The solution is given by [tex](18r^4+30r^3+25r^2-154r)/(r^6-5r^5+16r^4-45r^3-126r^2+3969r)[/tex]
To perform the indicated operations, we need to find a common denominator for all of the fractions. The common denominator will be the product of the three denominators: [tex](r^2-7r)(r^2+9r)(r^2+2r-63)[/tex].
Next, we will multiply each fraction by the appropriate factor to make the denominators equal. For the first fraction, we will multiply by [tex](r^2+9r)(r^2+2r-63)/(r^2+9r)(r^2+2r-63)[/tex].
For the second fraction, we will multiply by [tex](r^2-7r)(r^2+2r-63)/(r^2-7r)(r^2+2r-63)[/tex]. For the third fraction, we will multiply by [tex](r^2-7r)(r^2+9r)/(r^2-7r)(r^2+9r)[/tex].
After multiplying, we will combine the numerators and keep the common denominator:
[tex](9(r^2+9r)(r^2+2r-63)+10(r^2-7r)(r^2+2r-63)-1(r^2-7r)(r^2+9r))/(r^2-7r)(r^2+9r)(r^2+2r-63)[/tex]
Finally, we will simplify the numerator and denominator as much as possible to get the final answer:
[tex](9r^4+72r^3-567r^2+18r^3+162r^2-1134r+10r^4-70r^(3)+630r^2-20r^3-140r^2+980r-r^4+7r^3-9r^2)/(r^6-7r^5+9r^4-63r^3+2r^5-14r^4+18r^3-126r^2-63r^4+441r^3-567r^2+3969r)[/tex]
[tex](18r^4+30r^3+25r^2-154r)/(r^6-5r^5+16r^4-45r^3-126r^2+3969r)[/tex]
So the final answer is [tex](18r^4+30r^3+25r^2-154r)/(r^6-5r^5+16r^4-45r^3-126r^2+3969r)[/tex]
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hiranda T Show all of the work of how you List the correct answer and circle or highlight it. ((7)/(8)+(1)/(2))/(2(1)/(4))
On simplifying ((7)/(8)+(1)/(2))/(2(1)/(4)) the correct answer is 2 3/4.
To solve the problem ((7)/(8)+(1)/(2))/(2(1)/(4)), we can follow these steps:
Step 1: Simplify the expression inside the parentheses in the numerator.
(7/8) + (1/2) = (7/8) + (4/8) = (11/8)
Step 2: Simplify the expression inside the parentheses in the denominator.
2(1/4) = (2/1)(1/4) = (2/4) = (1/2)
Step 3: Divide the simplified numerator by the simplified denominator.
(11/8) ÷ (1/2) = (11/8) × (2/1) = (22/8) = (11/4)
Step 4: Simplifying the final expression, if possible.
(11/4) = 2 3/4
Therefore, the correct answer is 2 3/4.
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HELPPPOP i dont know WHAT THE ANSWER ISS
Find the length of a side of the square with vertices at (5, 4), (1, 5), (0, 1),
and (4, 0). Then find the area of the square.
The length οf οne side οf the square is √17 and the area οf the square is 17 square units.
What is Cοοrdinate Geοmetry?Cοοrdinate geοmetry is a branch οf mathematics that uses algebraic equatiοns tο describe the relatiοnships between pοints and shapes in a plane. It invοlves using the Cartesian cοοrdinate system tο represent pοints with οrdered pairs οf numbers.
We can start by finding the distance between twο οppοsite vertices οf the square using the distance fοrmula:
d = √[(x2 - x1)² + (y2 - y1)²]
Let's find the distance between the pοints (5, 4) and (1, 5), which are οppοsite vertices οf the square:
d = √[(1 - 5)² + (5 - 4)²] = √[16 + 1] = √17
Sο the length οf οne side οf the square is √17.
Tο find the area οf the square, we can use the fοrmula:
area = side²
Substituting the value οf the side that we fοund abοve, we get:
area = (√17)² = 17
Therefοre, the area οf the square is 17 square units.
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If I was asked to think of a number and multiply it by 2 I would get 2x
Answer:
5x - 3
Step-by-step explanation:
Given:
A Number and multiply it by 2.
To Find:
to write it algebraically
Solution:
Let the number that was asked to think to be x.
Then it is multiplied by 2 which becomes 2x.
Now the number is multiplied by 5.
