prime numbers divided by 4 remainder 1 proven true

Answer:

Numbers: 11, 7 and 44

Expressions: n, n - 4, 4n

Step-by-step explanation:

We have that A + B + C = 62, where A, B and C represent numbers.

We know that the 2nd number (B) is equal to the first number (A), diminished by 4. This means B = A - 4  

We also know that the 3rd number (C) is four times the first (A), that is to say 4*A. This means C = 4*A

If we replace this information in the first expression, we have

A + B + C = A + (A - 4) + (4*A) = 62

We change an equation of 3 variables to an equation of just one. Now we solve it putting together the terms that have the variable A:

A + A - 4 + 4*A = 62

2A + 4A - 4 + 4 = 62 + 4

6A = 66

A = 11

Our first number is 11, now we find B and C:

B = A - 4 = 11 - 4 = 7

C = 4A = 4*11 = 44

So, the numbers are 11, 7 and 44

If we change the A for a "n", the expressions that represent the numbers are n, n - 4 and 4n

Final answer:

The probability that at least two more coin tosses are required to get the first head, after ten tails in a row with a balanced coin, is 50%.

Explanation:

The student is asking about the probability of a specific outcome in a sequence of coin tosses. Given that we have already tossed a balanced coin ten times and obtained zero heads, we want to calculate the probability of having to toss the coin at least two more times to get the first head. The probability of getting a head on any given coin toss is 0.5. To get the first head in at least two more tosses means we need a tail (T) on the next toss and then any outcome (H or T) after that. Therefore, we calculate the probability as follows:

The combined probability is the product of the two individual probabilities, which is 0.5 * 1 = 0.5 or 50%. Hence, the probability that we must toss the coin at least two more times to obtain the first head is 50%.

The value of a in the point (a, 7) is 3

The slope intercept equation is as follows:

y = mx + b

where

m = slope

b = y-intercept

Therefore,

Let's find b using (6, -2)

m = -3

y = -3x + b

-2 = -3(6) + b

-2  = -18 + b

b = -2 + 18

b = 16

Therefore,

y = -3x + 16

let's find the value of a in (a, 7)

7 = -3x + 16

7 - 16 = -3x

-9 = -3x

x = -9 / -3

x = 3

Therefore,

a = 3

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The average rate of speed of Miguel when he biked on Saturday given the distance and time is 18 kilometers per hour.

Average speed is the rate at which an object moves with respect to time. Average speed is determined by dividing total distance travelled by time.

Average speed = total distance / time

36 / 2  = 18 kilometers per hour.

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12/3 = 4

 it was dilated with a scale factor of 4

Final answer:

The sentence 'The sum of 14 and Holly's age is 55' can be translated into the equation 14 + h = 55, where h represents Holly's age.

Explanation:

To translate the sentence into an equation, we identify the mathematical operations described by the sentence. The sentence 'The sum of 14 and Holly's age is 55' indicates an addition operation involving Holly's age and the number 14. Using the variable h for Holly's age, we form the equation 14 + h = 55. This is a simple linear equation where h represents an unknown value that can be solved for.

The minimum value of the function y=3x^2-12x+10 is found using the vertex form, and it is -2. This value corresponds to the y-coordinate of the vertex of the parabola.

To find the minimum value of the function y=3x^2-12x+10, we can use the vertex form of a quadratic equation, or complete the square method. We also know that a quadratic function with a positive coefficient for the x^2 term opens upwards and has a minimum value at its vertex. To get the vertex form, we complete the square:

y = 3(x^2 - 4x) + 10
y = 3(x^2 - 4x + 4 - 4) + 10
y = 3((x-2)^2 - 4) + 10
y = 3(x-2)^2 -12 + 10
y = 3(x-2)^2 - 2

Now, the vertex of this parabola is at (2, -2). As this is the vertex of a parabola that opens upwards, the y-coordinate of this point gives us the minimum value of the function, which is -2.

Final answer:

The determinant of a matrix is 0 means the matrix is singular, indicating no unique solution for the corresponding system of equations, or in geometry, that space is compressed to a lower dimension.

Explanation:

When the determinant of a matrix is 0, it signals that the matrix does not have an inverse and is considered singular. In practical terms, this means that the system of equations that corresponds to the matrix does not have a unique solution. For instance, in a square matrix, if the determinant is zero, it may indicate that the rows or columns are linearly dependent, meaning that at least one row or column is a scalar multiple of another, resulting in redundant information. In geometry, if we're dealing with transformation matrices, a determinant of zero implies that the transformation compresses the space into a lower dimension, effectively 'flattening' it, and thus, the original geometrical figure loses its volume.