For a real value of z, the general solution of above system of equations is x = -1/3 + (1/3)z, y = 5/3 + (1/3)z and z = z
In this question, we have been given a system of equations.
x+2y-z=3 ...........(1)
-2x-y+z=-1 ............(2)
6x-3y-z=-7 ...........(3)
We need to solve above system of equations using the elimination method.
We solve above system of equations by Gaussian elimination method.
First we write above system f equations in as an augmented matrix.
[tex]\begin{bmatrix}1 & 2 & -1 &|& 3 \\-2 & -1 & 1 &|& -1 \\6 & -3 & -1 &|& -7\end{bmatrix}[/tex]
We convert the augmented matrix into the row echelon form as:
[tex]\begin{bmatrix}1 & 2 & -1 & 3 \\0 & 3 & -1 & 5 \\0 & 0 & 0 & 0\end{bmatrix}[/tex]
So, we get an system of equations as:
x + 2y -z = 3 ............(4)
3y - z = 5 ...........(5)
From equation (5),
y = 5/3 + (1/3)z
Substitute above value of y in equation (4),
x = 3 - 2[5/3 + (1/3)z] + z
x = -(1/3) + (1/3)z
So, the general solution is,
[tex]\left(\begin{matrix}\frac{-1}{3}+\frac{1}{3}*z \\\\\frac{5}{3}+\frac{1}{3}*z \\\\z\end{matrix}\right)[/tex]
Therefore, for a real value of z, the general solution of above system of equations is x = -1/3 + (1/3)z, y = 5/3 + (1/3)z and z = z
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An atom has a diameter of 2.00 Å and the nucleus of that atom has a diameter of 9.50×10−5 Å . Determine the fraction of the volume of the atom that is taken up by the nucleus. Assume the atom and the nucleus are a sphere.
fraction of the volume of the atom taken by nucleus = 1.07*10^-13
Space occupied by the sphere is the volume of sphere.
A straight line passing from side to side through the center of the body especially a circle or sphere is known as diameter.
Volume of sphere = 4/3πr³
Diameter of an Atom = 2.00Â
Diameter of an nucleus = 9.50×10⁻⁵Â
Radius of an Atom = 1.00Â
[tex]\frac{4}{3}\pi r^3=\frac{4}{3}\pi (4.75*10^(-5))^3\\[/tex]
Volume of an Atom = [tex]\frac{4}{3}\pi r^3= \frac{4}{3}\pi 1^3=\frac{4}{3}\pi[/tex]
Volume of nucleus = [tex]\frac{4}{3} \pi r^3=\frac{4}{3} \pi (4.75*10^-5)^3\\=\frac{4}{3} \pi 107.1718*10^-15[/tex]
fraction of the volume of the atom taken by nucleus =
4/3π 107.17*10^-15/ 4/3π =107.17*10^-15 = 1.07 *10^-13
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order from greatest to least .7 .07 .072 7.2 7 .27
Order from greatest to least
0.7
0.07
0.072
7.2
7
0.27
Answer:
(largest) 7.2 , 7 , 0.7 , 0.27 , 0.072 , 0.07 (smallest)
A square field has an area of 22,500 square feet. To the nearest foot, what is the diagonal across the field?
The length of the diagonal across the field is 212.132 feet.
What is a square?A rectangle whose all sides are equal is called a square.
A square is always a rectangle, parallelogram, and a quadrilateral but its reverse statement may or may not be true.(ie it is not always necessary that a rectangle is a square or a parallelogram is a square).
Given that a square field has an area of 22,500 square feet. Now, the length of the side of the square field can be written as,
Area of the square field = (Length of square field)²
22,500 square feet = (Length of square field)²
Length of square field = √(22,500 sq. feet)
Length of square field = 150 feet
Now, the length of the diagonal of the field can be written as,
(Length of the diagonal)² = 2(Length of the side of square field)²
(Length of the diagonal)² = 2(150²)
(Length of the diagonal)² = 45,000 sq. feet
(Length of the diagonal) = 212.132 feet
Hence, the length of the diagonal across the field is 212.132 feet.
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A clothing pattern calls for 4.81 yd2 of fabric. how many square inches of fabric does the pattern use? (note: 1 yd = 3 ft., 1 ft = 12 in.)
Pattern uses 6233.76 inches² fabric.
Firstly we will convert units in required form. Therefore, converting equivalent of yard² in inches². As per the given information, 1 yard = 3 feet and 1 foot = 12 inches.