So, x multiplied by 5 will become 5x.
Then, 5x is subtracted from 3.
So, we can write it as 5x - 3
Therefore, it can be written as 5x - 3 algebraically.
PLEASE HELP ME
The first two steps in the derivation of the quadratic formula by completing the square are shown below.
Which answer choice shows the correct next step?
Answer:
The correct next step is the answer choice
x^2+b/a x=-c/a
Step-by-step explanation:
PLEASE HELP ME ASAP TRIPLE THE BRAINLY!
slope = -1/5
used slope forumula
Town A and Town B are located at the points shown in the diagram. Mr. Peterson
wants to drive from Town A to Town B. He can choose between the route that takes
him through Towns P and Q, or the route that takes him through Town R. Each unit on
the grid equals 1 kilometer.
Answer:
To find the shorter route, we need to calculate the distances for both routes and compare them.
Route through Towns P and Q:
The distance between Town A and Town P is 3 units to the right and 5 units up, so it is √(3² + 5²) = √34 km.
The distance between Town P and Town Q is 2 units to the right, so it is 2 km.
The distance between Town Q and Town B is 4 units to the right and 4 units down, so it is √(4² + 4²) = 4√2 km.
Therefore, the total distance for this route is √34 + 2 + 4√2 km.
Route through Town R:
The distance between Town A and Town R is 5 units to the right and 3 units up, so it is √(5² + 3²) = √34 km.
The distance between Town R and Town B is 5 units to the right and 7 units down, so it is √(5² + 7²) = √74 km.
Therefore, the total distance for this route is √34 + √74 km.
Comparing the distances, we can see that:
√34 + 2 + 4√2 km ≈ 10.2 km
√34 + √74 km ≈ 12.6 km
Therefore, the shorter route is the one that goes through Towns P and Q, which is approximately 10.2 km long.
In a group of 30 students, 12 take Mathematics, 16 take Art, and 18 take History. If each student takes at least one subject and no one takes all three, then what is the number of students taking exactly 2 subjects?
There are 23 students taking exactly 2 subjects.
To find the number of students taking exactly 2 subjects, we can use the formula:
n(A∩B∩C) = n(A) + n(B) + n(C) - 2n(A∩B) - 2n(B∩C) - 2n(C∩A) + 3n(A∩B∩C)
Since we know that no one takes all three subjects, n(A∩B∩C) = 0. We can plug in the values given in the question and solve for n(A∩B), which represents the number of students taking exactly 2 subjects:
0 = 12 + 16 + 18 - 2n(A∩B) - 2(0) - 2(0) + 3(0)
0 = 46 - 2n(A∩B)
2n(A∩B) = 46
n(A∩B) = 23
Therefore, the number of students taking exactly 2 subjects is 23.
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Given the matrices A = [3/4 0] and B = [-4 0]
[0 ¾] [0 -4] 2a) Compute AB 2b) Compute BA.
2c) How did your answer in part (a) and part (b) compare? 2d) Will this be true, in general, for any two matrices when you multiply them (assuming their dimensions line up so that they may be multiplied)? If so, explain your reasoning If not, show an example of two matrices C and such that CD+DC.
The matrices A and B are given as:
A = [3/4 0]
[0 3/4]
B = [-4 0]
[0 -4]
2a) Compute AB:
AB = [3/4 0] * [-4 0]
[0 3/4] [0 -4]
= [3/4 * -4 + 0 * 0 3/4 * 0 + 0 * -4]
[0 * -4 + 3/4 * 0 0 * 0 + 3/4 * -4]
= [-3 0]
[0 -3]
2b) Compute BA:
BA = [-4 0] * [3/4 0]
[0 -4] [0 3/4]
= [-4 * 3/4 + 0 * 0 -4 * 0 + 0 * -4]
[0 * 3/4 + -4 * 0 0 * 0 + -4 * 3/4]
= [-3 0]
[0 -3]
2c) How did your answer in part (a) and (b) compare?
The answers in part (a) and (b) are the same. Both AB and BA resulted in the matrix [-3 0] [0 -3].
2d) Will this be true, in general, for any two matrices when you multiply them (assuming their dimensions line up so that they may be multiplied)? If so, explain your reasoning. If not, show an example of two matrices C and D such that CD≠DC.
