if you're good at quadratics...

Yes, the number of linearly independent columns in a matrix A is equal to the number of linearly independent rows in A.

This is because the definition of linear independence states that no linear combination of a set of vectors can equal the zero vector, so if one set of vectors is linearly independent, the other set of vectors must also be linearly independent.

To explain further, the number of linearly independent vectors in a matrix A is determined by the rank of the matrix. The rank of a matrix is the number of linearly independent rows and columns in the matrix.

So if the rank of A is n, then the number of linearly independent columns and rows of A must be n. This is because if there are n linearly independent columns in the matrix, the number of linearly independent rows must also be n.

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Answer:

Step-by-step explanation:

[tex]\frac{2}{3}[/tex] +[tex]\frac{3}{4}[/tex]

=[tex]\frac{17}{12}[/tex]

As per the unitary method, Jackson needs 64 tablespoons of lemonade mix to make 4 quarts of lemonade. (option D).

First, we need to convert the 4 quarts of lemonade into ounces since we know the ratio in terms of ounces. One quart is equal to 32 ounces (8 oz per cup and 4 cups per quart). So, 4 quarts would be equal to 128 ounces (32 oz x 4).

Next, we need to determine how much lemonade mix is needed for every 6 ounces of water.

Therefore, the ratio of lemonade mix to water is 3 tablespoons/6 ounces.

Now that we have the ratio, we can use it to find out how much lemonade mix is needed for 128 ounces of water (4 quarts). We can set up a proportion using the ratio we just found:

3 tbsp / 6 oz = x tbsp / 128 oz

We want to solve for x, which represents the amount of lemonade mix needed for 128 ounces of water. To solve for x, we can cross-multiply and simplify:

3 tbsp x 128 oz = 6 oz x

384 tbsp oz = 6x

x = 64 tbsp

So the answer is D) 64 tbsp.

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8 is the maximum number of tablets the pharmacist will tell them they should take of this product per day.

According to FDA guidelines, the maximum daily dosage of acetaminophen for healthy adults should not exceed 4,000 milligrams. This equates to a maximum of eight 500 mg tablets per day. It's essential to remember that the recommended dosage may differ based on individual factors such as age, weight, and medical conditions. Therefore, it's crucial to consult with a healthcare provider or licensed pharmacist before taking any medication. Always follow the recommended dosage and avoid exceeding the daily limit to prevent potential health risks associated with acetaminophen overdose.

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The amount or fraction of work Jerry gets done in 1 hour is 1/3 when it takes him 3 hours to clean all the tank in his section and it take Hart x hours to clean all the tanks in his section.

We know that it takes jerry 3 hours to clean all the tank in his section and it take Hart x hours to clean all the tanks in his section,

when they work together, they can clean the tanks in 1 hour.

now let work be - 1

therefore,

work x Jerry = work x Hart

1×3 = 1×x

3 = 1x

3 = x

therefore in 1 hour work done by Jerry will be 1/3.

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Answer: 23

Step-by-step explanation:

Answer: Always true.

Step-by-step explanation: Use algebra. They end up being the same statements, which means x could be whatever and the equation would still be true.

From the given information provided, the percentage of children who travel by bicycle is 15%.

According to the given information, out of 20 children, 3 of them travel to school by bicycle. To calculate the percentage of children who travel to school by bicycle, we need to divide the number of children who travel by bicycle by the total number of children and multiply by 100.

Percentage of children who travel to school by bicycle = (Number of children who travel by bicycle / Total number of children) x 100%

= (3/20) x 100%

= 15%

Therefore, the percentage of children who travel to school by bicycle is 15%.

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lets use the two given points to form a system of equations and solve for the values of a and b.

Starting with the given equation:

y = ab

At point (2, 60):

60 = ab^(2)

At point (4, 960):

960 = ab^(4)

Now we can solve for a and b using algebraic manipulation.