As 1 foot = 12 inches, so, 3 feet = 36 inches, which is further equal to 1 yard.
Now, 1 yard² = (36 inches)²
1 yard² = 1296 inches²
Coming back to the problem, we will now calculate inches² from fabric measurement in yard². So, relating the units for conversion.
1 yard² = 1296 inches²
4.81 yard² = 1296 × 4.81
Performing multiplication
Measurement of 4.81 yard² in inches² = 6233.76
Hence, the pattern uses 6233.76 inches² fabric.
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Quadrilateral ABCD has the following angle measures:
m∠A = 105°, m∠B = ?°, m∠C = 34°, m∠D = 90°
What is m∠B?
pleeeeease help!!
Answer:
131°
Step-by-step explanation:
Sum of all angles of a quadrilateral = 360°
We can find the unknown angle ∠B, by subtracting the sum of the angles ∠A, ∠C, ∠D from 360°.
∠A + ∠B + ∠C + ∠D = 360
105 + ∠B + 34 + 90 = 360
∠B + 229 = 360
∠B = 360 - 229
∠B = 131°
Answer:
The measure of m<GHE is 49°, the correct option is B.
What is a Quadrilateral?
A quadrilateral is a polygon with four sides.
It is of various types,
Trapezoid, which have one pair of parallel sides,
Parallelogram, it has two pairs of parallel sides,
Rhombus, it has all sides equal,
Rectangle, opposite sides are parallel and equal, and,
Square, all sides are equal and opposite sides are parallel, and all angles have measure of 90 degree.
The quadrilateral ABCD is ≅ quadrilateral EFGH
When two figures are congruent then by CPCTC, corresponding parts of congruent figures are congruent.
The measure of the angle
B = m<BCD = 90°
m<BAD = 131°
As the quadrilaterals are congruent,
By CPCTC
∠F = ∠FGH = 90°
m ∠FEH = 131°
The measure of m<GHE has to be determined.
The sum of the measures of all angle so of a quadrilateral is 360 degree.
The ∠F + ∠FGH + ∠FEH + ∠GHE = 360
90 +90+ 131 +∠GHE = 360
∠GHE = 49°
Step-by-step explanation:
which expression shows the prime factorization of 40? 2 × 2 × 5 2 × 2 × 5 2 × 2 × 2 × 5 2 × 2 × 2 × 5 2 × 5 × 5 2 × 5 × 5 2 × 5 × 5 × 5
The expression 2×2×2×5 shows the prime factorization of 40.
Prime factorization:
The number 40 is equal to
2 x 2 x 2 x 5, or = 40
23 x 5. = 40
What does prime factorization serve as?In mental arithmetic, fractions, finding square roots, and computing the HCF and LCM of two smaller numbers, prime factorization is a highly helpful tool when working with whole numbers. A whole number higher than 1 whose only elements are 1 and itself is referred to as a prime number. A whole number that may be split evenly into another number is referred to as a factor.
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Answer:
2 × 2 × 2 × 5
Step-by-step explanation:
2 × 2 × 2 × 5 = 4 × 2 × 5 = 8 × 5 = 40
Miguel made up an equation for his friend to solve. He told his friend that three times the square of a number is 265", What is the number to the nearest tenth
The number after rounding off to the nearest tenth will be 9.4
In this question, we have been given that Miguel made up an equation in which three times of square of a particular number equals 265. We have to set up the equation according to this.
Taking the number as 'x'. We get the equation according to the condition :
=> 3x² = 265
By solving this equation we get :
=> 3x² = 265 => x² = 88.3
=> x = 9.39
We get x = 9.39. By rounding off x to the nearest tens, we get x = 9.4
Hence, the number to the nearest tenth will be 9.4
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Answer the question in photo [ Brainliest + easy points ]
x+12+4x-16= 6x-13
5x-4=6x-13
-4=x-13
x=9
9+12+4(9)-16
21+36-16
57-16= 41
JL=41 units
Largest possible value of a?
–9 – a = b
a2 – 6a – 9 = b
If the ordered pair (a, b) satisfies the system of equations above, what is the largest possible value of a?
Solving a quadratic equation, it is found that the largest possible value of a is of a = 5.
How to solve a quadratic equation?A quadratic equation is modeled by the following rule:
y = ax^2 + bx + c
The solutions of the equations are the values of x for which y = 0.