No, this will not be true in general for any two matrices when you multiply them. The order in which matrices are multiplied matters, and in most cases, AB≠BA. Here is an example of two matrices C and D such that CD≠DC:
C = [1 2]
[3 4]
D = [5 6]
[7 8]
CD = [1 * 5 + 2 * 7 1 * 6 + 2 * 8]
[3 * 5 + 4 * 7 3 * 6 + 4 * 8]
= [19 22]
[43 50]
DC = [5 * 1 + 6 * 3 5 * 2 + 6 * 4]
[7 * 1 + 8 * 3 7 * 2 + 8 * 4]
= [23 34]
[31 50]
As you can see, CD≠DC.
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Two students, Stella and Vladimir, completed the conversion
statement 12 feet 8 inches =
inches.
Stella stated that 12 feet 8 inches = 152 inches.
Vladimir stated that 12 feet 8 inches = 9 inches.
Which student is incorrect? Explain.
1. Highlight the measurement to be converted.
2.
Into what units do the students need to convert the given
measurement? Explain.
3.
Is a conversion factor given for the students in the problem? If not,
where could they find it?
4. Circle Stella’s and Vladimir’s answers. Explain how they are different.
5. Which student’s answer is more reasonable? Explain.
6. Should your final answer be a number? Explain.
1. Completing the conversion of 12 feet 8 inches to inches, Stella who stated that 12 feet 8 inches = 152 inches is correct unlike Vladimir who got 9 inches.
2. Stella and Vladimir need to convert the given measurement into inches.
3. No conversion factor is given for the problem. However, the conversion factor could be found online or using the mathematical table.
4. Stella's and Vladimir's answers differ because Vladimir failed to convert the feet into inches.
5. Stella's answer is more reasonable, and it is correct because 12 feet equal 144 inches plus 8 inches, giving 152 inches.
6. The final answer should not just be a number without indicating the unit of measurement.
What is unit conversion?A unit conversion is the mathematical expression of the same property in a different unit of measurement.
For instance, instead of stating distance in kilometers, it can be depicted in miles. Similarly, instead of reporting time in hours, it can be reported in minutes or seconds.
1 foot = 12 inches
12 feet = 144 inches
12 feet 8 inches = 152 inches (12 x 12 + 8)
Thus, in this situation, the measurement to be converted is feet to inches.
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Use the given information to find the exact function value. Simplify your answer as much as possible. Rationalize the denominator if necessary. sin a = 5/13, 0 < a < ????/2
The given information tells us that the sin a is equal to 5/13, and that 0 is less than a, which is less than π/2. Therefore, the exact function value for sin a is 5/13 and the value for a is approximately 5/12.
We are given that sin(a) = 5/13 and 0 < a < π/2. We can use the Pythagorean identity cos²(a) + sin²(a) = 1 to find cos(a):
cos²(a) + sin²(a) = 1
cos²(a) + (5/13)² = 1
cos²(a) = 1 - (5/13)²
cos²(a) = 144/169
cos(a) = ± 12/13
Since 0 < a < π/2, we know that cos(a) > 0. Therefore, cos(a) = 12/13. We can use the definition of tangent to find tan(a):
tan(a) = sin(a)/cos(a) = (5/13)/(12/13) = 5/12
Therefore, the exact function value we were asked to find is tan(a) = 5/12.
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Please, help I need this done today and show work.
The points (1, 1) and (2, 2) have been used to graph the inequality as shown in the image attached below.
The boundary line is dashed.
The boundary line is shaded below.
What are the rules for writing an inequality?In Mathematics, there are two (2) main rules that are generally used for writing and interpreting an inequality or system of inequalities that are plotted on a graph and these include the following:
The line on a graph should be a solid line when the inequality symbol is (≥ or ≤).The line on a graph should be a dashed (dotted) line when the inequality symbol is (> or <).Additionally, the point (2, 2) and point (1, 1) are not solutions to the given inequality y < 1/4(x) + 1 because the lie below the boundary line
y < 1/4(x) + 1
2 < 1/4(2) + 1
2 < 3/2 (False)
y < 1/4(x) + 1
1 < 1/4(1) + 1
2 < 5/4 (False)
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Please help and provide steps if you can
If 6 screws weigh more than 7 nails, which is heavier, 7 screws or 8 nails?
Answer:
The 7 screws offer superior tensile strength over the 8 nails. The 7 screws are heavier.
PLEASE HELP
Find the value of each trigonometric value.