First, we can divide the second equation by the first equation:

960/60 = (ab^(4))/(ab^(2))

16 = b^(2)

Taking the square root of both sides, we get:

b = 4

Substituting b = 4 into either of the two equations, we can solve for a:

60 = a(4^(2))

60 = 16a

a = 60/16

a = 15/4

Therefore, the values of a and b are:

a = 15/4

b = 4

So the exponential function is:

y = (15/4) * 4^x

"another way, same answer "

The linear function rule to model the cost y for any number of samples x is (y = x + 2.25) and this can be determined by forming the linear equation.

An international food festival charges for admission and for each sample of food.

Admission and 6 samples cost $8.25.

Admission and 8 samples cost $10.25.

The following steps can be used in order to determine the linear function rule to model the cost y for any number of samples x:

Step 1 - The linear equation is formed in order to determine the linear function.

Step 2 - The linear equation that represents the situation "admission and 6 samples cost $8.25" is:

8.25 - 6c = d   --- (1)

Step 3 - The linear equation that represents the situation "admission and 8 samples cost $10.25" is:

10.25 = 8c + d    --- (2)

Step 4 - Substitute the value of 'd' in equation (2).

10.25 = 8c + 8.25 - 6c

2 = 2c

c = 1

Step 5 - Substitute the value of 'c' in the equation (1).

d = 8.25 - 6

d = 2.25

Step 6 - So, the linear function rule to model the cost y for any number of samples x is:

y = x + 2.25.

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Answer:

the area of the photograph when it is put into the frame is 80 square inches, and the area of the frame is 88 square inches.

Step-by-step explanation:

Answer: 36 inches(2)

Step-by-step explanation:

Answer:

its 204cm3

Step-by-step explanation:

2x5x6=60

8x3x6=144

144+60=204

hope this helps!

the baker needs 15 gallons of milk to make 80 chocolate pies for the community festival. To find out how much milk is needed to make 80 chocolate pies, we need to first determine the total amount of milk required for one chocolate pie. We know that the baker needs 3 cups of milk to make one chocolate pie.

To convert cups to gallons, we need to divide by 16 since there are 16 cups in a gallon. So, one chocolate pie requires 3/16 gallons of milk.

To find out how much milk is needed to make 80 chocolate pies, we can multiply the amount of milk required for one pie by the number of pies. This gives us:

(3/16) gallons of milk per pie x 80 pies = 15 gallons of milk

Therefore, the baker needs 15 gallons of milk to make 80 chocolate pies for the community festival.

It's important to make sure we convert units correctly when solving these types of problems. In this case, we had to convert cups to gallons to make sure we were using the same unit of measurement for both the amount of milk needed for one pie and the total amount needed for 80 pies. By following these steps, we can accurately determine the amount of ingredients needed to fulfill an order, which is important for bakers and other food service professionals.

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The percentage of the student body that is at least 20 years old is 50%.

The ages of students at a university are normally distributed with a mean of 20. To find the percentage of the student body that is at least 20 years old solution to the is as follows: As the Mean of ages of students at a university, μ = 20. Age is normally distributed so it will be divided into two groups  Therefore, the mean age is equal to the median age. As the age is normally distributed, 50% of the population will have an age greater than or equal to 20 years old. Thus, the percentage of the student body that is at least 20 years old is 50%. Hence, the required percentage of the student body that is at least 20 years old is 50%.

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Answer:

sin(x) = 9/41

Step-by-step explanation:

The slope of the line through (-2,3) and (1,-3) is -2.

Y = mx + b, where m is the line's slope and b is the y-intercept, is the slope-intercept form of a linear equation (the point where the line intersects the y-axis). This form is helpful because, given a line's equation, it enables us to quickly determine the slope and y-intercept. By charting the y-intercept and utilising the slope to identify additional points on the line, we can also use this technique to graph a line. We can also create an equation for a line given its slope and y-intercept using this approach.

The slope of the line is given as:

slope = (y2 - y1) / (x2 - x1)

Substituting the value of the given coordinates we have:

slope = (-3 - 3) / (1 - (-2))

slope = -6 / 3

slope = -2

Hence, the slope of the line through (-2,3) and (1,-3) is -2.

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The top of the ladder is 12 feet above the ground, the bottom of the ladder will be moving away from the wall at a rate of 0.096 ft/s.

