The woman bought 21 cards that play a song when they are opened, and 79 cards that do not play music.
Let's assume that the woman bought x cards that play a song when they are opened, and 100-x cards that do not play music.
We know the cost of the cards that play a song is 30 cents each, so the cost of x of these cards is 0.3x dollars.
Similarly, the cost of the cards that do not play music is 5 cents each, so the cost of 100-x of these cards is 0.05(100-x) dollars.
The total cost of all the cards is $10.25, so we can set up the following equation
0.3x + 0.05(100-x) = 10.25
Simplifying the equation, we get
0.3x + 5 - 0.05x = 10.25
0.25x = 5.25
x = 21
Therefore, the woman bought 21 cards that play a song when they are opened, and 100-21 = 79 cards that do not play music.
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Suppose that you buy an exercise ball with diameter of 85 cm and your brother buys an exercise ball with diameter of 65 cm. What is the difference between the volume and surface area of both balls? Justify your answer and round to the nearest hundredth
The difference between the volume and surface area of both balls is: 108101.69 cubic centimeters.
The formula for the volume of a sphere is V = (4/3)π[tex]r^3[/tex], where r is the radius of the sphere, and the formula for the surface area of a sphere is A = 4π[tex]r^2[/tex].
Let's start by finding the radius of each ball. The diameter of the first ball is 85 cm, so its radius is 85/2 = 42.5 cm. The diameter of the second ball is 65 cm, so its radius is 65/2 = 32.5 cm.
The volume of the first ball is:
V1 = (4/3)π[tex](42.5)^3[/tex] = 256905.53 cubic centimeters
The volume of the second ball is:
V2 = (4/3)π[tex](32.5)^3[/tex] = 139379.44 cubic centimeters
The difference in volume is:
V1 - V2 = 256905.53 - 139379.44 = 117526.09 cubic centimeters
To find the difference in surface area, we plug in the radius values:
A1 = 4π[tex](42.5)^2[/tex] = 22698.16 square centimeters
A2 = 4π[tex](32.5)^2[/tex] = 13273.76 square centimeters
The difference in surface area is:
A1 - A2 = 22698.16 - 13273.76 = 9424.40 square centimeters
Therefore, the difference between the volume and surface area of both balls is: 117526.09 cubic centimeters - 9424.40 square centimeters, rounded to the nearest hundredth, is 108101.69 cubic centimeters.
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Sam and jeremy have ages that are consecutive odd interferes. The product of their ages is 783. Which equation could be used to find Jeremy’s age, j, if he is the younger man?
So, the equation we could use to find Jeremy's age, j, if he is the younger man, is: j = x + 2, where x satisfies the equation x(x + 2) = 783.
What is equation?An equation is a mathematical statement that asserts the equality of two expressions. In an equation, an expression is written on the left side of an equals sign (=), and another expression is written on the right side of the equals sign. The equals sign indicates that the two expressions have the same value.
Here,
Let's use algebra to solve this problem. We can start by representing Sam's age as x and Jeremy's age as x + 2, since they are consecutive odd integers. We know that the product of their ages is 783, so we can set up the following equation:
x(x + 2) = 783
We can simplify this equation by multiplying out the left side:
x² + 2x = 783
Now we have a quadratic equation in standard form. We can solve for x by using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 1, b = 2, and c = -783, so we can substitute these values into the formula:
x = (-2 ± √(2² - 4(1)(-783))) / 2(1)
Simplifying the expression under the square root, we get:
x = (-2 ± √(3136)) / 2
x = (-2 ± 56) / 2
So, we have two possible values for x:
x = 27 or x = -29
We can reject the negative value, since it does not make sense in the context of the problem. Therefore, we have:
x = 27
Now we can use this value to find Jeremy's age, which is x + 2:
j = x + 2
= 27 + 2
= 29
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please help i need this asap
All the value of angles are,
⇒ ∠a = 118°
⇒ ∠b = 62°
⇒ ∠q = 84°
⇒ ∠v = 84°
Given that;
Two parallel lines t₁ and t₂ are shown in image.
Here, we have;
Apply the definition of corresponding angles,
∠d = 180 - 62
∠d = 118°
Hence, By definition of vertically opposite angles,
⇒ ∠a = 118°
And,
∠b = 180 - 118°
∠b = 62°
Apply the definition of corresponding angles,
∠q = 180 - 96
∠q = 84°
Hence, By definition of vertically opposite angles,
⇒ ∠v = 84°
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Imaginé que usted recibe semanalmente $50 y de esos destina el 10% de ahorros para comprar un pantalón que cuesta $40
Cuántas semanas tendrá que ahorrar?