For this problem, the equation is:
a² - 6a - 9 = b.
b = -9 - a, hence we replace in the equation as follows;
a² - 6a - 9 = -9 - a
a² - 5a = 0.
Traditionally, the solutions are given using Bhaskara's formula, however, since the coefficient c is zero, we can solve the equation by elimination as follows:
a² - 5a = 0.
a(a - 5) = 0.
Then the solutions are:
a = 0.a - 5 = 0 -> a = 5.5 > 0, hence the largest possible value of a is of a = 5.
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(2) six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. the six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. an eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. what is the radius of this eighth sphere? (2013 amc 10a
The radius of 8th sphere is 3/2.
The 8th sphere's center and the hexagon's two opposing ends should form an isosceles triangle. Then put up another triangles between the point of tangency of the 7th and 8th spheres, and the point of tangency between the 7th sphere and 2 of the original spheres on different sides of the hexagon. Each triangle's side length should be expressed in terms of r (the sphere's 8 radius) and h. (the height of the first triangle). The values of h and r may then be determined by setting up two equations for the two triangles using the Pythagorean Theorem.
(1+r)² = [tex]2^{2}[/tex] + [tex]h^{2}[/tex]
(3√2)² =3² + (h+r)²
r= 3/2
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1. Is x = 2, y = 6, z = 1 a solution to the system of equations? Explain.
1.5x + y-z = 8
x + 3y = 18
2y + z = 13
2. If y = 6, find the values of and z.
x
4x+y=z=15
x + 2y + z = 38
The solution to the system of equation is x = 6, y = 4, z = 5
The equation is
1.5x + y - z = 8
x + 3y = 18
2y + z = 13
From this we have,
y = 8 - 1.5x + z
x + 3y = 18
2y + z = 13
x + 3(8 - 1.5x + z) = 18
2(8 - 1.5x + z) + z = 13
Apply the Distributive Property
x + 24 - 4.5x + 3z = 18
16 - 3x + 2z + z = 13
Combine like terms
-3.5x + 24 +3z = 18
16 - 3x + 3z = 13
Rearrange variables to the left side of the equation
-3.5x + 3z = 18 - 24
-3x + 3z = 13 - 16
Subtract the two equation
-3.5x + 3z - (-3x + 3z ) = 18-24 - ( 13-16)
-3.5x + 3z + 3x -3z = 18 - 24 - 13 + 16
-3.5x + 3x = 18 - 24 - 13 + 16
x = 6
Substitute the value of x in one of the equation
-3×6 + 3z = 13 - 16
-18 + 3z = -3
3z = 15
z = 5
Now put the value of x and z
y = 8 - 1.5x + z
y = 8 - 1.5×6 + 5
= 8 -9 + 5
= 4
Therefore the solution to the system of equation is x= 6, y = 4 and z = 5
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a line passes through the points (9, 16) and (0, 5). what is it’s equation in slope intercept form
Answer:
y = x + 5
Step-by-step explanation:
Slope: m = (y2-y1)/(x2-x1)
m = (16 - 5)/(9 - 0) = 9/9 = 1
Using m = 1, y = 5, x = 0
then y = mx + b
5 = 1(0) + b
b = 5
so y = x + 5
Find the measure of LBOD
m LBOD=
Classify LBOD
The measure of angle BOD is 65°
Calculating measure of anglesFrom the question, we are to determine the measure of angle BOD (m ∠ BOD)
The measure of angle BOD can be determine from the given protractor.
To determine the measure of angle BOD, we will read from line BO, in the counterclockwise direction, to line DO.
Reading in the counterclockwise direction, line BO is on 0°, and line DO is on midpoint of 60° and 70°.
The midpoint of 60° and 70° is 65°
Hence, the measure of angle BOD is 65°.
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If m ∠ E B C = 3 r + 10 ° and m ∠ A B E = 2 r − 20 ° , find m ∠ E B F .
The measure of ∠EBF is 56°.
Here, we are given that-
m ∠EBC = 3 r + 10 °
and m ∠ABE = 2 r − 20 °
Also we can observe that ∠EBC and ∠ABE are angles on a straight line and hence will be supplementary.
⇒ ∠EBC + ∠ABE = 180
⇒ 3r + 10 + 2r - 20 = 180
5r - 10 = 180
5r = 180 + 10
5r = 190
r = 190/5
r = 38
Thus, ∠ABE = 2(38) - 20
= 76 - 20
= 56
Now, from the figure, m ∠EBF = m ∠ABE
⇒ m ∠EBF = 56°
Thus, the measure of ∠EBF is 56°.