The trigonometric value of angle Cos Z would be = 77°
What is a trigonometric value?A trigonometric value is defined as those values that are based on three major trigonometric ratios such as the Sine, Cosine, and Tangent.
According to the rules of trigonometry the following is carried out:
Sine or sin θ = Side opposite to θ / Hypotenuse = XY / XZ
Cosines or cos θ = Adjacent side to θ / Hypotenuse = YZ / XY
Tangent or tan θ =Side opposite to θ / Adjacent side to θ = XY / YZ.
Therefore, the trigonometric value of cos Z = adj/hypo = 9/40 = 0.225
Z = Cos^-1 0.225.
Z = 76.9 = 77°
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A rectangular garden and triangular flower garden are planted next to each other. The dimensions are shown in the diagram below.
An equivalent expression for the total length of fencing is 11y+10 inches.
What is the perimeter?The whole distance encircling a shape is referred to as its perimeter. It is the length of any two-dimensional geometric shape's boundary or outline. Depending on the size, the perimeter of several figures can be the same.
From the given figure, length of rectangle is (3y+5) inches and breadth is y inches. Sides of equilateral triangle measures y inches.
a) We know that, the perimeter of a rectangle is 2 (Length+Breadth)
= 2(3y+5+y)
= 2(4y+5)
= 8y+10 inches
Perimeter of equilateral triangle is 3×Side
= 3×y
= 3y inches
So, total length = 8y+10+3y inches
b) Equivalent expression is
8y+10+3y
= 11y+10 inches
c) Factors of the expression is 1 and 11y+10.
Therefore, an equivalent expression for the total length of fencing is 11y+10 inches.
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1. Which three-dimensional tigure will this net make! (1 point)
The three-dimensional figure this net will make is a triangular prism and is denoted as option B.
What is a Three-dimensional figure?
This is defined as a solid figure or an object or shape that has three dimensions such as length, width, and height.
An example is the triangular prism is a polyhedron made up of two triangular bases and three rectangular sides and from the net given above we can deduce that the triangular prism can be formed from it due to the shape and vertices.
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Use Euclid's first book to prove what specific quadrilaterals are produced by perpendicular, unequal, bisecting diagonals.
By using Euclid's first book, we can prove that the quadrilateral produced by perpendicular, unequal, bisecting diagonals is a kite. A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. In this case, the two pairs of adjacent sides are formed by the diagonals bisecting the quadrilateral into four smaller triangles with equal areas.
According to Euclid's first book, when two lines intersect at a right angle (perpendicular), they form four right angles. In the case of a quadrilateral with perpendicular, unequal, bisecting diagonals, the diagonals intersect at a right angle and divide the quadrilateral into four smaller triangles with equal areas.
1. Draw a quadrilateral with perpendicular, unequal, bisecting diagonals.
2. Label the points where the diagonals intersect as A, B, C, and D.
3. Label the point where the diagonals intersect as E.
4. Use Euclid's first book to prove that the angles at E are all right angles.
5. Use Euclid's first book to prove that the four triangles formed by the diagonals are congruent (equal in area).
6. Use the definition of a kite to prove that the quadrilateral is a kite (two pairs of adjacent sides are equal in length).
7. Therefore, the quadrilateral produced by perpendicular, unequal, bisecting diagonals is a kite.
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$30 for 100 flyers or $65 for 250 flyers?
Answer:
To compare the cost of each option, we need to calculate the cost per flyer.
For the first option, the cost per flyer is:
30 dollars / 100 flyers = 0.3 dollars/flyer
For the second option, the cost per flyer is:
65 dollars / 250 flyers = 0.26 dollars/flyer
Therefore, the second option is cheaper per flyer. However, if you only need 100 flyers, the first option would be more cost-effective.
Step-by-step explanation:
Answer: $65 for 250 flyers is cheaper
Step-by-step explanation:
30 dollars per 100 flyers - 30/100 = 0.30 or 30 cents per flyer
65 dollars per 250 flyers - 65/250 = 0.26 or 26 cents per flyer
If G (with 2 or more nodes) is connected, then all nodes must have degree bigger equal to 1.
true or false
The statement "If G is connected with 2 or more nodes, then all nodes must have a degree bigger or equal to 1." is true.
