In a chess variant, a "lord" can move one space at a time, either upward, or to the right, or diagonally up and to the right. how many ways are there for a lord to move from the bottom left to top right corner of the 8 by 8 chessboard?

Final answer:

An independent system of linear equations has two lines that intersect at exactly one point. If the slope of one line is the negative reciprocal of the other, the lines are perpendicular, indicating they intersect once and the system is independent.

Explanation:

If the slope of one line in a system of linear equations is the negative reciprocal of the slope of the other line, this indicates that the two lines are perpendicular to each other. Perpendicular lines intersect at one point and therefore, the system of equations is independent. An independent system has a single unique solution, meaning that there is exactly one point where the two lines cross.

In terms of graphs and equations, the slope or 'm' in the equation y = mx + b represents the rate at which the dependent variable changes with respect to the independent variable. The negative reciprocal relationship between slopes of two lines means that one will rise (positive slope) and fall (negative slope) at the same rate but in opposite directions, ensuring they intersect just once unless the y-intercepts (b) are also identical, which would make them the same line (dependent system) rather than intersecting lines.

Simultaneous equations can be solved if the number of unknown variables are equal to the number of equations. In the geometry question, the equations are given by the relationship between the lines and angles

Part A

m = 7°, p = 10°, t = 5°, a = 6, and s = 2°

Part B

Arranging the variables from least to greatest gives; stamp

A stamp sits at the corner of an envelope but travels around the world

Part A

From the diagram, we have;

H is a vertical angle to the 90° angle

(7·m + 3)° + 90° + (8·m - 18)° = 180° (linear angles)

∴ (7·m + 3)° + (8·m - 18)°  = 180° - 90° = 90°

(15·m - 15)° = 90°

15·m = 90°+ 15° = 105°

m = 105°/15 = 7°

m = 7°

(7·m + 3)° = (5·p + 2)° (vertically opposite angle theorem)

∴ (5·p + 2)° = (7 × 7 + 3)° = 52°

p = (52° - 2°)/5 = 10°

p = 10°

(8·m - 18)° = (11·t - 17)° (vertically opposite angle theorem)

∴ (8 × 7 - 18)° = (11·t - 17)°

38° = (11·t - 17)°

11·t = 38° + 17° = 55°

t = 55°/11 = 5°

t = 5°

CD = 5·a + 12 and DE = 9·a - 12 are congruent segments

5·a + 12 = 9·a - 12 (Definition of congruent segments)

9·a - 5·a = 12 + 12 = 24

4·a = 24

a = 24/4 = 6

a = 6

(21·s + 6° and 48° are congruent congruent angles

21·s + 6° = 48° (Definition of congruent angles)

∴ s = (48° - 6°)/21 = 2°

s = 2°

Part B

Arranging the values of the variables from least to most gives;

s = 2°

t = 5°

a = 6

m = 7°

p = 10°

Which gives;

The answer to the riddle = s, t, a, m, p = Stamp

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The y-intercept tells us where a line intersects the y-axis and the rate of change (slope) shows how much 'y' changes for a unit change in 'x'. Given these, we can identify different line characteristics based on whether the lines have the same or different y-intercepts and slopes.

Let's start by understanding the concepts of y-intercepts and rate of change. The y-intercept describes where a line intersects the y-axis. For example, in Figure A1, the y-intercept is 9, which means the line crosses the y-axis at y=9. Similarly, the rate of change, also known as the slope, indicates how much 'y' changes for each change in 'x'. A straight line will have the same slope at all points along the line, as illustrated in the example of Figure A1 where the slope is 3.

Now, let's compare the items:
A. When the y-intercepts are the same but the rates of change are different, the lines start at the same point on the y-axis, but diverge as 'x' increases because the slope of each line is different.
B. Different y-intercepts and different rates of change means that the lines start at different points on the y-axis and vary in how steeply they climb or fall.
C. When items have the same y-intercept and the same rate of change, these lines are coincident; they will overlap perfectly because they have the same starting point on the y-axis and ascend or descend at the same rate.
D. When the rates of change are the same but the y-intercepts are different, the lines are parallel. They rise or fall at the same rate, but they do not start at the same point on the y-axis.

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The y-intercepts are the same, but the rates of change are different. The option (C) is correct.

To compare the y-intercepts and the rates of change of the given linear equation and the data in the table, we need to first understand each component:

  - The slope (rate of change) is the coefficient of [tex]\( x \),[/tex] which is [tex]\( 2 \).[/tex]

  - The y-intercept is the constant term, which is [tex]\( -4 \).[/tex]

Let's find the rate of change (slope) and the y-intercept for the data in the table.

