Use the properties of logarithms to rewrite and simplify the logarithmic expression.

Using Pivotal Condensation, the determinant of matrix A is 5.

Step 1: We start by initializing D as 1.

Step 2: We use the first row for pivotal condensation.
Row 0:  0 * D + 2 * 1 + 2 * 0 = 0
Row 1:  1 * D + 0 * 1 + 3 * 0 = 1
Row 2:  2 * D + 1 * 1 + 1 * 0 = 2

Step 3: We make the first row entries 0 by multiplying the entire row by (-2).
Row 0: -0 * D - 2 * 1 - 2 * 0 = 0
Row 1:  1 * D + 0 * 1 + 3 * 0 = 1
Row 2:  2 * D + 1 * 1 + 1 * 0 = 2

Step 4: We add row 0 to row 1 and row 0 to row 2.
Row 0:  0 * D + 2 * 1 + 2 * 0 = 0
Row 1:  1 * D + 0 * 1 + 3 * 0 = 1
Row 2:  0 * D + 3 * 1 + 3 * 0 = 3

Step 5: We make the entries of the second row 0 by multiplying the entire row by (-1/3).
Row 0:  0 * D + 2 * 1 + 2 * 0 = 0
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2:  0 * D + 3 * 1 + 3 * 0 = 3

Step 6: We add row 1 to row 0 and row 1 to row 2.
Row 0:  1/3 * D + 2 * 1 + 2 * 0 = 1/3
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2:  3/3 * D + 3 * 1 + 3 * 0 = 5

Step 7: We multiply the entries of the first row by (-3) to make the entries of the first row 0.
Row 0:  0 * D + 6 * 1 + 6 * 0 = 0
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2:  3/3 * D + 3 * 1 + 3 * 0 = 5

Step 8: We multiply the last row by D.
Row 0:  0 * D + 6 * 1 + 6 * 0 = 0
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2:  5 * D + 3 * 1 + 3 * 0 = 5D

Step 9: We subtract row 1 from row 0 and row 1 from row 2.
Row 0:  4/3 * D + 6 * 1 + 6 * 0 = 4/3D
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2:  4/3 * D + 3 * 1 + 3 * 0 = 4/3D

Step 10: We calculate the determinant by multiplying the last row entries.

Determinant of matrix A is 5.

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To determine whether a given vector is a solution to the equation \(A\vec{x}=\vec{0}\), we need to multiply the matrix A with the vector \(\vec{x}\) and check if the resulting vector is equal to the zero vector \(\vec{0}\).

Let's consider the first vector \(\vec{x_1}=\left[\begin{array}{c} 2 \\ 1 \\ 1 \end{array}\right]\). Multiplying the matrix A with this vector, we get:
\[ A\vec{x_1}=\left[\begin{array}{ccc} 3 & 6 & -6 \\ -1 & -2 & 5 \\ -1 & -2 & 2 \\ -1 & -1 & 0 \end{array}\right] \left[\begin{array}{c} 2 \\ 1 \\ 1 \end{array}\right]=\left[\begin{array}{c} 12 \\ -4 \\ -3 \\ -3 \end{array}\right] \]
Since the resulting vector is not equal to the zero vector \(\vec{0}\), the vector \(\vec{x_1}\) is not a solution to the equation \(A\vec{x}=\vec{0}\).
Similarly, we can check for the other vectors:
For the vector \(\vec{x_2}=\left[\begin{array}{c} -1 \\ 2 \\ -1 \end{array}\right]\):
\[ A\vec{x_2}=\left[\begin{array}{ccc} 3 & 6 & -6 \\ -1 & -2 & 5 \\ -1 & -2 & 2 \\ -1 & -1 & 0 \end{array}\right] \left[\begin{array}{c} -1 \\ 2 \\ -1 \end{array}\right]=\left[\begin{array}{c} 6 \\ 1 \\ 1 \\ 1 \end{array}\right] \]

Again, the resulting vector is not equal to the zero vector \(\vec{0}\), so the vector \(\vec{x_2}\) is not a solution to the equation \(A\vec{x}=\vec{0}\).
For the vector \(\vec{x_3}=\left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right]\):
\[ A\vec{x_3}=\left[\begin{array}{ccc} 3 & 6 & -6 \\ -1 & -2 & 5 \\ -1 & -2 & 2 \\ -1 & -1 & 0 \end{array}\right] \left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right]=\left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \]
In this case, the resulting vector is equal to the zero vector \(\vec{0}\), so the vector \(\vec{x_3}\) is a solution to the equation \(A\vec{x}=\vec{0}\).

