6. Prove that linear functions grow by equal differences over equal intervals.
Part I. This is the graph of. Use the graph to show that equal intervals of x-values have equal differences
of y-values.
a) Think about an interval on the x-axis starting with p and ending with p + k. What is the difference
between the x-values? What is the difference between the y-values for these x-values? Complete the
tables using.

Answer:

Slope = -7/3

Step-by-step explanation:

-7x = 6 + 3y is in the standard form of a line, whose general equation is

Ax = C + By (it's sometimes written in terms of C and is Ax + By = C, but in this problem, it's written in terms of Ax).

We can find the slope of the line by converting from standard form to slope-intercept form, whose general equation is y = mx + b, where

Step 1:  Subtract 6 from both sides:

(-7x = 6 + 3y) - 6

-7x - 6 = 3y

Step 2:  Divide both sides by 3 to isolate y:

(-7x - 6 = 3y) / 3

-7/3x - 2 = y

Thus, the slope of the line is -7/3

Answer:  Therefore the slope is [tex]-\frac{7}{3}[/tex].

Step-by-step explanation:

We can rewrite the equation -7x=6+3y in slope-intercept form y = mx + b, where the m is the slope of the line, and b is the y-intercept.

-7x = 6 + 3y

-6     -6        

-7x - 6 = 3y

[tex]\frac{-7x}{3}-\frac{6}{3} =\frac{3y}{3}[/tex]

[tex]\frac{-7}{3}x-2 =y[/tex]

Therefore the slope is [tex]-\frac{7}{3}[/tex].

The road trip will take approximately 5 hours.

Two one-step equations:

a) 2x + 5 = 13

In this equation, the variable 'x' represents an unknown number. By performing one step of subtraction, we can find the value of 'x' that makes the equation true.

The solution is x = 4.  

b) 3y - 7 = 16

Similar to the first equation, 'y' represents an unknown number.

By adding 7 to both sides of the equation, we can isolate the variable and solve for 'y.'  

The solution is y = 7.

Two equations with fractions:

a) (1/3)x + 2 = 5

Here, the variable 'x' is multiplied by a fraction.

To isolate 'x,' we can subtract 2 from both sides and then multiply both sides by the reciprocal of 1/3, which is 3/1.

The solution is x = 9.

b) (2/5)y - 3 = 1

In this equation, 'y' is multiplied by a fraction.

We can isolate 'y' by adding 3 to both sides and then multiplying both sides by the reciprocal of 2/5, which is 5/2.

The solution is y = 4.

One equation with the distributive property:

a) 2(x + 3) = 10

This equation demonstrates the distributive property.

By applying it, we multiply 2 by both x and 3, resulting in 2x + 6 = 10.

We can then solve for 'x' by subtracting 6 from both sides.

The solution is x = 2.

One equation with decimals:

a) 0.4x + 0.8 = 1.6

In this equation, 'x' is multiplied by a decimal.

To isolate 'x,' we subtract 0.8 from both sides and then divide both sides by 0.4.

The solution is x = 2.

Real-world problem:

Imagine you're planning a road trip.

The distance you'll be traveling is 250 miles, and your car's average speed is 50 miles per hour.

You want to determine how long the trip will take.

Let 't' represent the time in hours it will take to complete the trip.

The equation that represents this situation is:

50t = 250

By dividing both sides of the equation by 50, we find that t = 5.

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The figures are similar because the ratio of segment BC to segment EF is equal to the of segment AB to segment DE.

The scale factor is equal to 3/2.

In Mathematics and Geometry, two (2) triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Additionally, the lengths of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.

Based on the side, angle, side (SAS) similarity theorem, we can logically deduce the following:

Scale factor = BC/EF = AB/DE

Scale factor = 9/6 = 6/4

Scale factor = 3/2 = 3/2 (similar)

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Answer:

[tex](y - 4)^2 = -8(x - 2).[/tex]

Step-by-step explanation:

The equation of a parabola with a vertical axis of symmetry, vertex (h, k), and focus (h+a, k) is given by:

[tex](y - k)^2 = 4a(x - h)[/tex]

In this case, the vertex is (2, 4) and the focus is (0, 4).

Comparing this to the general equation, we have h=2, k=4, and h+a=0.

From h+a=0, we can solve for a:

a=-h = -2

Substituting the values of h, k, and p into the equation, we get:

[tex](y - 4)^2 = 4(-2)(x - 2)[/tex]

Simplifying further:

[tex](y - 4)^2 = -8(x - 2)[/tex]

Therefore, the parabola equation is[tex](y - 4)^2 = -8(x - 2).[/tex]

The solution set which include 3 are x > 1.4, x < 5.1 and x < 8.2.