The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\) and \((x_2, y_2)\)[/tex] is calculated as:

[tex]\[m = \frac{y_2 - y_1}{x_2 - x_1}\][/tex]

We'll calculate the slope using the points [tex]\((x, y)\)[/tex] from the table:

Using the points [tex]\((-4, -6)\) and \((-2, -5)\):[/tex]

[tex]\[m = \frac{-5 - (-6)}{-2 - (-4)} = \frac{-5 + 6}{-2 + 4} = \frac{1}{2}\][/tex]

Using the points [tex]\((-2, -5)\) and \((0, -4)\):[/tex]

[tex]\[m = \frac{-4 - (-5)}{0 - (-2)} = \frac{-4 + 5}{0 + 2} = \frac{1}{2}\][/tex]

We can see the rate of change is consistently [tex]\( \frac{1}{2} \).[/tex]

We can use the slope-intercept form [tex]\( y = mx + b \)[/tex]. We know [tex]\( m = \frac{1}{2} \)[/tex] and we can use any point from the table to find [tex]\( b \).[/tex]

Using the point [tex]\((0, -4)\):[/tex]

[tex]\[y = \frac{1}{2}x + b \\-4 = \frac{1}{2}(0) + b \\-4 = b\][/tex]

So the y-intercept [tex]\( b \)[/tex] is [tex]\( -4 \).[/tex]

Rate of change (slope):

 - Equation [tex]\( y = 2x - 4 \)[/tex] has a slope of [tex]\( 2 \).[/tex]

 - Table data has a slope of [tex]\( \frac{1}{2} \).[/tex]

Y-intercept:

Both the equation and the table data have a y-intercept of [tex]\( -4 \).[/tex]

Therefore, the correct comparison is (C) The y-intercepts are the same, but the rates of change are different.

The complete question is:

Compare the y-intercepts and the rates of change of the following items.

(A) The items have different y-intercepts and different rates of change.

(B) The rates of change are the same, but the y-intercepts are different.

(C) The y-intercepts are the same, but the rates of change are different.

(D) The items have the same y-intercept and the same rate of change.

Answer:

[tex]y = \frac{48}{x}[/tex]

Step-by-step explanation:

Inverse variation:

if [tex]y \propto \frac{1}{x}[/tex]

then the equation is in the form of:

[tex]y = \frac{k}{x}[/tex]                 ....[1]

where, k is the constant of variation.

As per the statement:

When x = 3, y = 16 and when x = 6, y = 8.

Substitute the value of x and y to find k.

Case 1.

When x = 3, y = 16

then;

[tex]16=\frac{k}{3}[/tex]

Multiply by 3 both sides we have;

48 = k

or

k = 48

Case 2.

When x = 6, y = 8

then;

[tex]8=\frac{k}{6}[/tex]

Multiply by 6 both sides we have;

48 = k

or

k = 48

In both cases, we get constant of variation(k) = 48

then the equation we get,

[tex]y = \frac{48}{x}[/tex]

Therefore, the inverse variation equation can be used to model this function is, [tex]y = \frac{48}{x}[/tex]

Answer:

The perimeter of a face of the new cube is:

108 cm

Step-by-step explanation:

Original edge length of cube=9 cm

Increased edge length of the cube=9×3 cm

                                                       =27 cm

The perimeter of a face of a cube=4s

where s is the edge length of the cube

Perimeter of a face of new cube=4×27 cm

                                                    = 108 cm

Hence, the perimeter of a face of the new cube is:

108 cm

We have to write 5.378 E9 in standard form.

We can write  5.378 E9 as [tex]5.378\times 10^9[/tex]

Now, in order to write it in standard form we can write [tex]10^9=1000000000[/tex]

Now, we can multiply 5.378 with 1000000000

[tex]5.378\times 1000000000\\=5378000000[/tex]

Therefore, the standard form of 5.378 E9 is given by 5378000000

By saving double the amount saved on the previous day starting with $2 in 8 days Joey will have $500.

An arithmetic sequence is a set of numbers in which every no. next to the previous number has the same common difference

d = aₙ - aₙ₋₁ = aₙ₋₁ - aₙ ₋₂.

In a geometric sequence numbers are written in the same constant ratio(r).

It means every next number is a multiple of a common constant and the previous number.

r = aₙ/aₙ₋₁ = aₙ-₁/aₙ₋₂.

Given, Joey saved $2 today. If he doubles the number of dollars he saves each day.

This can be formed as a geometric sequence with common ratios(r) = 2.

a₁ = 2, We know the sum of a geometric series is [tex]S_n = \frac{a_1(r^n - 1)}{r - 1}[/tex].