It will take 8 weeks to save up for $40 pair of trousers.
How many weeks will it take to save for $40?Savings means left over after subtracting spending from their disposable income.
We will calculate how money saved each week which is:
= 10% * $50
= $5.
So, this means we have $45 left after saving for a week.
To save up $40, we need to save for:
= 40/5
= 8 weeks.
Translated question "I imagine that you receive $50 a week and of that you use 10% of your savings to buy a pair of trouser that cost $40. How many weeks will you have to save?"
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What is the logarithmic form of the exponential equation 4 ^ 3 = (5x + 4)
Answer:
log base 4 (5x + 4) = 3
Step-by-step explanation:
a^b=c is loga(c)=b
mathsathomecom
Expand up to the 4th term
√1+3x
Answer:
1+3x
Step-by-step explanation:
[tex] \sqrt{1 } = 1 \\ 1 + 3x = 1 + 3x[/tex]
Factor completely. If the polynomial is not factorable, write prime.
a^8 - a^2 B^6
The polynomial, a⁸ - a²·B⁶ in factored form is; a²·(a³ - B³)·(a³ + B³)
What is a polynomial?A polynomial consists of the differences or sums of terms that are the product of powers of the same variable.
The specified polynomial can be presented as follows;
a⁸ - a²·B⁶
The common factor in the terms of the polynomial is a², therefore, we get;
a⁸ - a²·B⁶ = a⁶ × a² - a²·B⁶
a⁶ × a² - a²·B⁶ = a²·(a⁶ - B⁶)
(a⁶ - B⁶) = (a³ - B³) × (a³ + B³)
The polynomial is therefore; a⁸ - a²·B⁶ = a²·(a³ - B³)·(a³ + B³)
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x Which two choices are equivalent to this expression?
2√75 +3√50
x
x
A
25√6
B 10√3+15/2
C 25√3+25√2
D 2√25-3+3√25-2
E
3√25+2√25
A figure with parallel Lines m and n is shown.
The measure of angles A, B, and C for the given parallel lines will be 53°,90°, and 143° corresponding.
What is an example of a parallel line?
In terms of geometry, parallel lines are two separate lines that never cross each other and are located in the same plane. Both vertical and horizontal can be used. A zebra crossing, rows of notebooks and nearby railway tracks are just a few instances of parallel lines that we encounter every day.
As per the parallel lines m and n.
The adjacent angle of B = 37° (corresponding angle same)
m∠B = 180° - (53° + 37°) = 90°
m∠A = 53° (corresponding angle same)
m∠C = 180° - 37° = 143°
Hence "The measure of angles A, B, and C for the given parallel lines will be 53°,90°, and 143° corresponding".
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Can someone help me please
Answer:
A, 48 degrees
Step-by-step explanation:
180 - 42 - 90 = 48 (all interior angles add up to 48)
Answer:
it’s A
Step-by-step explanation:
it’s a because the right triangle is always 90°.As you can see there is 42° so you subtract to find the missing number.
Simplify m^8m^−6.
one over m to the forty eighth power
m^2
one over m squared
−m^14
It should be noted that m⁸m⁻⁶ is equivalent to (m²)¹, which is equal to m².
How to calculate the valueUsing the product of powers rule for exponents, we can simplify m⁸m as follows:
m⁸m⁻⁶ = m⁸⁻⁶) = m²
Therefore, m⁸⁻⁶ is equal to m².
Now, we can further simplify by expressing m² as (m²)¹. Multiplying the exponents, we get:
(m²)¹ = m²
We can say that m⁸m⁻⁶ is equivalent to (m²)¹, which is equal to m².
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please help
Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
The student council at Lakewood High School is making T-shirts to sell for a fundraiser, at a price of $11 apiece. The costs, meanwhile, are $10 per shirt, plus a setup fee of $54. Selling a certain number of shirts will allow the student council to cover their costs. What will the costs be? How many shirts must be sold?
Answer:
The system of equations to describe the situation is:
* $10x + 54 = 11y$
* $y = 11x$
To solve using substitution, we can substitute the second equation into the first equation. This gives us:
* $10x + 54 = 11(11x)$
* $10x + 54 = 121x$
* $-11x = -54$
* $x = 5$
We can then substitute this value of $x$ into the second equation to solve for $y$. This gives us:
* $y = 11(5)$
* $y = 55$
Therefore, the student council will need to sell 55 T-shirts to cover their costs. The costs will be $54 + $10(55) = $604.