The question is: a 13 foot ladder is leaning against a wall. If the top slips down the wall at a rate of 1.25 ft/s, how fast will the bottom of the ladder be moving away from the wall when the top is 12 feet above the ground?

To answer this question, we can use the concept of right triangle geometry. We know that when the top of the ladder is 12 feet above the ground, the bottom of the ladder is 13 feet away from the wall.

We can use the Pythagorean Theorem to calculate the hypotenuse of this right triangle, which is the total length of the ladder. This will allow us to calculate the rate at which the bottom of the ladder is moving away from the wall.

The Pythagorean Theorem states that

a^2 + b^2 = c^2,

where a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse.

In this case, a is 12 and b is 13.

Therefore,

122 + 132 = c^2, or c^2 = 169.

Therefore, c = 13.04 feet.

This is the total length of the ladder.

We now have the total length of the ladder, so we can calculate the rate at which the bottom is moving away from the wall.

Since we know the top is moving at a rate of 1.25 ft/s, and the total length of the ladder is 13.04 feet, the bottom must be moving at a rate of (1.25/13.04) = 0.096 ft/s.

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The polar curve r = 1.5sinθ in rectangular form is x² + y² - 1.5y = 0

Since we have the polar curve r = 1.5sinθ and we want to convert it to rectangular form, we proceed as follows

We know that in rectangular form the polar curve is

So, multiplying r = 1.5sinθ by r, have that

r = 1.5sinθ

r × r = r × 1.5sinθ

r² = 1.5rsinθ

Substituting this into equation (3), we have that

x² + y² = r² (3)

x² + y² = 1.5rsinθ (4)

From equation (2), y = rsinθ

So, substituting this into equation (4), we have that

x² + y² = 1.5rsinθ

x² + y² = 1.5y

x² + y² - 1.5y = 0

So, the rectangular form is x² + y² - 1.5y = 0  

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With given quadratic equation for time, it will take the rock approximately 4.47 seconds to reach the river.

What exactly are quadratic equations?

Quadratic equations are equations of the second degree, meaning they involve terms raised to the power of two. These equations are written in the form of ax² + bx + c = 0, where x is the unknown variable, and a, b, and c are constants (numbers).

The graph of a quadratic equation is a curve called a parabola, which can be either upwards or downwards depending on the sign of the coefficient a. Quadratic equations can have one, two, or zero real solutions, depending on the discriminant b² - 4ac.

Now,

In this equation, h represents the height of the rock above the river at time t in seconds. When the rock hits the river, its height h will be zero. So we can set h to zero and solve for t:

h = -16t² + 320

0 = -16t² + 320 (substituting h = 0)

16t² = 320

t² = 20

t = ±√20

Since time cannot be negative, we take the positive square root:

t = √20 ≈ 4.47 seconds

Therefore, it will take the rock approximately 4.47 seconds to reach the river.

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Answer:

28 is NOT a given solution to the equality

Step-by-step explanation:

For this inequality, we substitute 28 for t and see if the left side is less than the right side.

[tex]t - 7 < 10, t = 28\\(28) - 7 < 10\\21 < 10 \\False[/tex]

The statement is false, therefore the number provided is not a solution

Therefore, the value of θ for the acute angle in a right triangle is 46°.We can use the fact that the sine and cosine of complementary angles are equal to find θ.

To find the value of θ for the acute angle in a right triangle, we need to use one of the trigonometric ratios. In this case, we are given the sine of θ and the cosine of 44°.  Recall that the sine of an angle θ is defined as the ratio of the opposite side to the hypotenuse in a right triangle:

sin(θ) = opposite / hypotenuse

And the cosine of an angle φ is defined as the ratio of the adjacent side to the hypotenuse:

cos(φ) = adjacent / hypotenuse

In a right triangle, the two acute angles are complementary, which means that their sum is 90°. That is,θ + 90° = 90°

θ = 90° - 44°.Now we can use the given equation sin(θ) = cos(44°) and substitute θ: sin(90° - 44°) = cos(44°) => sin(46°) = cos(44°)

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Answer:

$32.50

Step-by-step explanation:

20% * y = 6.5 =>

y =6.5 ÷ 20% =

6.5 ÷ (20 ÷ 100) =

(100 × 6.5) ÷ 20 =

650 ÷ 20 =

32.5

Percentage of 20% of what number = 6.5?