Therefore, only the vector \(\vec{x_3}=\left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right]\) is a solution to the equation \(A\vec{x}=\vec{0}\).

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The solution is, the initial amount that you invested is $1200.

Interest is the price you pay to borrow money or the cost you charge to lend money. Interest is most often reflected as an annual percentage of the amount of a loan. This percentage is known as the interest rate on the loan.

here, we have,

Let the initial amount that you invested be $ x.

We are told that the new balance after investing $500 is $1760.

Thus, the balance you had before the deposit $500 us;

$1760 - $500 = $1260

So, when the amount invested was $x, the balance was $1260.

Since this money earned 5% interest on the amount you initially

Thus;

$1260 = initial deposit + (5% of inital deposit)

Thus;

x + (5% * x) = 1260

x + (0.05x) = 1260

1.05x = 1260

x = 1260/1.05

x = $1200

Hence, The solution is, the initial amount that you invested is $1200.

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Answer:

c

Step-by-step explanation:

it makes sense

The pair of integers to be used to rewrite the middle term when factoring 6t^(2)+5t-4 by grouping is (5t - 4) and (6t^2 + 5t). By factoring by grouping, we can factor the middle term out of the polynomial and then factor the remaining terms separately.

First, the middle term is factored out of the equation. This is done by multiplying the first and last terms together, which in this case is (6t^2)(-4). This results in the equation 6t^2 + 5t - 4 being rewritten as 6t^2 + (5t - 4)(-4).

Next, the remaining terms are factored separately. The first term, 6t^2, is a perfect square and can be factored as (3t)(2t). The second term, (5t - 4)(-4), can be factored by taking out a common factor from each term. In this case, the common factor is (-4). This results in the equation being rewritten as (3t)(2t) + (-4)(5t - 4).

The final step is to group the terms together and factor out the greatest common factor. In this equation, the greatest common factor is (3t)(-4). Thus, the equation 6t^2 + 5t - 4 can be rewritten as (3t)(-4)(2t + 5).

In conclusion, the pair of integers used to rewrite the middle term when factoring 6t^2 + 5t - 4 by grouping is (5t - 4) and (6t^2 + 5t).

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Therefore, the probability of Blake making more than 7 out of 25 shots in practice is 0.9984.

To determine the probability of Blake making more than 7 out of 25 shots in practice, we can use the binomial probability formula: P(X = x) = (n choose x) * p^x * (1-p)^(n-x), where n is the number of trials, x is the number of successes, and p is the probability of success.

In this case, n = 25, p = 0.52, and we want to find the probability of x > 7. We can calculate this by finding the probability of x ≤ 7 and subtracting it from 1.

P(X > 7) = 1 - P(X ≤ 7)

= 1 - [(25 choose 0) * (0.52)^0 * (0.48)^25 + (25 choose 1) * (0.52)^1 * (0.48)^24 + ... + (25 choose 7) * (0.52)^7 * (0.48)^18]

= 1 - 0.0016

= 0.9984

Therefore, the probability of Blake making more than 7 out of 25 shots in practice is 0.9984.

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Answer:

Linear equation to represent the total cost:

Let x be the number of kgs of tomatoes Mail bought, and y be the number of kgs of potatoes Mail bought. The cost of x kgs of tomato at Rs. 50 per kg is 50x, and the cost of y kgs of potato at Rs. 20 per kg is 20y. Therefore, the total cost of the vegetables is:

Total cost = 50x + 20y

Substituting the value of total cost as Rs. 200, we get:

50x + 20y = 200

This is the required linear equation to represent the total cost.

Finding the value of x:

Let's assume that Mail bought x kgs of tomato and 2.5 kgs of potato. Using the equation derived above:

50x + 20(2.5) = 200

Simplifying the equation:

50x + 50 = 200

50x = 150

x = 3

Therefore, Mail bought 3 kgs of tomato.

Finding the point at which the graph of 5x+2y=20 cuts x-axis:

To find the point at which the graph of 5x+2y=20 cuts the x-axis, we need to set y = 0 in the equation and solve for x:

5x + 2(0) = 20

5x = 20

x = 4

Therefore, the point where the graph of 5x+2y=20 cuts the x-axis is (4,0).

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

multiply x by y multiply x by y multiply x by y 3 plus 5 minus 5 to the power of 2 then divide 3 plus 87 then you give your mom and dad a high five then go make out with a girl for an hour and then your answer is 72 to the power of x multiplyed by y.

