consider the system of equations 2x −5y =1 4x −10y =3 how many solutions does this system have? explain.

Answer:

deductive reasoning

Step-by-step explanation:

The lengths of the diagonals of rectangle is 66 units

The properties of a rectangle is given as;

From the information given, we have that;

$WY=14x+10

$XZ=11x+22

Since the diagonals are of the same length, we have;

WY = XZ

14x + 10 = 11x + 22

collect like terms

14x - 11x = 22 - 10

subtract the like terms

3x = 12

Make 'x' the subject of the formula

x = 4

WY = 54 + 10 = 66

XZ = 66

Hence, the values are 66

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The probability that a subject, with a true cholesterol level of 196, takes the test twice and both tests classify the subject incorrectly depends on several factors.

The factors that matter, such as the accuracy of the test, the threshold for normal cholesterol levels, the population being tested, and the distribution of cholesterol levels in that population. Without more information, it is not difficult to determine the exact probability.

However, it is generally unlikely for a test to classify a subject's cholesterol level incorrectly twice in a row, especially if the test has a high level of accuracy. Cholesterol levels in a population tend to follow a normal distribution, making it unlikely for a single subject to have significantly different readings on two separate tests.

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The value of m∠ABC in the triangle ABC is 55°

An equation is an expression showing the relationship between two or more numbers and variables. An equation can either be linear, quadratic, cubic and so on depending on the degree.

From the diagram:

m∠ABC + m∠ABD = 180° (sum of angles on a straight line)

Substituting gives:

(6x + 5) + m∠ABC = 180

m∠ABC = 175 - 6x

m∠ABC + m∠BAC + m∠ACB = 180 (angle in a triangle)

Substituting:

(3x) + (x + 45) + (175 - 6x) = 180

2x = 40

x = 20

m∠ABC = 175 - 6x = 175 - 6(20) = 55°

m∠ABC is equal to 55°

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Answer:

2

Step-by-step explanation:

Let ( x1, y1) represent the first point (0,0)  and  (x2, y2) represent the second point (2,4). Slope (m) (y2 - y1) / ( x2 - x1)= (4 - 0) / (2-0) =2

Their total determines which angles are equivalent. Alternative Corresponding Angles Parallel Corresponding angles

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750=y²-5y is the equation that can be used to solve for y, the length of the room.

The area of Rectangle is length times of width.

Given that the area of a rectangular room is 750 square feet.

The width of the room is 5 feet less than the length of the room.

Let y be the length of the room

Width=y-5.

Area=Length×Width

750=y(y-5)

750=y²-5y

Hence, 750=y²-5y is the equation that can be used to solve for y, the length of the room.

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The requried price of an eraser and ruler is given as 0.15 Tetris and 0.25 Tetris respectively.

Simultaneous linear equations are two- or three-variable linear equations that can be solved together to arrive at a common solution.

1 GEL and 25 tetris = 1.25
1 GEL and 5 tetris = 1.05

We can set up two equations to find the unknown values of x and y:

1.25 = 5x + 2y  - - - - - -(1)

1.05 = 2x + 3y  - - - - - (2)

To solve the system of equations, we can substitute the value of y in the first equation using the second equation:

y = (1.25 - 5x) / 2

Substituting this value of y in the second equation:

1.05 = 2x + 3((1.25 - 5x) / 2)

x = 0.15

Now,
Put x = 0.15 equation 1,
e get
y = 0.25

Thus, the requried price of an eraser and ruler is given as 0.15 Tetris and 0.25 Tetris respectively.

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Aiden brings money about $40 on the trip.

Let's call the amount of money Aiden brings on the trip "x".

       0.45x = 18

        x = 18 / 0.45

        x = 40

The subject of mathematics that the problem is related to is algebra. Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and study relationships between variables. In this problem, we use algebraic equations and operations to find the value of the unknown variable x, which represents the amount of money Aiden brings on the trip.

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Answer:

3a - 7.5

Step-by-step explanation:

4.5a−1.5(a+5)

4.5a - 1.5a - 7.5

3a - 7.5

So, the answer is 3a-7.5

the values of x and y will be 60 and 25 respectively.

Parallel lines are lines in a plane that are always the same distance apart. Parallel lines never intersect. Perpendicular lines are lines that intersect at a right (90 degrees) angle.

Given, ∠2y + 25  and ∠3y are alternate interior angles

Alternate interior angles are the angles formed when a transversal intersects two coplanar lines.

Thus,

2y + 25  = 3y

y = 25

And since, When two lines intersect each other, then the opposite angles, formed due to the intersection, are called vertical angles or vertically opposite angles. Also, a pair of vertical angles are always congruent.

thus,

∠x + 15 = ∠3y

x + 15 = 3y

x + 15 = 75

x = 60

therefore, the values of x and y will be 60 and 25 respectively.

