and this would be the domain of the graph. But if we wanted it to not be in factored form we would just put back what we had so we would have:
[tex]y=\sqrt{-\frac{1}{25}(4x^2-100)}[/tex]
then distribute the -1/25 to get:
[tex]y=\sqrt{-\frac{4}{25}x^2+4}[/tex]
A bakery makes 40 different flavors of muffins. 25% of the flavors have chocolate as one of the ingredients. draw a tape diagram to show how many flavors have chocolate and how many dont
The flavors that have chocolate as one of the ingredients will be 10 and the flavors that don't have chocolate as one of the ingredients will be 30.
How to calculate the number of chocolate?A percentage simply has to do with the a value or ratio which can be stated as a fraction of 100. It should be noted that when we want to we calculate a percentage of a number, we simply divide it and then multiply the value that is gotten by 100.
In this case, the bakery makes 40 different flavors of muffins. 25% of the flavors have chocolate as one of the ingredients
The flavors that have chocolate as one of the ingredients will be:
= 25% × 40
= 10
The flavors that don't have chocolate as one of the ingredients will be:
= 40 - 10
= 30
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∠A and ∠B are vertical angles. If angle A=(8x-23) and angle B=(6x-5) then find the measure of ∠A.
The measure of ∠A is 49.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
Vertical angles are equal.
∠A = 8x - 23
∠B = 6x - 5
Now,
8x - 23 = 6x - 5
8x - 6x = -5 + 23
2x = 18
x = 9
Now,
∠A.
= 8x - 23
= 8 x 9 - 23
= 72 - 23
= 49
Thus,
∠A is 49.
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if a coin is tossed five times: find the probability of getting at least one tail. give answer as simplified fraction.
The probability of obtaining at least one tail is equal to one minus the probability of getting all heads when a coin is tossed five times.
The probability of obtaining at least one tail is equal to one minus the probability of getting all heads when a coin is tossed five times.
3/8
The probability of obtaining at least one tail is equal to 1 minus the probability of obtaining all heads. The probability of obtaining all heads is 1/2^5 = 1/32. Therefore, the probability of obtaining at least one tail is 1 - 1/32 = 31/32. This simplifies to 3/8.
The probability of obtaining at least one tail is equal to one minus the probability of getting all heads when a coin is tossed five times. The probability of obtaining all heads is one out of two to the power of five, which is equal to one out of thirty-two. Therefore, the probability of obtaining at least one tail is one minus one out of thirty-two, which is equal to thirty-one out of thirty-two. This simplifies to three out of eight.
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Briley answered 85% of the 60 quiz questions correctly. How many questions did she answer correctly?
Answer:
51 questions she answered correctly.
Step-by-step explanation: 85/100 x 60
= 510/10
= 51
Answer:
51 questions
Step-by-step explanation:
To do this we need to realize that 85% is equal to 85/100. This can be simplified in fraction form to .85.
Looking at the question, we see that it says "Briley answered 85% of the 60 quiz..." A key word here is of. Of always means multiply. So we can see if we were going to form an equation, we would be multiplying.
Briley answered correctly = (85%) times (60 questions)
Mathematically, ?=(.85)x(60)=51
So we know she answered 51 correctly, and that 51 is 85% of 60.
I hope this helps you!
the slope field for the given differential equation is provided. sketch both branches of particular solution curve that passes through the point
The solution curve for the differential equation dy/dx = (y - y²) / x is plotted in the slope field.
What is a slope field?
The solutions to a scalar function's first-order differential equation are shown graphically by slope fields, also known as direction fields. Functions depicted as solid curves are solutions to a slope field. In order to determine the estimated tangent slope at a point on a curve, where the curve is some solution to the differential equation, one can use a slope field, which displays the slope of a differential equation at specific vertical and horizontal intervals on the x-y plane.
The slope field is a cartesian grid where you draw lines in various directions to represent the slopes of the tangents to the solution. Therefore by drawing a curve through consecutive slope lines, we can find a solution to the differential equation.
The differential equation is dy/dx = (y - y²) / x.
The solution curve is drawn using the solution points (1,2) for the differential equation.
Therefore, the solution curve is obtained.
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ap stats the random variable x has mean 12 and standard deviation 3 the random variable w is defined as w
The mean of the random variable w is 15, and the standard deviation is 3.46.