Answer:

Step-by-step explanation:

W = w

L = 3w

H = w+4

Now V = LHW

          = (3w)(w+4)(w)

         = (3w²+12w)(w)

        = 3w³+12w²

Answer:

Your answer is 4

Step-by-step explanation:

[tex]a^2+b^2=c^2[/tex]

c = 2sqrt(5)

b = 2

a = length

[tex]a^2 = c^2 -b^2\\\\a =\sqrt{c^2-b^2} \\\\a=\sqrt{(2\sqrt{5} )^2-2^2}\\\\a=\sqrt{(2*2*5)-4}\\\\a=\sqrt{20-4}\\\\a=\sqrt{16}\\\\a=4[/tex]

A) The Rational Root Theorem states that the only rational zeros of p(x) = x4+12x-5 must be a factor of -5 divided by a factor of 1. Therefore, the four possible rational zeros are -5, -1, 1, and 5.

B) The other three zeros is 1 - 2i, 1 + 2i, 1 - √(7), 1 + √(7)

A) The Rational Root Theorem states that if a polynomial with integer coefficients has a rational root r = p/q (where p and q have no common factors), then p must divide the constant term of the polynomial and q must divide the leading coefficient.

The constant term of p(x) = x^4 + 12x - 5 is -5, which has the factors ±1 and ±5. The leading coefficient is 1, which has the factors ±1. Therefore, the possible rational roots are:

±1/1, ±5/1, ±1/5, ±5/5 (which simplifies to ±1)

B) If r = 1 - 2i is a zero of p(x), then its complex conjugate r* = 1 + 2i is also a zero of p(x), since p(x) has real coefficients. Therefore, we can factor p(x) as:

p(x) = (x - r)(x - r*)(x²+ bx + c)

where b and c are the coefficients of the quadratic factor. We can expand this and compare coefficients to get:

x⁴ + 12x - 5 = (x - 1 + 2i)(x - 1 - 2i)(x² + bx + c)

Expanding the first two factors gives:

(x - 1 + 2i)(x - 1 - 2i) = x² - 2x + 5

Therefore, we have:

x⁴ + 12x - 5 = (x² - 2x + 5)(x² + bx + c)

Expanding the right side and comparing coefficients, we get:

b = -2 and c = -6

So the quadratic factor is:

x² - 2x - 6

We can find its roots using the quadratic formula:

x = [2 ± √(4 + 4(6))] / 2

x = 1 ± √(7)

Therefore, the four zeros of p(x) are:

1 - 2i, 1 + 2i, 1 - √(7), 1 + √(7)

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Jane can buy 20 melons for the price of 32 apples.

To find out how many melons Jane can buy for the price of 32 apples, we need to use the given ratios to find the equivalent value of melons in terms of apples.

First, we know that 6 apples are equivalent to 3 bananas. So, we can simplify this ratio to 2 apples per 1 banana.

Next, we know that 4 bananas are equivalent to 5 melons. So, we can simplify this ratio to 4/5 of a banana per 1 melon.

Now, we can use these ratios to find the equivalent value of melons in terms of apples. If 2 apples are equivalent to 1 banana, and 4/5 of a banana is equivalent to 1 melon, then we can multiply these ratios together to find the equivalent value of melons in terms of apples:

2 apples/1 banana × 4/5 banana/1 melon = 8/5 apples/1 melon

This means that 1 melon is equivalent to 8/5 apples, or 1.6 apples.

Finally, we can use this ratio to find out how many melons Jane can buy for the price of 32 apples:

32 apples × 1 melon/1.6 apples = 20 melons

Therefore, Jane can buy 20 melons for the price of 32 apples.

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Answer:

19/3

Step-by-step explanation:

The equation of the line satisfying the given conditions. It passes through (1,6); parallel to the line y = 9x - 4 is y = 9x - 3.

To find an equation of the line that satisfies the given conditions, we need to use the slope-intercept form of the equation of a line, which is y = mx + b, where m is the slope and b is the y-intercept.

Since the line we are looking for is parallel to the line y = 9x - 4, it will have the same slope, which is 9. Therefore, the equation of the line we are looking for will be y = 9x + b.

Now, we need to find the value of b. We can do this by plugging in the given point (1,6) into the equation and solving for b:

6 = 9(1) + b
6 = 9 + b
b = -3

So the equation of the line that satisfies the given conditions is y = 9x - 3.