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The solution to this system is (x,y)=(10/3, 20/3 + 4).

An equation is a mathematical statement that describes the relationship between two or more variables. It is a statement that states the equality of two expressions. Equations can be used to describe anything from the motion of a pendulum to the growth of a population. They are used in all areas of mathematics, physics, and other sciences. Equations are written using symbols, such as = (equal to), > (greater than), and < (less than).

The solution to this system of equations is the set of values that make both equations true at the same time. To solve this system, we can use the substitution method. First, we solve for one of the variables in one equation, then substitute that value into the other equation.

For this system, we will solve for x in the second equation. We can do this by subtracting 3x from both sides, which gives us 10 = 3x. We then divide both sides by 3, which gives us x = 10/3.

Next, we substitute 10/3 into the first equation and solve for y. We can do this by replacing x with 10/3, which gives us y = (10/3)² -2(10/3) + 4. Simplifying this equation gives us y = 20/3 + 4.

Therefore, the solution to this system is (x,y)=(10/3, 20/3 + 4).

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Bill has 6 hats in total.

According to the question

In order to solve the issue, an inequity that depicts Bill and Jim's condition and their hat collections must be eliminated. Jim has 45 hats, Bill has a certain amount. According to the issue, even if Bill purchases 8 additional hats and increases his total by 3, he would still possess fewer hats than Jim.

Let's use the variable "x" to represent Bill's total number of hats.

We are aware that when Bill purchases 8 additional hats, his total will be x + 8.

The result of multiplying this new sum by three is 3(x + 8).

This number is less than 45, thus we can create an inequality since we know that:

3(x + 8) < 45

Extending the sentence on the left:

3x + 24 < 45

24 is subtracted from both sides:

3x < 21

by 3 and dividing both sides:

x < 7

The least positive integer solution to this inequality is 7, as x, the number of hats Bill owns, must be a positive integer. Bill thus needs six hats.

The difference between Jim and Bill is that Jim has 45 hats, but Bill owns less than 7, even after purchasing 8 additional hats and multiplying the total by 3.

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Bill has 6 hats in total.

According to the question

In order to solve the issue, an inequity that depicts Bill and Jim's condition and their hat collections must be eliminated. Jim has 45 hats, Bill has a certain amount. According to the issue, even if Bill purchases 8 additional hats and increases his total by 3, he would still possess fewer hats than Jim.

Let's use the variable "x" to represent Bill's total number of hats.

We are aware that when Bill purchases 8 additional hats, his total will be x + 8.

The result of multiplying this new sum by three is 3(x + 8).

This number is less than 45, thus we can create an inequality since we know that:

3(x + 8) < 45

Extending the sentence on the left:

3x + 24 < 45

24 is subtracted from both sides:

3x < 21

by 3 and dividing both sides:

x < 7

The least positive integer solution to this inequality is 7, as x, the number of hats Bill owns, must be a positive integer. Bill thus needs six hats.

The difference between Jim and Bill is that Jim has 45 hats, but Bill owns less than 7, even after purchasing 8 additional hats and multiplying the total by 3.

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A percent error calculation is used to determine how close a measured value is to a true value.

         Percentage Error = ((Estimated Number – Actual Number)/ Actual

         number) x 100.

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The probability that a plain pencil from the first box and a color pencil from the second box are selected is 3/8.

The Probability in mathematics is the possibility of an event in time. In simple words, how many times that incident is happening in any given time interval.

Given:

A box contains 5 plain pencils and 5 pens.

A second box contains 3 color pencils and 1 crayon.

One item from each box is chosen at random.

The probability that a plain pencil from the first box and a color pencil from the second box are selected,

= 5/10 x 3/4

= 3/8

Therefore, the probability is 3/8.

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When y = 92, the value of x is approximately 4.07 hours. This means that if a student spends approximately 4.07 hours on their homework each week, we would expect their quiz score to be 92.

What is the linear equation?

Linear equations are the equations of degree 1. It is the equation for the straight line. The standard form of linear equation is ax+by+c =0, where a ≠ 0 and b ≠ 0.

If y = 92, we can solve for x by substituting 92 for y in the equation y = 61 + 7.6x:

92 = 61 + 7.6x

7.6x = 92 - 61

7.6x = 31

x = 31 / 7.6

x = 4.07

Therefore, when y = 92, the value of x is approximately 4.07 hours. This means that if a student spends approximately 4.07 hours on their homework each week, we would expect their quiz score to be 92.