Given the inequalities that include 3 in the following inequalities

x > 1.4, x < 2.6, x > 4.2, x < 5.1 and x < 8.2.

To find the solution set which include 3, write the solution set which consists of integer.

The solution set of x > 1.4 is { 2, 3, 4, 5, 6,     ........}

The solution set of x < 2.4 is { 2, 1. 0, -1, ...............}

The solution set of x > 4.2 is { 5, 6. 7, 8, ...............}

The solution set of x < 5.1 is { 5, 4, 3, 2, 1, ...............}

The solution set of x < 8.2 is { 8, 7, 6, 5, 4, 3, ...............}

Hence, the solution set which include 3 are x > 1.4, x < 5.1 and x < 8.2.

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The percentage of scores that fall between 65 and 75 is approximately [tex]2 \times 34 = 68%.[/tex]%

To determine the percentage of scores that fall between 65 and 75, we can use the properties of the normal distribution.

Mean score = 75

Standard deviation = 5

We know that the normal distribution is symmetric around the mean, and approximately 68% of the data falls within one standard deviation from the mean.

This means that about 34% of the scores fall between the mean and one standard deviation above the mean.

To calculate the percentage of scores between 65 and 75, we need to determine how many standard deviations away from the mean 65 and 75 are.

For 65:

(65 - 75) / 5 = -2

For 75:

(75 - 75) / 5 = 0

From the calculations, we can see that 65 is 2 standard deviations below the mean, and 75 is at the mean.

Since the distribution is symmetric, we can consider the percentage of scores between the mean and one standard deviation above the mean (34%) and double it to account for the scores between the mean and one standard deviation below the mean.

Therefore, the percentage of scores that fall between 65 and 75 is approximately 2 [tex]\times[/tex] 34% = 68%.

However, none of the given answer options match the calculated result. Therefore, none of the provided answer options accurately represent the percentage of scores between 65 and 75 based on the given information.

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The coefficient h in the expression means the time taken for the job to be completed in hours.

The custom cake baker charges a flat fee for each job, plus an hourly rate for the number of hours the job takes to complete. The total amount of her bill to the customer can be expressed by 40 + 25h where h is the hours it takes for the job.

Therefore, the coefficient h in the expression can be interpreted as follows:
total amount of bills = 40 + 25h

where

40 dollars is the flat fee for each job

Then the coefficient h means the hourly rate for the time taken for the job to be completed.

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Answer:

-6

Step-by-step explanation:

Because two lines that are parallel have the same slope

Answer:

(a) - [tex]y=-2x+18[/tex]

(b) - [tex]y=\frac{1}{2} x-\frac{9}{2}[/tex]

Step-by-step explanation:

Given the equation of a line, which we'll call line 1, find the following.

(a) - The equation of a line, which we'll call line 2, that is parallel to line 1 and travels through the point (9,0)

(b) - The equation of a line, which we'll call line 3, that is perpendicular to line 1 and travels through the point (9,0)

Given:

[tex]3y+6x=-6[/tex]

(1) - Write the equation of line 1 in slope-intercept form

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Slope-Intercept Form:}}\\y=mx+b\\\bullet \ m \ \text{is the slope of the line}\\\bullet \ b \ \text{is the y-intercept of the line}\end{array}\right}[/tex]

[tex]3y+6x=-6\\\\\Longrightarrow 3y=-6-6x\\\\\Longrightarrow y=-\frac{6}{3} -\frac{6}{3}x \\\\\therefore \boxed{y=-2x-2}[/tex]

Thus, we can conclude the slope of line 1 is -2.

(2) - Answering part (a)

To find a line that is parallel to line 1, the slopes must be the same, -2. Use the point-slope form for a line to find the equation for line 2.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Point-Slope Form:}}\\y-y_1=m(x-x_1)\\\bullet \ m \ \text{is the slope of the line}\\\bullet \ (x_1,y_1) \ \text{is a point the line passes through}\end{array}\right}[/tex]

[tex]y-y_1=m(x-x_1); \ \text{Recall that} \ m=-2 \ \text{and} \ (x_1,y_1)=(9,0)\\\\\Longrightarrow y-0=-2(x-9)\\\\\therefore \boxed{\boxed{ y=-2x+18}}[/tex]

Thus, the equation for line 2 is found.

(2) - Answering part (b)

To find a line that is perpendicular to line 1, the slope of line 3 must be the opposite-reciprocal of line 1's. Once again, use the point-slope form of a line to find the equation of line 3.