∴ 500 = 2(2ⁿ - 1)/(2 - 1).

2ⁿ - 1 = 250.

2ⁿ = 251.

[tex]log_22^n = log_2251[/tex].

n = 7.97 Or approximately in 8 days.

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Answer:

182.77

Step-by-step explanation:

gradpoint

Answer:

1) distributive property

2) subtraction property of equality

3) division property of equality

Step-by-step explanation:

Given : If [tex]\frac{1}{2}(8x+14)=39[/tex] , then x=8

To complete the proof :

Solution :

Following are the complete steps or proof of solving the given expression,

[tex]\frac{1}{2}(8x+14)=39[/tex]  is given

[tex]4x + 7 = 39[/tex] - distributive property

[tex]4x = 32[/tex] - subtraction property of equality

[tex]x = 8[/tex] - division property of equality

The correct answer is A. Crossing the street.

Explanation

Being an observer refers to observing or looking at every detail of the surrounding environment or situation. This implies when you are an observer you understand a situation by noticing and understanding every detail. In this task, individuals mainly use their sense of sight, which allows individuals to understand visually elements, although observing often involves other senses such as hearing. This is necessary in the cases we need to remember information, perform precise actions, among others.

According to this, it is necessary to be an observer when crossing a street because this task or process requires concentration and precise actions and by observe you make sure that there is no danger of being hit by a car before crossing. So, the correct answer is A. Crossing the street

The correct answers are a,b,e,f

By applying the Intermediate Value Theorem to the function f(x)=x^4+6x^3-x^2-30x+4, we find that the intervals containing at least one zero, or root, are [-4,-3], [-3,-2], [-1,0], and [0,1].

In mathematics, particularly in calculus, the Intermediate Value Theorem (IVT) is used to show that a given interval has at least one zero, or root, of a function. In this case, the function is f(x)=x^4+6x^3-x^2-30x+4.

To apply IVT, we calculate the function value at the endpoints of the intervals. If the function values at the endpoints of an interval have different signs, then by IVT, there is at least one zero in that interval.

A change in sign between the function values at the endpoints of an interval indicates there is at least one root in that interval. Thus, by checking the signs of the consecutive function values, we can conclude that the intervals which contain at least one root are [-4,-3], [-3,-2], [-1,0], and [0,1].

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The gradient of the line y=4x+7 is 4 and the y-intercept is 7. This line rises by 4 units vertically for every 1 unit it moves horizontally, and it intersects the y-axis at the point (0,7).

In the given line equation y=4x+7, the coefficient of x is the gradient (or slope) of the line and the constant term is the y-intercept. So, in this case, the gradient of the line is 4 and the y-intercept is 7. This means the line rises by 4 units vertically for every 1 unit it moves horizontally, and it intersects the y-axis at the point (0,7).

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Answer:

The correct option is 3.

Step-by-step explanation:

The table of values is

Number of days (x)   Rent (dollars) (y)

               0                         12

               1                          21

               2                         30

               3                         39

               4                         48

It is given that Thomas wants to rent a lawn mower. he has to pay a fixed base cost plus a daily rate for renting the lawn mower.  It means the relationship between x and y is linear.

The slope intercept form of a line is

[tex]y=mx+b[/tex]              .... (1)

where, m is slope and b is y-intercept.

Slope formula:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

The line passes through (0,12) and (1,21). The slope of the line is

[tex]m=\frac{21-12}{1-0}=9[/tex]

The slope of the line is 9. From the given table it is clear that the y-intercept or initial value of the function is 12.

Substitute m=9 and b=12 in equation (1).

[tex]y=9x+12[/tex]

The required equation is y = 9x + 12. Therefore the correct option is 3.

Answer:

Too little

Step-by-step explanation:

Final answer:

Tracy's employer pays $23.75 each month towards her life insurance premium. The annual value of this benefit to Tracy is $285, calculated by multiplying the monthly employer contribution by 12.

Explanation:

The question is asking to calculate the annual value of the benefit that Tracy receives from her employer for her life insurance. Tracy pays $8.50 monthly and her employer covers the remainder, making the total monthly premium $32.25. To find the annual value of the benefit, we need to first determine how much her employer pays each month and then multiply this by 12 to get the annual value.

First, we subtract Tracy's contribution from the total premium:
$32.25 - $8.50 = $23.75

This is the monthly contribution by the employer. Now, to find the annual value, we multiply the monthly employer contribution by 12:
$23.75 × 12 = $285. Therefore, the annual value of the benefit that Tracy receives is $285.