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The volume of the solid is approximately 54.8 cubic cm.

Integration is a fundamental concept in calculus that deals with finding the area under a curve and calculating the accumulation of values over an interval. Integration can be thought of as the reverse of differentiation, which is the process of finding the rate of change of a function at a given point.

There are two main types of integration: definite and indefinite integration. Definite integration involves finding the exact area between a curve and the x-axis over a specified interval. Indefinite integration involves finding the general antiderivative of a function, which is a function that, when differentiated, will yield the original function.

The process of integration can be represented symbolically as ∫ (the integral symbol) and the result of an integration is typically represented as an antiderivative. The fundamental theorem of calculus states that differentiation and integration are inverse operations, so the derivative of an antiderivative is the original function.

The paraboloid z = 8 - 2x² - 2y² can be written in polar coordinates as z = 8 - 2r². The volume of the solid can be found by using the double integral of the height of the paraboloid over the region defined by the polar coordinates.

In polar coordinates, the region is defined by 0 <= r <= √4 and 0 <= θ <= [tex]\frac{\pi }{2}[/tex]The volume of the solid is given by:

V=[tex]\int\limits \int\limits {(8-2r^{2} )} r\, dr dy[/tex]

= [tex]\int\limits^0_\frac{\pi }{2} {} \, dx\int\limits^0_\sqrt{4} \,}(8-2r^{2})r }dr[/tex]

= [tex]\int\limits^0_\frac{\pi }{2} dx[ {4r^{2} -\frac{2}{3}r^{3} }] \,[/tex] from 0 to sqrt(4)

= [tex]\frac{\pi }{2} [4(4)-(\frac{2}{3}4^{\frac{3}{2} } ][/tex]

= [tex]\frac{\pi }{2}[16-\frac{2}{3}(16^{\frac{1}{2} } )][/tex]

= [tex]\frac{\pi }{2}[16-8({\frac{2^{} \frac{1}{2} }{3} }) ][/tex] cubic cm

So the volume of the solid is approximately 54.8 cubic cm.

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The coordinate of point C' after transformation is; C'(4, 4)

Dilation is defined as a transformation, that is used to resize the object. Dilation is used to make the objects either larger or smaller. That is a reduction or an enlargement.

Usually when we dilate an object by a scale factor, what we simply do is to multiply the original coordinates by the scale factor to get the new dilated coordinates.

Since the quadrilateral is dilated by a scale factor of 2, we will multiply the original coordinate of point C by 2.

Coordinate of C = (2, 2)

Thus;

Dilated coordinate; C' = (2*2, 2*2) = 4,4

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For c = 1, we have f(t,y) =-y = 1. Rearranging this equation yields y = 1, which simplifies to y = et. The isoclines for c = -1, 0, 1 are y = e-t, y = 0, and y = et, respectively.

c = -1: y = e-t

c = 0: y = 0

c = 1: y = et

For c = -1, we have f(t,y) = -y = -1. Rearranging this equation yields y = -1, which simplifies to y = e-t.

For c = 0, we have f(t,y) = -y = 0. Rearranging this equation yields y = 0.

For c = 1, we have f(t,y) =-y = 1. Rearranging this equation yields y = 1, which simplifies to y = et.

The isoclines for c = -1, 0, 1 are y = e-t, y = 0, and y = et, respectively.

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f(x) =3x-8 and g(x) = √x+2 are the function and their inverses intersect.

A function is a mathematical representation that maps one set of values to another set of values. The inverse of a function is a reflection of the original function over the line y=x, where the input and output are reversed.

For the function f(x) = 3x - 8, the inverse can be found by switching the input and output values, then solving for x.

In this case, we have

=> y = 3x - 8 and x = (y + 8)/3.

The two functions will intersect when the input value is equal to the output value, or when

=> 3x - 8 = (y + 8)/3.

Solving for x, we find that x = 4. This is the only intersection point between the two functions.

For the function g(x) = √x + 2, the inverse can be found by switching the input and output values, then solving for x.

In this case, we have

=> y = √x + 2 and x = y² - 2.

The two functions will intersect when the input value is equal to the output value, or when

=> √x + 2 = y² - 2.

Solving for x, we find that x = 4. This is the only intersection point between the two functions.

Therefore option (a) and (b) is correct.

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Answer:

$212

Step-by-step explanation:

The statement is True: the null space of a is the solution set of the equation ax = 0.

The collection of all solutions to the homogeneous equation makes up a matrix's null space. Any matrix's null space contains the zero vector at all times. The null space is denoted as Null A.

We know that The null space of an m × n matrix A, written as Nul A, is the set of all solutions to the homogeneous equation Ax = 0.