The formula for the mean of a random variable is μ = E(x), and the formula for the standard deviation of a random variable is σ = sqrt(Var(x)). In this case, the mean of the random variable x is 12, and the standard deviation is 3. We can calculate the mean and standard deviation of the random variable w as follows:
E(w) = E(x) + 3 = 12 + 3 = 15
Var(w) = Var(x) + 9 = 9 + 9 = 18
SD(w) = sqrt(Var(w)) = sqrt(18) = 3.46
Therefore, the mean of the random variable w is 15, and the standard deviation is 3.46.
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Last year, Chris opened an investment account with 5200 . At the end of the year, the amount in the account had increased by 7%
Answer:
Step-by-step explanation:
By the way you've worded this, I'm guessing that you're looking for how much money Chris made at the end of the year.
When we're dealing with annual income, we must always consider how much was already present in savings before actually performing any operation, because that will give us the final answer.
Since Chris started with $5200, we do [tex](0.07)($5200) = $364[/tex] to find that Chris made $364 by the end of the year.
Lastly, to find his total income at the end of the year, we add [tex]5200 + 364 =[/tex] $5,564 at the end of the year.
Someone help me please
in a company 70% of the workers are men is 1380 women work for the company how many workers are there in all?
Round 38.297 to the nearest hundredth
If p is a prime number greater than 3, which of the following is not a factor of 6p
When p is a prime number greater than 3, the. option that is not a factor of 6p is A. p^2 (B
What is the factor about?The process of factorization simply has to do with the division of arithmetic algebraic expressions into the sum of their factors. in this case, multiplying the variables will yield the original expression. It means the writing of a number into smaller or simpler numbers that give the original number.
6*p / 6*p = 1
6*p / 3*p = 2
6*p / 2*p = 3
6*p / 3 = 2*p
6*p / p^2 = 6 / p which is not divisible by any prime number greater than 3.
The correct option is A.
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Complete question
If p is a prime number greater than 3, which of the following is NOT a factor of 6p ?
(A) p^2 (B) 6p (C) 3p (D) 2p (E) 3
Can you work out at what height the ladders cross?
Are you surprised that we don't need to know the distance between the
two walls?
Suppose now the first ladder is 10m above the
base of the second wall, with the second ladder
remaining as before.
Can you work out at what height the ladders
cross now?
5m
Suppose the first ladder meets the wall at a
height of a metres, as in the diagram.
At what height do the ladders cross now?
Suppose the second ladder meets the wall at a
height of b metres, as in the diagram.
At what height do the ladders cross now?
5m
nrich.maths.org/510
© University of Cambridge
10m
a
Three degrees of freedom remain after applying five equations to eight variables. The other five variables may theoretically be solved for by specifying any three of them.
What is the height of ladder ?Five equations link these eight variables together. There are two equations given by the Pythagorean Theorem.
[tex]a^ 2=x^ 2+w^2 \ b^2=y^2+w^{\} 2[/tex]\
Two more are provided by the side ratios of similar triangles.
[tex]x/w=c/v[/tex]
[tex]y/w=c/u[/tex]
The two triangles are connected by the fifth equation, which is a straightforward linear relationship.
[tex]w=u+v[/tex]
Three degrees of freedom remain after applying five equations to eight variables. The other five variables may theoretically be solved for by specifying any three of them. But it might or might not be simple. What comes next will involve two distinct circumstances.
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In this problem, we will look at properties of solids with base on the xy-plane given by the circle x2+y2=a2 and its interior, where a>0
is to be given in parts (i) and (ii) below.
(i). Let a=4, and assume that cross-sections of thesolid perpendicular to the x -axis are circular disks with diameters lying in the xy-plane. Find the area A(x) of the cross-section at location x.
(ii). Let a=9, and assume that cross-sections of the solid perpendicular to the x-axis are equilateral triangles with bases in the xy-plane. Find the area A(x) of the cross-section at location x.
(iii). Find the volume of the solid described in (i) above.
The area A(x) of the cross-section at location x for a=4 is given by A(x)=π(4-x)2, where x is the distance from the origin. This can be derived from the equation A(x)=π(a2-x2).
The area A(x) of the cross-section at location x for a=9 is given by A(x)= 3√3(9-x)2/4, where x is the distance from the origin. This can be derived from the equation A(x)=3√3(a2-x2)/4.
(iii). The volume of the solid described in (i) can be calculated using the equation V=∫A(x)dx, where A(x) is the area of the cross-section at location x, and the integral is taken from 0 to 4. Substituting A(x)=π(4-x)2, we have V=π∫(4-x)2dx, which gives V=π(16 - 8x + x3/3) evaluated from 0 to 4, or V=64π/3.