20% × ? = 6.5

20% of 32.5 = 6.5

Answer: y = -3x + 7

Step-by-step explanation:

slope = -3

y intercept = 7

In the following question, Yes, there is sufficient evidence to support the claim that the computer-aided instruction (CAI) technique performs differently than the traditional method.

To determine this, a two-tailed hypothesis test can be performed using the sample data given. The null hypothesis is that the computer-aided instruction technique has the same mean as the traditional method (101 hours). The alternative hypothesis is that the computer-aided instruction technique has a different mean than the traditional method (101 hours). With a sample mean of 100 hours, a sample size of 140 students, a known population standard deviation of 6 hours, and a level of significance of 0.01, the hypothesis test is conducted and results in a p-value of less than 0.01. This suggests that there is sufficient evidence to reject the null hypothesis and support the alternative hypothesis, that the computer-aided instruction technique performs differently than the traditional method. The conclusion is that the computer-aided instruction technique has a different mean than the traditional method.

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There is sufficient evidence to support the claim that the new license training method using computer-aided instruction performs differently than the traditional method. Therefore, the researcher can proceed with the new method.

Hypothesis testing is a technique used to make inferences about the population based on the sample data. It is divided into two types, one-tailed and two-tailed. A two-tailed test involves testing for the difference between the two population means, while a one-tailed test involves testing for a higher or lower population mean. A level of significance is predetermined to accept or reject the null hypothesis.
Given, using traditional methods, it takes 101 hours to receive a basic flying license. A new license training method using computer-aided instruction (CAI) has been proposed. A researcher used the technique with 140 students and observed that they had a mean of 100 hours. Assume the standard deviation is known to be 6.
We have to determine if there is sufficient evidence to support the claim that the technique performs differently than the traditional method at a level of significance of 0.01.
Null Hypothesis H₀: μ₁ = μ₂  (The new technique and traditional method do not perform differently.)
Alternate Hypothesis H₁: μ₁ ≠ μ₂  (The new technique and traditional method perform differently.)
The test statistic for testing the difference in two population means with known standard deviation is given by:
z = (x₁ - x₂) / (σ/√n)
Where,
x₁ = sample mean of the new technique
x₂ = sample mean of the traditional method
σ = known standard deviation
n = sample size
z = test statistic
z = (100 - 101) / (6 / √140)
z = -2.624
The corresponding p-value can be obtained from the Z-table.
p-value = P(z < -2.624) + P(z > 2.624)
p-value = 0.0088 + 0.0088
p-value = 0.0176
The level of significance α = 0.01
Since p-value < α, we can reject the null hypothesis.
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Answer:

Compound angle identities are trigonometric identities that relate the trigonometric functions of the sum or difference of two angles to the trigonometric functions of the individual angles.

The main compound angle identities are:

1. sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)

2. cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)

3. tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))

where A and B are any two angles.

These identities can be used to simplify trigonometric expressions, solve trigonometric equations, and prove other trigonometric identities.

For example, we can use the compound angle identity for sin(A + B) to find sin(π/4 + π/6):

sin(π/4 + π/6) = sin(π/4)cos(π/6) + cos(π/4)sin(π/6)

sin(π/4 + π/6) = (√2/2)(√3/2) + (√2/2)(1/2)

sin(π/4 + π/6) = (√6 + √2) / 4

Compound angle identities are an important part of trigonometry and are used in various fields such as physics, engineering, and mathematics.

Answer: 30 album pages

Step-by-step explanation:

cards = 10

teams = 24

total cards = 240

240 / 8 = 30

The equatiοn οf the line οf best fit fοr this data-set is given as fοllοws:

y = 10x + 60.

The slοpe-intercept definitiοn οf a linear functiοn is given as fοllοws:

y = mx + b.