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Answer:

132.7cm²

Step-by-step explanation:

area of circle =  πr²

6.5² * π

=132.7cm²

Sky can run 3 miles per hour faster than her sister rose can walk. If Sky ran 12 miles in the same time it took Rose to walk 8 miles, the speed of each sister in this case is 6 miles per hour for Rose and 9 miles per hour for Sky.

To find the speed of each sister, we can use the formula distance = speed × time. We can set up a system of equations to solve for the speeds of Sky and Rose. Let s be the speed of Sky and r be the speed of Rose. Then we have:

12 = s × t (equation 1)
8 = r × t (equation 2)

We are also told that Sky can run 3 miles per hour faster than Rose, so we can write:

s = r + 3 (equation 3)

Now we can substitute equation 3 into equation 1 and solve for t:

12 = (r + 3) × t
t = 12 / (r + 3)

Next, we can substitute this value of t into equation 2 and solve for r:

8 = r × (12 / (r + 3))
8(r + 3) = 12r
8r + 24 = 12r
24 = 4r
r = 6

So the speed of Rose is 6 miles per hour. We can use equation 3 to find the speed of Sky:

s = 6 + 3
s = 9

So the speed of Sky is 9 miles per hour. Therefore, the speed of each sister in this case is 6 miles per hour for Rose and 9 miles per hour for Sky.

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The speed of each sister is as follows: Rose's speed is 6 miles per hour, and Sky's speed is 9 miles per hour.

Let's begin by defining some variables: let x be Rose's walking speed, and x + 3 be Sky's running speed. Since we know that Sky ran 12 miles and Rose walked 8 miles in the same amount of time, we can write an equation to represent this:

12 / (x + 3) = 8 / x

Cross-multiplying and simplifying gives us:

12x = 8x + 24

4x = 24

x = 6

So Rose's walking speed is 6 miles per hour, and Sky's running speed is 6 + 3 = 9 miles per hour.

Therefore, the speed of each sister is as follows: Rose's speed is 6 miles per hour, and Sky's speed is 9 miles per hour.

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(B) 0.9%. We can use the formula for compound interest to find the value of the investment after 2 years:

A = P(1 + r/n)^(nt)

where A is the amount of money after the time period, P is the principal (initial investment), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

Plugging in the given values, we get:

A = 1000(1 + 0.018/2)^(2*2)

= 1000(1 + 0.009)^4

= 1000(1.009)^4

≈ 1083.02

So the investment is worth approximately $1083.02 after 2 years.

To find the interest rate per year, we can use the formula:

r = n[(A/P)^(1/nt) - 1]

Plugging in the values we know, we get:

r = 2[(1083.02/1000)^(1/(2*2)) - 1]

= 2[(1.08302)^(1/4) - 1]

≈ 0.9%

Therefore, the answer is (B) 0.9%.

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Answer:

The length of the short leg is 2 inches, the length of the long leg is 8 inches, and the length of the hypotenuse is 10 inches.

In a right triangle, if the length of the long leg is 2 inches more than the length of the short leg and the hypotenuse is 8 inches less than three times the length of the short leg, then the length of the short leg is x and the length of the long leg is [tex]x+2[/tex] and the hypotenuse is [tex]3x-8[/tex].

We can use the Pythagorean theorem to solve for the lengths of each side. The Pythagorean theorem states that [tex]a^{2} = b^{2} + c^{2}[/tex], where a and b are the lengths of the legs and c is the length of the hypotenuse.

So, [tex]x^{2} + (x+2)^{2} = (3x-8)^{2}[/tex]

We can solve for x, which gives us x = 6. Therefore, the length of the short leg is 2 inches, the length of the long leg is 8 inches, and the length of the hypotenuse is 10 inches.

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The expression 6x - x + 5 as 6 + 5 are not equivalent expression because one of the expression contain a variable while the other doesn't.

In Mathematics, an expression is sometimes referred to as an equation and it can be defined as a mathematical equation which is typically used for illustrating the relationship that exist between two (2) or more variables and numerical quantities (number).

Based on the information provided above, we have the following mathematical expression:

Expression = 6x - x + 5

Expression = 5x + 5

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Multiply/divide both sides by the same non-zero constant (option C).

Both equations can be considered equivalent (yes).

To solve the equation, we can start by simplifying equation A by opening the brackets:

3(x + 2) = 18

3x + 6 = 18

We can then solve for x by dividing both sides of the equation by a non-zero constant (3):

3x + 6 = 18

3x = 18 - 6

3x = 12

x = 4

From this solution, we can simplify equation A to obtain equation B:

x + 2 = 6

x = 6 - 2

x = 4

Hence, equation B can be obtained from equation A.