Here is the solution in a table format:
| Variable | Value |
|---|---|
| $x$ | 5 |
| $y$ | 55 |
| Cost | $604 |
I hope this helps!
Step-by-step explanation:
Troy Engines, Limited, manufactures a variety of engines for use in heavy equipment. The company has always produced all of the necessary parts for its engines, including all of the carburetors. An outside supplier has offered to sell one type of carburetor to Troy Engines, Limited, for a cost of $30 per unit. To evaluate this offer, Troy Engines, Limited, has gathered the following information relating to its own cost of producing the carburetor internally:
Per Unit 19,000 Units Per Year
Direct materials $ 12 $ 228,000
Direct labor 10 190,000
Variable manufacturing overhead 3 57,000
Fixed manufacturing overhead, traceable 3* 57,000
Fixed manufacturing overhead, allocated 6 114,000
Total cost $ 34 $ 646,000
*One-third supervisory salaries; two-thirds depreciation of special equipment (no resale value).
Required:
1. Assuming the company has no alternative use for the facilities that are now being used to produce the carburetors, what would be the financial advantage (disadvantage) of buying 19,000 carburetors from the outside supplier?
2. Should the outside supplier’s offer be accepted?
3. Suppose that if the carburetors were purchased, Troy Engines, Limited, could use the freed capacity to launch a new product. The segment margin of the new product would be $190,000 per year. Given this new assumption, what would be the financial advantage (disadvantage) of buying 19,000 carburetors from the outside supplier?
4. Given the new assumption in requirement 3, should the outside supplier’s offer be accepted?
Troy Engines, Limited, could use the freed capacity to launch a new product with a segment margin of $190,000 per year, the financial advantage of buying 19,000 carburetors from the outside supplier can be calculated as follows:
Total cost of producing 19,000 carburetors internally = $646,000
Cost of purchasing 19,000 carburetors from outside supplier = $30 x 19,000 = $570,000
Savings from purchasing carburetors from outside supplier = $646,000 - $570,000 = $76,000
Additional revenue from launching new product = $190,000
Therefore, the total financial advantage of buying 19,000 carburetors from the outside supplier and launching the new product would be $266,000 ($76,000 in savings plus $190,000 in additional revenue).
Based on this analysis, it would be financially advantageous for Troy Engines, Limited, to accept the outside supplier's offer and purchase the carburetors. The savings from purchasing the carburetors would be significant, and the additional revenue from launching the new product would further increase the company's profitability.
However, it's important for the company to consider other factors as well, such as the quality and reliability of the outside supplier's carburetors and the potential impact on the company's reputation.
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A backyard is 40. 5 feet long and 25 feet wide. in order to install a pool, the yard needs to be reduced by a scale of one-third. what is the area of the reduced yard? feet2.
If A backyard is 40. 5 feet long and 25 feet wide then, the area of the reduced yard is approximately 450.09 ft².
The area of the original backyard is:
40.5 ft x 25 ft = 1012.5 ft²
To reduce the yard by a scale of one-third, we need to multiply the length and width by 2/3:
40.5 ft x 2/3 = 27 ft
25 ft x 2/3 = 16.67 ft (rounded to two decimal places)
The area of the reduced yard is:
27 ft x 16.67 ft = 450.09 ft² (rounded to two decimal places)
Therefore, the area of the reduced yard is approximately 450.09 ft².
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Write a number equivalent to x to the power of -3 using a positive exponent.
The number equivalent to x to the power of -3 using a positive exponent is 1/x³.
How can we express x to the power of -3 as a positive exponent?When a number is raised to a negative exponent, it means the reciprocal of that number is being raised to the corresponding positive exponent. In other words, x⁻³ can be written as 1/x³.
To understand why this is the case, consider the following example:
If we have x²/x⁵, we can simplify it by dividing the numerator and denominator by x². This results in 1/x³.
Therefore, any number raised to a negative exponent can be rewritten as its reciprocal raised to the corresponding positive exponent. So, x⁻³ can be rewritten as 1/x³.
When we raise a number to an exponent, we are essentially multiplying that number by itself a certain number of times. For example, 2³ means 2 multiplied by itself 3 times, which is equal to 8.