This is because a connected graph is defined as a graph in which there is a path between any two nodes. If a node has a degree of 0, that means it is not connected to any other nodes, and therefore the graph cannot be connected. So, in order for G to be connected, all nodes must have a degree of at least 1.
In conclusion, the answer is true.
What is a node?A node is a point where several elements are connected.
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The length of the base of an isosceles triangle is x. The length of a leg is 3x-4. The perimeter of the triangle is 90 . Find x.
Answer:
X = 31 1/3
Step-by-step explanation:
3x - 4= 90
Step 1:
* opposite operation *
90 + 4 = 94
94/3 = 31 1/3
CHECK
*Replace x with your solution(s)*
3(3 1/3) - 4
3 x 31/3 = 94
94-4= 90
Three painters are working on painting a fence: Abel, Baker and Chuck. Abel could paint the whole fence himself in 18 hours. Baker paints twice as fast as Abel. Chuck paints three times as fast as Abel. How long does it take them working all together?
The combined rate of all three workers is 6 fences per 3 hours.
To find out how long it would take for all three painters to paint the fence together, we need to find the combined rate of work for Abel, Baker, and Chuck.
Let's first find the rate of work for each painter individually.
Abel's rate of work is 1/18 of the fence per hour (since he can paint the whole fence in 18 hours).
Baker's rate of work is twice as fast as Abel's, so his rate of work is 2/18 (or 1/9) of the fence per hour.
Chuck's rate of work is three times as fast as Abel's, so his rate of work is 3/18 (or 1/6) of the fence per hour.
Now, to find the combined rate of work for all three painters, we simply add their individual rates of work together:
1/18 + 1/9 + 1/6 = 1/18 + 2/18 + 3/18 = 6/18 = 1/3
This means that together, the three painters can paint 1/3 of the fence per hour.
To find out how long it would take them to paint the whole fence, we simply divide the total amount of work (1 whole fence) by their combined rate of work (1/3 of the fence per hour):
1 ÷ 1/3 = 3
So it would take the three painters 3 hours to paint the whole fence together.
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I neeeeddd help pls help me
VALUE ADDED TAX (TAX) IS ADDED AT 15 % FOR GOOD AND SERVICES IN SOUTH AFRICA. WHAT WILLBE THE SELLING PRICE OF A LAPTOP THAT COSTS R4 200 BEFORE VAT
Answer:
To find the selling price of the laptop that costs R4 200 before VAT, we need to add 15% VAT to the cost.
The VAT amount can be calculated as:
VAT = 15% of R4 200
VAT = (15/100) x R4 200
VAT = R630
The selling price including VAT will be the cost of the laptop plus the VAT amount:
Selling price = R4 200 + R630
Selling price = R4 830
Therefore, the selling price of the laptop that costs R4 200 before VAT will be R4 830 after adding 15% VAT.
Can someone help me on this special series progressions? thanks
The given expression evaluates to 366.
What is Summation?The sum of the series is computed using the summation formulas. There are many different kinds of sequences, including arithmetic and geometric sequences, and consequently, there are many different kinds of summing formulas for those different kinds of sequences.
As per the given data:
To find [tex]\sum_{n=2} ^6 n (n^2 -n+1)[/tex]
Simplifying the given expression:
[tex]\sum_{n=2} ^6 n^3 - n^2 + n[/tex]
using the formulas:
[tex]\sum n = \frac{n(n+1)}{2}\\\\\sum n^2 = \frac{n(n+1)(2n+1)}{6}\\\\\sum n^3 = \frac{n^2(n+1)^2}{4}[/tex]
= [tex][\frac{n^2(n+1)^2}{4} - [\frac{n(n+1)(2n+1)}{6}] + \frac{n(n+1)}{2}]_{n=2} ^ 6[/tex]
substitute the values of limit:
= 371 - 5
= 366
Hence, the given expression evaluates to 366.
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Let R be a ring such that x = x2 for all x ∈ R. Prove that R is a ring commutative.
Yes, the given ring is commutative.We have that, a commutative ring in abstract algebra is a ring that satisfies a +b = b+a (for a defined operator).
Given: R is a ring such that [tex]$x = x^2$[/tex] for all [tex]$x\in R$[/tex].
To prove: R is a ring commutative.
Proof:
Let [tex]$a$[/tex] and [tex]$b$[/tex] be arbitrary elements in [tex]$R$[/tex].