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Your question was incomplete. Check for the missing figure below.
For a party, Carlos buys 8 bags of popcorns and 4 bags of pretzels on sale for $10.36. During the party, he goes back to the store after the sale is over to buy 4 more bags of popcorns and 2 bags of pretzels for $20.72. Let x represent the price of each bag of popcorn and let y represent the price of each bag of pretzels. Which system of linear equations can you solve to find the price of each bag ofpopcorn and each bag of pretzels?
Answer:
addingadddddddddddddddddd
Four cylindrical barrels each have a capacity of 100 L but have different diameters and heights. Determine the height of the barrel for each diameter. a) 40.0 cm b) 45.0 cm c) 50.0 cm d) 55.0 cm
you are selling flowers to raise money for a field trip. For each flower you sell you earn $5 toward the trip the total cost of the field trip including the bus ride is $75 you must sell enough flowers to cover the cost of the $15 bus ride and you plan to stop selling flowers if you earn enough to cover the total cost of the trip. Describe the possible numbers of flowers you sell.
The possible number of flowers you sell is 3 to 25
How to describe the possible numbers of flowers you sell.?The given parameters are:
Selling price = $5
Total cost of trip = $75
Bus ride = $15
If you earn enough to cover the total cost of the trip, the maximum numbers (Max) of flowers you sell is
Max = Total cost of trip/Selling price
This gives
Max = 75/5
Max = 25
If you earn enough to cover the bus ride, the minimum numbers (Min) of flowers you sell is
Min = Bus ride/Selling price
This gives
Min = 15/5
Min = 3
Hence, the possible number of flowers you sell is 3 to 25
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Mr. Thompson has a class of 14 students. He can spend $25 on each student to buy math supplies for the year. He first buys all of his students calculators,
which costs a total of $84. 14. After buying the calculators, how much does he have left to spend on each student?
Answer:
he has $6 left.
Step-by-step explanation:
Answer:
$18.99 for each student
Step-by-step explanation:
First we need to find out how much he spent on each student for the calculators.
We can do that by dividing 84.14, the total, by the number of students, 14:
84.14/14 = 6.01
He spent 6.01 on each student and has 25 for each student, so we can subtract that:
25 - 6.01 = 18.99
Find FH
FH = {Blank}
Answer:
FH = 24
Step-by-step explanation:
HG = 13
FH = 37 - 13
= 24
Answer:
The value of FH is 24.
Step-by-step explanation:
The equation will be,
→ FHG = FH + HG
Then the value of FH will be,
→ 37 = FH + 13
→ FH = 37 - 13
→ [ FH = 24 ]
Hence, the value of FH is 24.
2. The sum of the interior angles in a quadrilateral is 360°. Determine the value of x in
the diagram on the right.
Answer:
267/5 ( or ) 53.2
Step-by-step explanation:
Note : -
□ means 90°
2x - 42 + 3x + 45 + 90 = 360
2x + 3x + 45 - 42 + 90 = 360
5x + 3 + 90 = 360
5x + 93 = 360
5x = 360 - 93
5x = 267
x = 267/5 ( in fraction )
( or )
x = 53.2° ( in decimal )
-2w+w+7w−2w+w+7w i need help please i dont know what to do 8th grade basic integers thanks.?
The expression is simplified to 6w
What are algebraic expressions?Algebraic expressions are expressions made up of terms, variables, factors, coefficients and constants.
They also consist of arithmetic operations such as addition, subtraction, multiplication, division, etc
Given the expression;
-2w+w+7w
Using the BODMAS rule, we have to add
-2w + 8w
6w
Thus, the expression is simplified to 6w
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Write the specified quantity as a function of the specified variable. It will help in each case to draw a picture. The height of a right circular cylinder equals its diameter. Write the volume of the cylinder as a function of its radius.
The volume of the cylinder as a function of its radius is V = 2πr^3
How to write the volume of the cylinder as a function of its radius?The given parameters are
Diameter = Height
This can be represented as
d = h
The radius of a cylinder is calculated as
r = d/2
This means that
d = 2r
Substitute d = 2r in d = h
h = 2r
The volume of a cylinder is
V = πr^2h
Substitute h = 2r in V = πr^2h
V = πr^2(2r)
Evaluate
V = 2πr^3
Hence, the volume of the cylinder as a function of its radius is V = 2πr^3
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suppose M is the midpoint of FG. find each missing measure.