[tex]y-y_1=m(x-x_1); \ \text{Recall that} \ m=\frac{1}{2} \ \text{and} \ (x_1,y_1)=(9,0)\\\\\Longrightarrow y-0=\frac{1}{2}(x-9)\\\\\therefore \boxed{\boxed{ y=\frac{1}{2} x-\frac{9}{2} }}[/tex]

Thus, the equation of line 3 is found.

The solution of the given expression is,

x = 1/2.

The given expression is,

[tex]36^{3x} = 216[/tex]

Since we know that,

As the name indicates, exponents are utilized in the exponential function. An exponential function, on the other hand, has a constant as its base and a variable as its exponent, but not the other way around (if a function has a variable as its base and a constant as its exponent, it is a power function, not an exponential function).

Now we can write it as,

⇒ [tex]6^{2^{3x}} = 6^3[/tex]

⇒  [tex]6^{6x}} = 6^3[/tex]

Now equating the exponents we get,

⇒ 6x = 3

⇒   x = 3/6

⇒   x = 1/2

Hence,

Solution is, x = 1/2.

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Answer:

18

Step-by-step explanation:

is a 1/3 scale

6-12-?

8-16-24

simple :)

The area of the Rhombus-shaped keychain is 3 square centimeters.

The area of the rhombus-shaped keychain,

we can use the formula:

Area = (diagonal1 * diagonal2) / 2

Given that the horizontal diagonal has a length of 2 centimeters and

the vertical diagonal has a length of 3 centimeters,

we can substitute these values into the formula:

Area = (2 * 3) / 2

    = 6 / 2

    = 3 cm^2

Therefore, the area of the rhombus-shaped keychain is 3 square centimeters.

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In 6 years Juan will have the double of Carlos age.

The ages of each one are:

Carlos = 18 years old.

Juan = 42 years old.

In x years, they will have:

C = 18 + x

J = 42 + x

Carlos will have the double of Juan's age when:

J = 2*C

Replacing the equations we will get the linear equation:

42 + x = 2*(18  + x)

Solving for x we will get:

42 +x = 36 + 2x

Solving for x:

42 - 36 = 2x - x

6 = x

So Juan will have the double of Carlos age in 6 years.

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The width of the rectangle is 5k² and the length of the rectangle is 3k² + 7k + 4.

The greatest common monomial factor of 15k⁴, 35k³, and 20k² is 5k². So the width of the rectangle is 5k².

The area of the rectangle is the product of its length and width. So the length of the rectangle is the area divided by the width. The area is 15k⁴ + 35k³ + 20k² and the width is 5k². So the length is:

= (15k⁴ + 35k³ + 20k²) / (5k²)

= 3k² + 7k + 4.

Therefore, the width of the rectangle is 5k² and the length of the rectangle is 3k² + 7k + 4.

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The required equation to find the distance of any star from the Sun is d = 1/ tan [tex]\theta[/tex].

Given that, the astronomer is finding the distance(d) of star to the sun in astronomical unit (AU) and tan [tex]\theta[/tex] = 0.000001389.

To find the equation by using the trigonometric function that is

tan a = perpendicular / base.

By using the data and the tangent trigonometric function, the equation is

tan [tex]\theta[/tex] = 1/d.

On simplifying gives,

Thus, d = 1/tan [tex]\theta[/tex].

Hence, the required equation to find the distance of any star from the Sun is d = 1/ tan [tex]\theta[/tex]

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Answer:

28 meters Aprox

Step-by-step explanation:

To find the width of the rectangular park, we can use the Pythagorean theorem. The diagonal running from one corner to the opposite corner forms a right triangle with the length and width of the park.

Given:

Length of the park (L) = 45 meters

Diagonal distance (d) = 53 meters

Using the Pythagorean theorem:

d² = L² + W²

(53 meters)² = (45 meters)² + W²

2809 = 2025 + W²

W² = 2809 - 2025

W² = 784

Taking the square root of both sides:

W ≈ √784

W ≈ 28

Therefore, the width of the rectangular park is approximately 28 meters.

a) We can factor out a common factor of two from the denominator and a half from the numerator: (6p - 7q)/(12p - 14q) = (3p - 7q/2)/(6p - 7q)

b) The numerator and denominator can be factored to provide a common factor of2x²: (6x⁵ - 8x²)/(2x) = 2x³ - 4

c) A common factor of 5 may be extracted from the numerator: (10a + 15b)/5 = 2a + 3b

d) By grouping factors, we may factor the numerator: p² - 6q + 8/3p - 6 = [(p² - 6q) + 8]/[3(p - 2)] = (p - 2)(p - 4)/(p - 2) = p - 4 (where p ≠ 2)

Factorising is the process of employing brackets to represent an expression as the product of its components. We do this by eliminating any elements that are shared by all of the expression's terms. Maths. Algebra. We must eliminate any factors that are shared by each word in an expression before we can factorize it. Expanding brackets is the process's reverse.