In mathematics, we can also use exponents to represent the reciprocal of a number.
The reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of 2 is 1/2, and the reciprocal of 5 is 1/5.
Now, when we raise a number to a negative exponent, we are essentially raising its reciprocal to the corresponding positive exponent. This may seem a little confusing at first, but let me explain with an example:
x⁻³ = 1/(x³)
Let's verify this by simplifying the expression 1/(x³):
1/(x³) = 1/(xxx) = (1/x)(1/x)(1/x) = x⁻¹ * x⁻¹ * x⁻¹ = x⁻³
So we can see that x⁻³ is equivalent to 1/(x³), which is the reciprocal of x raised to the power of 3.
This concept of negative exponents is very useful in mathematics, as it allows us to simplify expressions and manipulate them in different ways.
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What is 133/14 simplify
Answer:
19/2
Step-by-step explanation:
133 = 7 × 19
14 = 7 × 2
133/14 = 19/2
Hence Simplified
A die is rolled twice. What is the probability of rolling a 5 or getting an even number?
2/3
1/12
3/1
Answer:
4/6 or 2/3
Step-by-step explanation:
probability is successful out of total. The total is 1,2,3,4,5,6, or 6 ways, and the successful is 2,4,5,6, or 4 ways
point (4, -13) lies on the graph of the equation y = kx + 7
what is value of k?
Answer:
-5
Step-by-step explanation:
(4, -13) = (x, y)
y = kx + 7
-13 = k(4) + 7
4k = -13-7
4k = -20
k = -5
#CMIIWTriangle A B C is shown. Side A C has a length of 27. Side C B has a length of 54.
Based on the diagram, which expresses all possible lengths of segment AB?
AB = 25
27 < AB < 81
AB = 85
AB< 27 or AB > 81
Step-by-step explanation:
Based on the diagram and the given lengths of sides AC and CB, we can use the triangle inequality theorem to determine the possible lengths of side AB.
According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore:
AC + CB > AB
27 + 54 > AB
81 > AB
Also, using the same theorem, we can see that:
AB + AC > CB
AB + 27 > 54
AB > 54 - 27
AB > 27
Therefore, we have:
27 < AB < 81
So, the answer is: 27 < AB < 81
Answer:
Based on the information given, the possible lengths of segment AB can be determined using the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we can apply the theorem to sides AC and CB to find the possible range of values for side AB.
Since AC has a length of 27 and CB has a length of 54, we can write two inequalities based on the Triangle Inequality Theorem:
AB + 27 > 54 AB + 54 > 27
Solving each inequality for AB, we get:
AB > 27 AB > -27
Since the length of a side of a triangle must be positive, we can disregard the second inequality. Therefore, the possible range of values for side AB is AB > 27.
However, it’s important to note that you mentioned a diagram in your message but did not provide it. Without seeing the diagram, I cannot determine if there are any additional constraints on the length of side AB.
Step-by-step explanation:
Help with problem in photo pls
Check the picture below.
A group of 6 friends went to the movies. In addition to their tickets, they bought a large bag of popcorn to share for $9. 50. The total was $54. 50. Complete parts a and b
Answer:
Step-by-step explanation:
a) Let x be the cost of each ticket. Then the total cost of the tickets for the 6 friends is 6x. The total cost, including the popcorn, is $54.50. So we can set up an equation:
6x + $9.50 = $54.50
Solving for x, we get:
6x = $45.00
x = $7.50
So each ticket costs $7.50.
b) To find the cost per person, we need to add up the cost of each ticket and divide by the number of people. The cost of all 6 tickets is:
6 tickets × $7.50/ticket = $45.00
Adding the cost of the popcorn, we get:
$45.00 + $9.50 = $54.50
So the total cost for the 6 friends is $54.50, and the cost per person is:
$54.50 ÷ 6 people = $9.08 per person.
Therefore, each person contributed $9.08 towards the total cost of the tickets and popcorn.
88 POINTS!!!!!
Joe and Tommy are playing a game of chicken. Tommy has a mass of 91 kg, Joe has a mass of 97 kg. Tommy is running towards Joe with a velocity of 5 m/s, Joe is running towards Tommy with a velocity of 6 m/s. Neither "chickens" out. After collision, Joe is standing still. How fast does Tommy bounce off Joe? Round to four decimal places
Tommy bounces off Joe at a speed of 5.9451 m/s after the collision.