Hence, the statement is True: the null space of a is the solution set of the equation ax = 0.

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Sixth grade students prefer action movie is 0.194 & Eight grade students prefer action movie is 0.333

Given that the information below:

Sixth grade students prefer animated movie - 45

Eight grade students prefer animated movie - 40

Sixth grade students prefer action movie - 35

Eight grade students prefer action movie - 60

Total sixth grade students - 80

Total eight grade students - 100

Total students prefer animated movie - 85

Total students prefer action movie - 95

Proportion of sixth grade students prefer action movie =

Sixth grade prefer action movie/Total students

= 35/180 = 0.194

Proportion of eight grade students prefer action movie =

Eight grade prefer action movie/Total students

= 60/180 = 0.333

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The answers for all triangle problems are:

10. ΔABC is a equilateral triangle, then:

∠A = ∠B = ∠C = 8x - 44

3 (8x - 44) = 180

24x - 132 = 180

24x = 48

x = 2

11. ΔRST is an isosceles triangle, then:

∠R = ∠T

15x - 31 = 9x + 11

6x = 42

x = 7

∠S + ∠R + ∠T = 180°

∠S + 2∠T = 180°

∠S + 2( 9x + 11) = 180°

∠S + 2(9(7) + 11) = 180°

∠S + 2 (63 + 11) = 180°

∠S + 2 (74) = 180°

∠S = 180 - 144

∠S = 36°

12. Based on the given data:

∠P = 5x - 14

∠Q = x - 5

∠R = 2x - 9

Since ΔPQR is a triangle, then:

∠P + ∠Q + ∠R = 180°

(5x - 14) + (x - 5) + (2x - 9) = 180°

8x - 28 = 180°

8x = 208

x = 26

∠P = 5x - 14 --> 5(26) - 14 = 116°

∠Q = x - 5 --> (26) - 5 = 21°

∠R = 2x - 9 --> 2(26) - 9 = 43°

13. ΔJKL is a triangle, where:

∠J = ∠L - 7

∠K = 2∠L - 21

∠J + ∠K + ∠L = 180

(∠L - 7) + (2∠L - 21) + ∠L = 180°

4∠L - 28 = 180°

4∠L = 208

∠L = 51°

∠J = ∠L - 7 --> (51) - 7 = 44°

∠K = 2∠L - 21 --> 2(51) - 21 = 81°

14. In a triangle ΔBCD:

BD = BD

∠B = 13x - 35

∠C = 5x - 19

∠D = 2x + 14

Since BC = BD; then:

∠C = ∠D

5x - 19 = 2x + 14

3x = 33

x = 11

∠C = ∠D = 2x + 14 --> 2(11) + 14 = 36°

∠B + ∠C + ∠D = 180°

∠B + 36° + 36° = 180°

∠B = 108°

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Answer:

9^(x)=(1)/(27)

Step-by-step explanation:

The statement using the specified property i.e  Associative Property of Multiplication will be 14y.

One's identity property is its capacity for multiplication.

7(2y)

The phrase is given as follows:

7(2y)

=14y

The one has the multiplication property, which says:

Any number or expression that is multiplied by one doesn't alter in any way.

The result is: 1xa=a

The identification property is another name for this particular attribute.

The general rule is shown as follows:

Examples include

1 * 2 = 2

1 * x = x

1 * ab = ab

and so forth

So, the property of multiplying by one is an identity property.

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The number of points scored by each girl is 4 and each boy is 1 point.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

here, we have,

9 girls and 5 boys scored a total of 51 points.

The difference between the number of points scored by the 9 girls and the number of points scored by the 5 boys is 21 .

Each girl scored the same number of points and each boy scored the same number of points.

Let x be the number of points the 9 girls scored.

=> 51 - x is the number of points the 5 boys scored.

We have x + 21 = 51 - x

           or x - 21 = 51 - x.

=> 2x = 30 or 2x = 72

=> x = 15 or x = 36

We reject x = 15

as 15 is not a multiple of 9

So x = 36,

each girl scored 36/9 = 4 points.

and,

=> 41 - (36) = 5,

each boy scored 5/5 = 1 points.

Hence, the number of points scored by each girl is 4

and each boy is 1 point.

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The approximate value of the correlation coefficient for the given graph is D. -1.

The correlation coefficient is a statistical measure that indicates the strength and direction of the linear relationship between two variables. A correlation coefficient of -1 indicates a perfect negative linear relationship, meaning that as the value of one variable increases, the value of the other variable decreases at a constant rate. In this case, as the number of years increases, the price of the car decreases, so the correlation coefficient is negative.