Finding an expression's highest common factor (HCF), or the largest number or letter that each term can be split by, is necessary to ensure that it has been properly factorized. Otherwise, there will still be common factors inside the bracket, preventing the expression from being properly factorized.

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The measure of angle AEC is 225 degree.

Given ABCD square then all sides are equal and all angles are of 90°

AB=BC=CD=DA

and also CDE is an equilateral triangle

CD=DE=EC and all angles are of 60°

Now, In ΔADE, ∵AD=AE ⇒ ∠DAE=∠AED

∠ADE=∠ADC-∠EDC=90°-60°=30°

By angle sum property of triangle

∠ADE+∠DAE+∠AED=180°

30°+2∠AED=180°

2∠AED=150°

∠AED=75°

and, ∠EAB = ∠DAB-∠DAE = 90°-75° = 15°

Reflex angle ∠AEC= 360°-∠AED-∠DEC

=360° - 75° -60°

=225°

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The complete and correct question is attached below:

| P | Q | ~Q | ~Q → P | ~P | (~Q → P) ⋀ ~P |

|---|---|----|--------|----|----------------|

| T | T | F  | T      | F  | F              |

| T | F | T  | T      | F  | F              |

| F | T | F  | T      | T  | T              |

| F | F | T  | F      | T  | F              |

To construct a truth table for the logical statement (~Q → P) ⋀ ~P, we need to consider all possible truth values for the variables Q and P. The symbol "~" represents negation or "not" in logic, so ~Q denotes "not Q" and ~P denotes "not P".

In the above table, we first list all possible truth values for P and Q and then determine the truth values for ~Q, ~Q → P, and ~P based on these values. Finally, we evaluate the logical statement (~Q → P) ⋀ ~P based on the truth values for (~Q → P) and ~P to determine the overall truth value of the statement for each combination of P and Q.

The output of the above truth table shows that the statement (~Q → P) ⋀ ~P is true only in one case when P is false and Q is true. In all other cases, the statement is false. Therefore, we can infer that the statement is not always true and hence it is not a tautology. The statement is only true in one specific case where P is false and Q is true.

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The exponential regression equation that best fits the data is y = 2^x.

To determine which exponential regression equation best fits the given data points (2, 4), (3, 8), (4, 16), and (5, 32), let's analyze the options for regression equations:

Option 1: y = 2^x

Option 2: y = 3^x

Option 3: y = 4^x

Option 4: y = 5^x

To find the best fit, we need to compare the predicted y-values from each equation with the actual y-values of the given data points.

The equation that produces the least amount of error or the closest predicted y-values to the actual ones will be the best fit.

Let's calculate the predicted y-values for each option using the given x-values:

Option 1: [tex]y = 2^2, 2^3, 2^4, 2^5 = 4, 8, 16, 32[/tex]

Option 2:[tex]y = 3^2, 3^3, 3^4, 3^5 = 9, 27, 81, 243[/tex]

Option 3:[tex]y = 4^2, 4^3, 4^4, 4^5 = 16, 64, 256, 1024[/tex]

Option 4:[tex]y = 5^2, 5^3, 5^4, 5^5 = 25, 125, 625, 3125[/tex]

By comparing the predicted y-values with the actual y-values of the data points, we can determine which option provides the closest fit.

Based on the given data points, we can observe that the predicted y-values from Option [tex]1 (y = 2^x)[/tex] match the actual y-values most closely.

The predicted values for Option 1 are 4, 8, 16, and 32, which exactly match the given data points.

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The complete question may be like: Consider the following data points: (2, 4), (3, 8), (4, 16), (5, 32). We need to determine which exponential regression equation best fits this data. Please provide the options for the regression equations so that I can assist you in finding the best fit.

Considering the definition of an equation and the way to solve it, there are 84 green counters in the box.

An equation is the equality existing between two algebraic expressions connected through the equals sign in which one or more unknown values appear.

The solution of a equation means determining the value that satisfies it. To solve an equation, keep in mind:

Knowing that:

the equation in this case is:

2/9 × total counters on the box =24

Solving:

total counters on the box =24÷ 2/9

The first step in dividing by a fraction is to find the reciprocal (reverse the numerator and denominator) of the second fraction.