We need to use the conservation of momentum, which states that the total momentum of a closed system remains constant. In this case, the closed system is the two players, Joe and Tommy.
We can start by calculating the initial momentum of the system, which is given by:
[tex]p_{initial} = m_{Tommy} * v_{Tommy} + m_{Joe} * v_{Joe}[/tex]
where m_Tommy and m_Joe are the masses of Tommy and Joe, respectively, and v_Tommy and v_Joe are their initial velocities.
Plugging in the given values, we get:
[tex]p_{initial} = 91 kg * 5 m/s + 97 kg * 6 m/s[/tex]
p_initial = 1123 kgm/s
After the collision, Joe is standing still, which means his velocity is zero. Let's call Tommy's velocity after the collision v_Tommy', which we need to find.
The final momentum of the system is given by:
[tex]p_{final} = m_{Tommy} * v_{Tommy'} + m_{Joe} * 0[/tex]
where we set Joe's velocity to zero since he's standing still.
Since the momentum is conserved, we can equate p_initial and p_final:
p_initial = p_final
[tex]m_Tommy * v_Tommy + m_Joe * v_Joe = m_Tommy * v_Tommy'[/tex][tex]m_{Tommy} * v_{Tommy} + m_{Joe} * v_{Joe} = m_{Tommy} * v_{Tommy'}[/tex]
91 kg * 5 m/s + 97 kg * 6 m/s = 91 kg * v_Tommy'
Solving for v_Tommy', we get:
[tex]v_{Tommy'} = (91 kg * 5 m/s + 97 kg * 6 m/s) / 91 kg[/tex]
v_Tommy' = 5.9451 m/s (rounded to four decimal places)
In summary, we used the conservation of momentum to calculate the velocity of Tommy after bouncing off Joe. We found that he bounces off Joe at a speed of 5.9451 m/s.
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Is The number of insects feeding on a tree leaf discrete or continious
The number of insects feeding on a tree leaf is a discrete variable.
The number of insects feeding on a tree leaf is a countable variable that can only take on integer values (0, 1, 2, 3, etc.). It cannot take on fractional or continuous values. This is because each insect can either feed on the leaf or not, and there cannot be a fractional or continuous number of insects feeding on the leaf.
Therefore, the number of insects feeding on a tree leaf is a discrete variable. This is in contrast to a continuous variable, which can take on any value within a certain range. For example, the weight of the insects on the leaf would be a continuous variable since it can take on fractional values.
In mathematical terms, the number of insects feeding on a tree leaf can be represented as a discrete random variable X, where X can take on any non-negative integer value.
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PLS HELP____________
Answer:
the answer is the 1st one
2+2=4
3+1=4
4+0 = 4
You buy a movie ticket for $5.25
and popcorn for $2.98
.
You pay with a $10
bill.
Carlotta purchased a whole life insurance policy with an annual premium of $780. In the first year, 60% of the annual premium is allocated to the insurance component and 40% to the investment component. The investment earns 2. 2% interest, compounded annually. How much will Carlotta have in the investment portion of her policy after the first year? Round to the nearest cent.
Omg help me please
After the first year, Carlotta will have $124.80 in the investment portion of her policy.
This is calculated by taking 40% of her annual premium
($780 x 0.40 = $312),
2.2% ($312 x 0.022 = $6.84).
So the total amount in the investment portion is
$312 + $6.84 = $318.84, rounded to the nearest cent, which is $124.80.
The whole life insurance policy that Carlotta purchased has both an insurance component and an investment component. In the first year, 60% of the annual premium is allocated to the insurance component, which means that $468 of her $780 premium goes towards the cost of the insurance.
The remaining 40% is allocated to the investment component, which is what Carlotta will earn interest on.
At a rate of 2.2%, compounded annually, the investment portion of Carlotta's policy earns $6.84 in interest after the first year. This is added to the $312 that was allocated to the investment portion, giving a total of $318.84.
This means that Carlotta has $124.80 in the investment portion of her policy after the first year. It's important to note that this amount will continue to grow over time as Carlotta pays her premiums and earns interest on her investment.
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Read the following question:
Answer: 5 hours of service
Step-by-step explanation:
25 + x(15) = 20x
⇒ 25 = 20x - 15x
⇒ 25 = 5x
⇒ x = 5
Let x and y be the numbers represented on the number line.
1. Ifp is the product of x and y, what point can represent p on the number line?
2. Use the information from (1) to find the point representing qon the number line if q is the
quotient of x and y. Explain your reasoning.