FM= 8a + 1, FG = 42, a =?
The value of a is 2.5
As FG is the line and M is the midpoint of the line
F ______________·_________________G
M
So by this, we have that
FG = FM + MG
Where FM = MG as M is the midpoint of the line.
Therefore FG = 2FM
As it is given that FM = 8a + 1 and FG = 42
Now putting the values we have
42 = 2( 8a + 1)
21 = 8a + 1
20 = 8a
a = 2.5
Therefore the value of a is 2.5.
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(2x² + 4x³) + (-4x² + 6x³ - 2)
I help I need help please someone help me !!!!!
Answer:
answer for this question is alternative A (80, 60) & (95,85)
Consider the system of equations 2x + 5y = 5 and -3x + 7y = 7. What number would you multiply to eliminate the y?
Answer:
(0, 1 )
Step-by-step explanation:
2x + 5y = 5 → (1)
- 3x + 7y = 7 → (2)
to eliminate y multiply (1) by 7 and (2) by - 5 , then add the result
14x + 35y = 35 → (3)
15x - 35y = - 35 → (4)
add (3) and (4) term by term to eliminate y
29x + 0 = 0
29x = 0 ⇒ x = 0
substitute x = 0 into (1) and solve for y
2(0) + 5y = 5y
0 + 5y = 5
5y = 5 ( divide both sides by 5 )
y = 1
solution is (0, 1 )
The figure below is not drawn to scale. What is the value of x?
Can someone please help mee (20 points + brainliest!!!)
Answer:
a. Price needs to decrease by 6 Bozats.
[tex]\sf b. \ \ \sf x =-\dfrac{1}{6} p+\dfrac{100}{3}[/tex]
Explanation:
Equation: p = 200 - 6x
where:
p = price per kilogramx = number of kilogram of snig solda.
To sell an additional kilogram of snig that is (x + 1) kilogram of snig.
current price = 200 - 6(x + 1) = 200 - 6x - 6
b.
[tex]\rightarrow p = 200 - 6x[/tex]
[tex]\rightarrow - 6x = p-200[/tex]
[tex]\rightarrow x =\dfrac{ p-200 }{-6}[/tex]
[tex]\rightarrow x =-\dfrac{1}{6} p+\dfrac{100}{3}[/tex]
an independent group of food service personnel conducted a survey on tipping practices in a large metropolitan area. they collected information on the percentage of the bill left as a tip for 3535 randomly selected bills. the average tip was 12.7.7% of the bill with a standard deviation of 2.8%2.8%. assume that the tips are approximately normally distributed. construct an interval to estimate the true average tip (as a percent of the bill) with 99% confidence. round the endpoints to two decimal places, if necessary.
The interval to find the true average tip is (14.62,13.78).
Average is a number that is calculated by adding quantities together and dividing the total by the number of quantities.
The Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range
Given:
We know that the confidence interval = sample mean ± standard deviation substitute the values in it then confidence interval = 14.2 ±2.5/(35)^1/2
confidence interval = 14.2 ± 2.5/5.91
confidence interval = 14.2 ± 0.42
confidence interval = 14.2 + 0.42 ,
confidence interval = 14.2 - 0.42
confidence interval = (14.62,13.78)
So, an interval to estimate the true average tip is (14.62,13.78).
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An investigator wants to test for the presence of simple sugars in a sample. She adds reagent to her sample and also to a sample of distilled water. What reagent should she have added?.
The investigator can test the presence of simple sugars in the sample by using Benedict's solution.
What is Benedict's test ?Simple carbs are tested for using Benedict's Test. With free ketone or aldehyde functional groups, reducing sugars (monosaccharides and some disaccharides) are recognized by the Benedict's test.
What is Benedict's solution?Benedict's solution turns orange or brick red when heated in the presence of simple carbohydrates. The reducing characteristic of simple carbs is what triggers this response. The Benedict's solution changes color as a result of the reduction of copper (II) ions to copper (I) ions.
The produced red copper(I) oxide precipitates out of solution because it cannot be dissolved in water. This explains how the precipitate was created. The closer the final hue is approaching brick-red and the more precipitate is generated as the reducing sugar concentration rises. Copper oxide, a brick-red solid, can occasionally precipitate out of a solution and gather at the test tube's bottom.
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