Explanation

Let's calculate the value of 3x – 4y – z using x = 2, y = -3, z = -5:

3(2) – 4(-3) – (-5)

= 6 + 12 + 5

= 23

Explanation

A prime number has only two distinct factors, 1 and itself (e.g., 2, 3, 5, 7).

It cannot be divided by any other number without leaving a remainder.

Explanation

The arithmetic mean (average) is calculated by dividing the sum of the numbers by the count of numbers.

Given:

Mean = 77

Count of numbers = 13

Sum of numbers = Mean × Count of numbers

= 77 × 13

= 1001

Explanation

To add these numbers, convert them to base 10, add, and then convert back to base 5:

(34)₅ = 3 × 5¹ + 4 × 5⁰ = 15 + 4 = 19

(44)₅ = 4 × 5¹ + 4 × 5⁰ = 20 + 4 = 24

Add the base 10 equivalents:

19 + 24 = 43

Convert the sum back to base 5:

43 = 3 × 5² + 1 × 5¹ + 3 × 5⁰

= (133)₅

Explanation

To find the value of k, use the Factor Theorem:

If a polynomial f(x) is completely divisible by (x - a), then f(a) = 0.

In this case, the polynomial is:

f(x) = 3x³ + 9x² - (3k - 4)x + 2

Since it's divisible by (x - 1), substitute x = 1:

f(1) = 3(1)³ + 9(1)² - (3k - 4)(1) + 2

= 3 + 9 - 3k + 4 + 2

= 18 - 3k

Set f(1) = 0:

18 - 3k = 0

Solve for k:

3k = 18

k = 6

Explanation

The characteristic of a decimal number in scientific notation is the exponent of 10 when expressed as N × 10^n.

Converting 0.006735 to scientific notation: 6.735 × 10⁻³, so the characteristic is -3.

Explanation

3, 4, and 5

Since all three conditions of the Triangle Inequality Theorem are satisfied:

3 + 4 = 7 > 5

3 + 5 = 8 > 4

4 + 5 = 9 > 3

Explanation

To find the range of the binary relation R, substitute each value of x from the domain {0, 4, 8} into the equation y = -2x + 1:

For x = 0:

y = -2(0) + 1

= 1

For x = 4:

y = -2(4) + 1

= -7

For x = 8:

y = -2(8) + 1

= -15

The corresponding y-values are {1, -7, -15}. Adding 2 to each element in the set {-1, -9, -17} gives the range.

Explanation

Given the quadratic equation:

x² - 7x + k = 0

Using the sum and product of roots formula:

Sum of roots: α + β = -b/a = 7/1 = 7

Product of roots: αβ = c/a = k/1 = k

So, the value of 𝑘 is the product of the roots, which is:

𝑘 = 12

Explanation

To find the roots of the quadratic equation x² - 7x - 9 = 0:

Use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

In this case, a = 1, b = -7, and c = -9:

x = (7 ± √((-7)² - 4(1)(-9))) / 2(1)

= (7 ± √(49 + 36)) / 2

= (7 ± √85) / 2

The two roots are:

x₁ = (7 + √85) / 2

x₂ = (7 - √85) / 2

The positive difference between the roots is:

|x₁ - x₂| = |((7 + √85) / 2) - ((7 - √85) / 2)|

= |(7 + √85 - 7 + √85) / 2|

= |2√85 / 2|

= √85