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the 16-cm diameter disk in (figure 1) can rotate on an axle through its center. what is the net torque about the axle?
The net torque about the axle is zero, as no external forces are acting on the disk. Since there are no external forces acting on the disk, the sum of all the torques must be zero. Therefore, the net torque about the axle is zero.
The net torque about the axle is calculated by summing up all the torques about the axle due to external forces. Since there are no external forces acting on the disk, the sum of all the torques must be zero. Therefore, the net torque about the axle is zero.
The net torque about the axle of the 16-cm diameter disk in is zero, since there are no external forces acting on the disk. To calculate the net torque about the axle, we need to take the sum of all the torques about the axle due to external forces. Since there are no external forces, this sum must be zero. This means that the net torque about the axle is zero. This is because there is no force causing the disk to rotate and therefore, the net torque about the axle must also be zero. This result is consistent with the law of conservation of angular momentum, which states that when no external forces are acting on an object, the total angular momentum of the system must remain constant. Thus, the net torque about the axle of the 16-cm diameter disk in is zero.
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Find each angle measure to the nearest degree
sin x = 0.7547
The each angle measure of the trigonometric ratios are obtained.
Define the term trigonometric ratios?A mathematical discipline called trigonometry examines connections between triangles' sides and angles. Due to the fact that every straight-sided form can be decomposed into a group of triangles, trigonometry may be found across all of geometry.The given values is:
sin x = 0.7547,
Using trigonometric properties,
cos x = √(1 - sin²x)
= √(1 - 0.7547²)
cos x = 0.6560
Cosec x = 1 / sin x
= 1 / 0.7547
Cosec x = 1.3250
Now,
1 + cot²x = cosec²x
cot x = √(cosec²x - 1)
= √(1.3250² - 1)
cot x = 0.8692
tan x = 1 / cot x
= 1 / 0.8692
tan x = 1.1544
Thus, the each angle measure of the trigonometric ratios are obtained.
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Patrick purchased some supplies for his guitar students. He bought 25 guitar picks and 25 guitar strings since they only cost $1 more than the guitar picks. He spent a total of $40. If c is the cost of one guitar pick, find the cost of one guitar string.(1 point)
The cost of the guitar string is given by the equation b = $ 1.30
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
Let the cost of one guitar pick be = c
Let the cost of one guitar string be = b
And , Patrick bought 25 guitar picks and 25 guitar strings for = $ 40
Substituting the values in the equation , we get
25c + 25b = 40 be equation (1)
Now , cost of one guitar string = 1 + c
b = 1 + c be equation (2)
Substituting the value of b in equation (1) , we get
25c + 25 ( 1 + c ) = 40
On simplifying the equation , we get
25c + 25 + 25c = 40
Subtracting 25 on both sides of the equation , we get
50c = 15
Divide by 50 on both sides of the equation , we get
c = $ 0.30
So , the cost of one guitar string = $ 1 + $ 0.30 = $ 1.30
Hence , the cost of one guitar string is $ 1.30
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Round your answer to the nearest thousandth.
The correlation coefficient for the data in the table is given as follows:
r = 0.724.
How to obtain the correlation coefficient for the data-set?The coefficient is obtained inserting the points in a data-set in a correlation coefficient calculator.
The input and output variables are given as follows:
Input: initial approval rating.Output: final approval rating.Hence the points that compose the data-set are given as follows:
(48, 34), (70, 46), (81, 38), (81,49), (86,49).
Inserting these points into a calculator, the correlation coefficient for the data is given as follows:
r = 0.724.
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What is the greatest common factor of {42,10}
Answer:
5
Step-by-step explanation:
(-1,-4) that is perpendicular to the line Y=-2/3x+1
the equation of line perpendicular to y=-2/3x+1 will be y = 3/2x -5/2.
What is equation straight line?
Y = mx + c is the general equation for a straight line, where m denotes the line's slope and c the y-intercept. It is the version of the equation for a straight line that is used most frequently in geometry. There are numerous ways to express the equation of a straight line, including point-slope form, slope-intercept form, general form, standard form, etc. A straight line is a geometric object with two dimensions and infinite lengths at both ends. The formulas for the equation of a straight line that are most frequently employed are y = mx + c and axe + by = c. Other versions include point-slope, slope-intercept, standard, general, and others.