In which the cοefficients are given as fοllοws:

m is the slοpe, representing the rate οf change οf the οutput οf the functiοn relative tο the input.

b is the y-intercept, representing the value assumed by the functiοn the input assumes a value οf zerο.

The functiοn gοes thrοugh pοint (0,60), hence the intercept b οf the line οf fit is given as fοllοws:

b = 60.

When x increases by 2, frοm 0 tο 2, y increases by 20, frοm 60 tο 80, hence the slοpe m is οbtained as fοllοws:

m = 20/2 = 10.

Hence the line οf fit is given as fοllοws:

y = 10x + 60.

Meaning that the third οptiοn is cοrrect.

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Complete question:

Data was cοllected οn the amοunt οf time that a randοm sample οf 8 students spent studying fοr a test and the grades they earned οn the test. A scatter plοt and line οf fit were created fοr the data.scatter plοt titled students' data, with pοints plοtted at 1 cοmma 75, 2 cοmma 70, 2 cοmma 80, 2 cοmma 90, 3 cοmma 80, 3 cοmma 100, 4 cοmma 95, and 4 cοmma 100, and a line οf fit drawn passing thrοugh the pοints 0 cοmma 60 and 2 cοmma 80

Determine the equatiοn οf the line οf fit.

y = 5x + 60

y = 5x + 80

y = 10x + 60

y = 10x + 80  

Answer:

The equatiοn οf the line οf best fit fοr this data-set is given as fοllοws:

y = 10x + 60.

Step-by-step explanation:

The width of the bookshelf will be 6 inches when the area and length are 216 in and  36 inches.

What is the width?

Width is a dimension of a geometrical figure that tells about the structure of the figure. In general, width is used to measure the object.

What is formulae of width?

Width is obtained by dividing area and length.

                               W=A/L

substitute values, 216/36= 6inches.

The width is gonna be 6 because all you have to do is divide 216 and 36

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The equation to  the given graph is y= (1/2) * 2ˣ. So correct option is B.

A parabola is a type of curve that results from the intersection of a cone with a plane that is parallel to one of its sides. It is a conic section, which means it is a geometric shape that can be formed by cutting a cone with a plane.

The shape of a parabola is similar to that of a U-shaped curve, but it is symmetric around a vertical line called the axis of symmetry. The point where the axis of symmetry intersects the parabola is called the vertex.

The equation of a parabola can be written in different forms, but the most common form is:

y = ax^2 + bx + c

where "a", "b", and "c" are constants that determine the shape, position, and orientation of the parabola. If "a" is positive, the parabola opens upward, and if "a" is negative, it opens downward.

We can observe that the given graph passes through the points (-1,1), (0,2), (1,4), (2,8). This suggests that the graph is a parabola that opens upwards.

We can use the general form of a quadratic equation to write the equation of the parabola in the form y = ax² + bx + c, where a, b, and c are constants. Since the parabola opens upwards, we know that the coefficient of the x² term is positive.

To find the values of a, b, and c, we can substitute the coordinates of the given points into the equation and solve the resulting system of equations.

Using the point (-1,1), we get:

1 = a(-1)² + b(-1) + c

1 = a - b + c

Using the point (0,2), we get:

2 = a(0)² + b(0) + c

2 = c

Using the point (1,4), we get:

4 = a(1)² + b(1) + c

4 = a + b + c

Using the point (2,8), we get:

8 = a(2)² + b(2) + c

8 = 4a + 2b + c

Substituting c = 2 from the second equation into the third and fourth equations, we get:

4 = a + b + 2

8 = 4a + 2b + 2

Simplifying these equations, we get:

a + b = 2

2a + b = 3

Solving for a and b, we get:

a = 1

b = 1

Therefore, the equation of the parabola is y = x² + x + 2.

To check which of the given options matches this equation, we can simplify the equation:

y = x² + x + 2

y = (x + 1/2)² + 7/4

This shows that the given parabola is a vertical shift of the standard parabola y = x² by 7/4 units upward.

Out of the given options, the only equation that matches this form is option B, y = (1/2) * 2ˣ. Therefore, the answer is B) y= (1/2) * 2ˣ.

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