Part 2: Equivalent equations are algebraic equations that have the same solutions.

To determine if both equations are equivalent, we can solve them to their simplest form:

Equation A:

3(x + 2) = 18

3x + 6 = 18

3x = 18 - 6

3x = 12

x = 4

Equation B:

x + 2 = 6

x = 6 - 2

x = 4

Since both equations have the same solution, we can conclude that they are equivalent.

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The future value of $2000 earning 18% interest, compounded monthly, for 4 years is $6,116.23.

To calculate the future value, we use the formula:

[tex]FV = P(1 + r/n)^{nt}[/tex]

Where:

FV is the future value

P is the principal amount ($2000)

r is the annual interest rate (18%)

n is the number of times the interest is compounded per year (12 for monthly compounding)

t is the number of years (4)

Plugging in these values, we get:

[tex]FV = 2000(1 + 0.18/12)^{12*4}[/tex]

FV = 6116.23

Therefore, the future value of $2000 earning 18% interest compounded monthly for 4 years is approximately $6,116.23. This means that if you invest $2000 today and earn 18% interest compounded monthly for 4 years, your investment will grow to $6,116.23 at the end of the 4-year period. It's important to note that the actual value may vary depending on the exact compounding method used by the bank or financial institution.

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Answer:

28

Step-by-step explanation:

So for this problem, you need to use the midsegment formula. The midsegment formula is [tex]SG = \frac{1}{2} (b_{1} + b_{2})[/tex]. In this case, [tex]b_{1}[/tex] is 21 and [tex]b_{2}[/tex] 35.

Adding these together get 56. The next part of the formula is dividing by two, so [tex]\frac{56}{2} = 28[/tex] so you answer is x=28

7 x 4 = 28

than you need to do y + 5x

than you get you answer 6

Let's say that the number of metal paperweights is "x", then the number of glass paperweights is "2x" since she has twice as many glass paperweights as metal.

We know that she has a total of 12 paperweights, so we can set up an equation:

x + 2x + 3 = 12

Simplifying the equation, we get:

3x + 3 = 12

Subtracting 3 from both sides, we get:

3x = 9

Dividing both sides by 3, we get:-

x = 3

So Teri has 3 metal paperweights, and since she has twice as many glass paperweights as metal, she has:

2x = 2(3) = 6 glass paperweights.

Therefore, Teri has 3 metal paperweights, 6 glass paperweights, and 3 wood paperweights in her collection.

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To simplify 7-5/6•7-7/6, we need to follow the order of operations (PEMDAS) and perform the multiplication and division before addition and subtraction.

7-5/6•7-7/6

= 7 - (5/6) * 7 - (7/6) (Multiplication first)

= 7 - (35/6) - (7/6) (Simplify the multiplication)

= (42/6) - (35/6) - (7/6) (Convert 7 to a fraction with a common denominator)

= (42 - 35 - 7) / 6 (Subtract the numerators)

= 0 / 6

= 0

Therefore, 7-5/6•7-7/6 simplifies to 0.

Answer: 0

Step-by-step explanation:

7 - 5 / 6 • 7 - 7 / 6

7 - 35 / 6 - 7 / 6

7 - 42 / 6

7 - 7

0

55% of the pins were purple if Omar ordered a set of purple and red pins. He received 60 pins in all. 33 of the pins were purple.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants. An algebraic expression is a mathematical phrase that can contain numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

To find the percentage of purple pins, we need to divide the number of purple pins by the total number of pins and multiply by 100.

Number of purple pins = 33

Total number of pins = 60

Percentage of purple pins = (Number of purple pins / Total number of pins) x 100

= (33 / 60) x 100

= 55%

Therefore, 55% of the pins were purple.

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In the word problem, The account have left $67 in 13 weeks.

Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.

Here the given expression represents the medical savings account.

=> y=-24x+379

Here y represent  of money and x represent number of weeks.                                                                                                                            

Here Amount of money y = $67 then,

=> 67=- 24x+ 379

=> -24x = 67-379

=> -24x= -312

=> x = -312/-24

=> x = 13

Hence The account have left $67 in 13 weeks.

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The value of the function (f·g)(-9) is 186.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

The given functions f(x)=2x²-4x-15 and g(x)=x+12.

Here, (f·g)(x)=f(x)+g(x)

= 2x²-4x-15+x+12

= 2x²-3x-3

(f·g)(-9)=2(-9)²-3(-9)-3

= 2×81+27-3

= 162+27-3

= 162+24

= 186

Therefore, the value of (f·g)(-9) is 186.

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