The line given y=-2/3x+1
here the slope is -2/3
So the slope of the required line will be m2 = - 1/m1
m2 = 3/2
The required line equation will be [tex]\frac{y+4}{x+1}[/tex] = 3/2
2(y+4) = 3(x+1)
2y+8= 3x+3
y = 3/2x -5/2
Hence the equation of line perpendicular to y=-2/3x+1 will be y = 3/2x -5/2.
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Find the x-intercept of the function `y=3x+6`
Find the y-intercept of the function `y=3x+6`
Question 1 : find x intercept
y=0
0=3x+6
subtract 6 from both sides, after doing that, the equation would be : -6=3x
Then divide by -3 on both sides, the answer would be : x=3
Therefore the x intercept as a coordinate is (3,0)
Question 2 : find the y intercept
x=0
y=3(0)+6
y=6
Therefore the y intercept as a coordinate would be (0,6)
Hope this helped =)
-YinAhara
Draw the graph of 5-3x-2x^2=0
This is a answer of graph format. (1 - x) • (2x + 5) = 0
what is graph?
In mathematics, a graph is a visual representation or diagram that shows facts or values in an ordered way.The relationships between two or more items are frequently represented by the points on a graph.A network with paths connecting any two of its vertices. Any two vertices or nodes in a disconnected graph are those that are separated by a path.Graph that completes a cycle is called a cycle graph. Complete Graph is a graph is said to be complete when every pair of vertices is connected by an edge.A graph is a type of non-linear data structure composed of nodes, also known as vertices and edges. Any two nodes, often referred to as vertices, in a network are connected by edges.Pull out like factors : [tex]-2x^{2} - 3x + 5 = -1 • (2x^{2} + 3x - 5)[/tex]
Factoring [tex]2x^{2}[/tex] + 3x - 5
The first term is, [tex]2x^{2}[/tex] its coefficient is 2 .
The middle term is, +3x its coefficient is 3 .
The last term, "the constant", is -5
Multiply the coefficient of the first term by the constant 2 • -5 = -10
Find two factors of -10 whose sum equals the coefficient of the middle term, which is 3 .
-10 + 1 = -9
-5 + 2 = -3
-2 + 5 = 3 That's it
Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -2 and 5
[tex]2x^{2} - 2x + 5x - 5[/tex]
Add up the first 2 terms, pulling out like factors :
2x • (x-1)
Add up the last 2 terms, pulling out common factors :
5 • (x-1)
Add up the four terms of step 4 :
(2x+5) • (x-1)
Which is the desired factorization.
Equation :
(1 - x) • (2x + 5) = 0
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X = y+9, y = 4x+10 solve for X and y
The value of x and y for the given equations are - (19/3) and -46/3 respectively.
What are equations?A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("="). For instance, 2x - 5 = 13.
Given the two equations are as follows:
x = y + 9 and y = 4x + 10
Substitute the value of y in the first equation:
x = y + 9
x = 4x + 10 + 9
Isolate the value of x:
x - 4x = 19
-3x = 19
x = - (19/3)
Substitute the value of x in the first equation:
x = y + 9
- 19/3 = y + 9
Isolate the variable y:
y = -19/3 - 9
y = - 46/3
Hence, the value of x and y for the given equations are - (19/3) and -46/3 respectively.
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In contrast to Supervised learning, Unsupervised learning has more models and more evaluation meth used in order to ensure the outcome of the model is accurate. This statement is overall a false statement.
An unsupervised learning model might give a less accurate outcome if compared to supervised learning.
1. Algorithms of unsupervised learning are taught using the data that are unlabeled.
2. The model of unsupervised learning doesn't take any feedback.
3. In unsupervised learning, we see that only input data is supplied to the model, no output data.
Assess reasonableness a monument will be at the circumcenter of a triangular plot of land at the state capital. on the coordinate grid, the vertices of the triangular plot of land are a(1, 5) , b(7, 5) , and c(8, 0) .
Overall, the statement that Unsupervised learning has more models and more evaluation methods used in order to ensure the outcome of the model is accurate is false. Unsupervised learning algorithms are taught using data that may be labeled, but in most cases the labels are not used to infer the model. The model of unsupervised learning does not take any feedback, as it is not supervised, and only input data is supplied to the model, no output data. The evaluation of unsupervised learning models relies on different metrics such as clustering accuracy, silhouette coefficient, etc. Unsupervised learning algorithms are used for tasks such as clustering, anomaly detection and density estimation.
assess reasonableness a monument will be at the circumcenter of a triangular plot of land at the state capital. on the coordinate grid, the vertices of the triangular plot of land are a(1, 5) , b(7, 5) , and c(8, 0) .
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