To find the point representing the product p of x and y on the number line, locate x and y on the number line and find their product. To find the point representing the quotient q of x and y on the number line, locate x and the reciprocal of y (1/y) on the number line, and find their product.
If p is the product of x and y, the point on the number line that represents p can be found by locating x and y on the number line and then finding their product. For example, if x is at 2 and y is at -3, then their product p is (-6) and is located at the point on the number line that corresponds to -6 which corresponds to point P.
To find the point representing q on the number line if q is the quotient of x and y, we can use the fact that the quotient is the same as the product of x and the reciprocal of y.
In other words, q = x / y = x * (1/y). Therefore, if we locate x and 1/y on the number line, their product gives us the point representing q. For example, if x is at 4 and y is at -2, then 1/y is -1/2 and is located at -2 on the number line. The product of 4 and -1/2 is -2, which corresponds to the point on the number line that represents q.
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A store sells tvs for x$ they are doing a black friday sale which is 42% off, call that function f(x). they are also giving all customers a $100 rebate, call that function g(x). what is f(g(x))? and what does it mean?
The final price a customer would pay for a TV after both the 42% Black Friday discount and the $100 rebate have been applied.
Let x represent the original price of the TVs. The store is offering a Black Friday sale of 42% off, which we can represent as a function f(x) = 0.58x (since 100% - 42% = 58%). They are also giving a $100 rebate to all customers, represented by the function g(x) = x - 100.
Now, we want to find f(g(x)), which means applying the function f(x) to the result of the function g(x). So, f(g(x)) = f(x - 100).
To do this, plug in (x - 100) for x in the f(x) function: f(x - 100) = 0.58(x - 100).
This function, f(g(x)), represents the final price a customer would pay for a TV after both the 42% Black Friday discount and the $100 rebate have been applied.
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find the general solution of the following linear system. y′ = [2 2 −4 2 −1 −2 4 2 −6] y with λ = −1,−2,−2
To find the general solution of the linear system y' = [2 2 -4; 2 -1 -2; 4 2 -6] y, we need to first find the eigenvectors and eigenvalues of the coefficient matrix A = [2 2 -4; 2 -1 -2; 4 2 -6].
Using the characteristic equation, we can find the eigenvalues:
det(A - λI) = 0
=> det([2-λ 2 -4; 2 -1-λ -2; 4 2 -6-λ]) = 0
=> (2-λ)[(-1-λ)(-6-λ) - 4] - 2[(-2)(-6-λ) - 8] + 4[2(-1-λ) - 4] = 0
=> λ^3 - + 8λ - 4 = 0
=> (λ-1)(λ-2[tex])^2[/tex] = 0
Thus, λ = 1, 2 (with multiplicity 2). For each eigenvalue, we need to find a corresponding eigenvector.
For λ = 1, we need to find the null space of the matrix (A - λI):
A - λI = [1 2 -4; 2 -2 -2; 4 2 -7]
=> R2 <- R2 - 2R1, R3 <- R3 - 4R1
[1 2 -4; 0 -6 6; 0 -6 9]
=> R3 <- R3 - R2
[1 2 -4; 0 -6 6; 0 0 3]
So, we have a basic eigenvector of the form [4,-2,1]^T. To obtain a linearly independent eigenvector, we use the method of generalized eigenvectors. We need to find a vector v such that (A - λI) v = u, where u is the basic eigenvector.
(A - λI) v = u
=> [1 2 -4; 2 -2 -2; 4 2 -7] v = [4; -2; 1]
=> R2 <- R2 - 2R1, R3 <- R3 - 4R1
[1 2 -4; 0 -6 6; 0 -6 9] v = [4; -2; 1]
=> R3 <- R3 - R2
[1 2 -4; 0 -6 6; 0 0 3] v = [4; -2; 1]
=> -6v2 + 6v3 = -2
=> 3v3 = 1
=> 2v2 - 4v3 = -2
=> v2 = 0
So, we have v = [0; 1/3; 2/3[tex]]^T[/tex] as the second eigenvector corresponding to λ = 1.
For λ = 2, we need to find the null space of the matrix (A - λI):
A - λI = [0 2 -4; 2 -3 -2; 4 2 -8]
=> R1 <-> R2
[2 -3 -2; 0 2 -4; 4 2 -8]
=> R3 <- R3 - 2R1
[2 -3 -2; 0 2 -4; 0 8 -12]
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