Then, the two numerators and the two denominators must be multiplied and, if necessary, the fractions are simplified.

total mountain bikes =24× 9/2= 24/1× 9/2

total mountain bikes =(24×9)/ (1×2)

total mountain bikes =216/2

total mountain bikes =108

Then, there are 108 green and yellow counters in the box.

So, the amount of green counters in the box is calculated as:

Amount of green counters= 7/9× 108

Amount of green counters= 7/9× 108

Amount of green counters= 84

Finally, there are 84 green counters in the box.

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The value of angle BCD is determined as 108⁰.

option B is the correct answer.

The value of angle BCD is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.

Also this theory states that arc angles of intersecting secants at the center of the circle is equal to the angle formed at the center of the circle by the two intersecting chords.

arc AB =  m∠ACB

m∠ACB =  72⁰

The value of angle BCD is calculated as follows;

angle BCD = 180 - 72⁰ (sum of angles in a circle)

angle BCD = 108⁰

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Answer:

The missing variable θ is equal to (11π)/8.

Step-by-step explanation:

Given:

ω = π/8 radian per min

t = 11 min

Step 1: Identify the formula and the missing variable.

The formula is ω = (θ/t), and we are trying to find the value of θ.

Step 2: Rearrange the formula to solve for the missing variable.

To isolate θ, we can multiply both sides of the equation by t:

ω * t = θ

Step 3: Substitute the given values.

Substituting the given values into the rearranged formula, we have:

θ = (π/8) * 11

Step 4: Simplify the expression.

To multiply fractions, we multiply the numerators and multiply the denominators:

θ = (π * 11) / (8 * 1)

θ = (11π) / 8

Step 5: Finalize the answer.

The value of the missing variable θ is (11π)/8. This is the exact answer unless otherwise indicated.

Therefore, the step-by-step process shows that the missing variable θ is equal to (11π)/8.

Answer:

Step-by-step explanation:

z=6

The correct answer is 86.08, as the octagon is given and the apothem given here is 13 units. The calculation after putting the value in formula is  86.08.

An octagon is a polygon with eight sides and eight angles. It is a two-dimensional geometric shape.  Each angle in a regular octagon measures 135 degrees, and all sides of a regular octagon are of equal length.

The formula is given below,

P= side length ×n

Apothem of octagon =13 units,

side length is = tan (360° / (2 × 8)) = (n/2) ÷ 13

= tan (360° / (2 × 8)) = n/26

tan 22.5°= n/26

n/26 =  0.4142

n = 10.76

perimeter of octagon = 8 ×  10.76 = 86.08

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The ballroom dance couple can arrange their performance in 28 different ways by selecting 6 routines out of the 8 they have learned.

The idea of combinations can be used.

The number of ways to select k items from a set of n items is given by the formula for combinations:

C(n, k) = n! / (k! * (n - k)!)

In this case, n = 8 (the total number of routines they have learned) and k = 6 (the number of routines they will perform).

Using the formula, we can calculate:

C(8, 6) = 8! / (6! * (8 - 6)!)

= 8! / (6! * 2!)

The factorial function is represented in this case by the exclamation symbol (!).

The sum of all positive integers from 1 to n is the factorial of the number n.

Calculating the factorials involved:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320

6! = 6 * 5 * 4 * 3 * 2 * 1 = 720

2! = 2 * 1 = 2

Plugging in these values:

C(8, 6) = 40,320 / (720 * 2)

= 40,320 / 1,440

= 28

Therefore, the ballroom dance couple can arrange their performance in 28 different ways by selecting 6 routines out of the 8 they have learned.

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Answer:

Naomi will need to score at least 89% on her fourth test to have an average of 90% in her math class.

Step-by-step explanation:

To find out what score Naomi needs to earn on her fourth test in order to have an average of 90%, we can set up an equation.

Let's denote the score on the fourth test as "x". Naomi has taken three tests, and their scores are 87%, 98%, and 86%. To find the average, we sum up all the scores and divide by the number of tests:

(87 + 98 + 86 + x) / 4 = 90

Now we can solve for x:

(87 + 98 + 86 + x) = 4 * 90

271 + x = 360

x = 360 - 271

x = 89

Naomi will need to score at least 89% on her fourth test to have an average of 90% in her math class.

Based on the figure, if I and K are parallel lines, the value of x+y in degrees is 62°.

The correct answer choice is option A.

From the diagram

x + 149° = 180° (Sum of angle on a straight line)

x = 180° - 149°

x = 31°

Also,

y = 31° (alternate angles are equal)

Therefore,

x + y = 31° + 31°

= 62°

Hence, the sum of angle x and angle y is 62°.

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