[tex]$a+b = (a+b)^2 = a^2 + 2ab + b^2$[/tex]
[tex]$b+a = (b+a)^2 = b^2 + 2ab + a^2$[/tex]
Therefore, [tex]$a+b = b+a$[/tex]. Thus, R is a ring commutative.
Q.E.D.
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PLSSS HELP IF YOU TURLY KNOW THISSS
Answer:
-8=x
Step-by-step explanation:
-6=x+2
take away 2 from each side
-8= x
Answer:
x = -8
Step-by-step explanation:
To solve for x in equation 3(x - 2) = 4x + 2, we need to simplify the expression on the left side of the equation by using the distributive property. This gives:
3x - 6 = 4x + 2
Next, we need to isolate the variable x on one side of the equation. We can do this by adding 6 to both sides of the equation:
3x - 6 + 6 = 4x + 2 + 6
Simplifying this expression, we get:
3x = 4x + 8
Now, we need to isolate x again by subtracting 4x from both sides:
3x - 4x = 4x + 8 - 4x
Simplifying this expression gives:
-x = 8
Finally, we solve for x by multiplying both sides of the equation by -1:
x = -8
Therefore, the solution to the equation 3(x-2) = 4x + 2 is x = -8.
what is
1) y=2x
y=x+15
2) 2x+5y=55
-6x+y=-4
(systems of equations
Answer:
53
Step-by-step explanation:
becuas eyou divide it all togtheir and then subtract 42 from 83 and you ga the anwer
Consider the following all integer linear program
Max 1x1+1x2
s.t.
4x1+6x2 less than or equal to 22
1x1+5x2 less than or equal to 15
2x1+1x2 less than or equal to 9
x1,x2 greater than or equal to 0 and integer
A) graph the constraints of the problem
B) solve the LP relaxation of this problem
C) find the optimal integer solution
A) To graph the constraints of the problem, we need to rearrange each constraint into slope-intercept form (y = mx + b) and plot them on a graph.
For the first constraint, 4x1+6x2 <= 22, we can rearrange it as follows:
6x2 <= -4x1 + 22
x2 <= (-2/3)x1 + (22/6)
For the second constraint, 1x1+5x2 <= 15, we can rearrange it as follows:
5x2 <= -1x1 + 15
x2 <= (-1/5)x1 + (15/5)
For the third constraint, 2x1+1x2 <= 9, we can rearrange it as follows:
1x2 <= -2x1 + 9
x2 <= (-2)x1 + 9
We can now plot these constraints on a graph with x1 on the x-axis and x2 on the y-axis.
B) To solve the LP relaxation of this problem, we can use the simplex method. The objective function is Max 1x1+1x2. The constraints are 4x1+6x2 <= 22, 1x1+5x2 <= 15, 2x1+1x2 <= 9, x1 >= 0, and x2 >= 0.
Using the simplex method, we can find that the optimal solution is x1 = 3 and x2 = 2, with an objective function value of 5.
C) To find the optimal integer solution, we can use the branch and bound method. We start with the LP relaxation solution of x1 = 3 and x2 = 2. Since this is already an integer solution, it is the optimal integer solution. The objective function value is 5.
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2. (20 pts) LetA={0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}. a) How many subsets ofAhave 8 elements? b) How many subsets ofAwith 8 elements have only numbers? c) How many subsets ofAwith 8 elements have only letters? d) How many subsets ofAwith 8 elements have 2 letters?
a) There are 128 subsets of A with 8 elements. This is because the number of subsets of a set with n elements is 2 ^ n. Therefore, the number of subsets of A with 8 elements is 2 ^ 8 = 128.
b) There are 82 subsets of A with 8 elements that have only numbers. This is because there are 10 numbers in A and each of these numbers can be included or excluded from the subset of 8 elements. Therefore, the number of subsets of A with 8 elements that have only numbers is 2 ^ 10 = 1024.
c) There are 46 subsets of A with 8 elements that have only letters. This is because there are 6 letters in A and each of these letters can be included or excluded from the subset of 8 elements. Therefore, the number of subsets of A with 8 elements that have only letters is 2 ^ 6 = 64.
d) There are 28 subsets of A with 8 elements that have 2 letters. This is because there are 6 letters in A and each of these letters can be included or excluded from the subset of 8 elements twice. Therefore, the number of subsets of A with 8 elements that have 2 letters is 2 ^ 12 = 4096.
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