The two equations represent two lines in the plane that are parallel and never intersect, so there is no point that satisfies both equations simultaneously. No solution equations.
Describe Equation?In mathematics, an equation is a statement that two expressions are equal. It consists of two sides separated by an equal sign (=). The expression on the left side of the equal sign is usually called the left-hand side (LHS), while the expression on the right side is called the right-hand side (RHS).
Equations can take many different forms and can involve various types of functions and operators, such as addition, subtraction, multiplication, division, exponentiation, logarithms, trigonometric functions, and more. They can also involve one or more variables, which can be solved for to obtain a specific value or range of values that make the equation true. Equations are used extensively in mathematics, science, engineering, economics, and many other fields.
We can solve this system of equations using the substitution method by solving one of the equations for one variable in terms of the other, and then substituting that expression into the other equation.
Let's solve the second equation for y in terms of x:
-x + y = 4
y = x + 4
Now we substitute this expression for y into the first equation and solve for x:
-2x + 2y = 7
-2x + 2(x + 4) = 7
-2x + 2x + 8 = 7
8 = 7
This last equation is a contradiction, which means that there is no solution to the system of equations. Geometrically, the two equations represent two lines in the plane that are parallel and never intersect, so there is no point that satisfies both equations simultaneously.
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Joe D. Curios is exploring a river with a strong curent. He paddles upstream by day taking measurements and readings of water quality going 5 miles a day. He drops an anchor at night, but the current is so strong is still pushes his boat 3 miles downstream each night. His destination is the dam, which is 10 miles upstream from his starting point . If he starts his journey on june 1, when will he reach the dam?
CAN U PLEASE WRITE THE WHOLE EQUATION OUT ON A PIECE OF PAPER PLEASE>>>>>>>>
Joe will reach the dam on June 6, which is 5 days after he starts his journey on June 1.
What is distance?
Distance is the measure of how far apart two objects or locations are from each other.
Let's first calculate the net distance Joe travels each day:
Net distance = Distance paddled upstream - Distance pushed downstream by the current
Net distance = 5 miles - 3 miles
Net distance = 2 miles
This means that Joe moves 2 miles closer to the dam each day.
To reach the dam, Joe needs to cover a distance of 10 miles upstream from his starting point. Since he covers 2 miles per day, he will take:
Days taken to reach the dam = Distance to cover ÷ Distance covered each day
Days taken to reach the dam = 10 miles ÷ 2 miles per day
Days taken to reach the dam = 5 days
Therefore, Joe will reach the dam on June 6, which is 5 days after he starts his journey on June 1.
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what is the probability of rolling a value higher than eight with a pair of fair dice? question 15 options: 6/36 18/36 10/36 8/36 12/36
Explanation:
We want a dice sum larger than 8. Meaning we want the sum to be either: 9, 10, 11, or 12
Here are the four ways to add to 9
3 + 6 = 9
4 + 5 = 9
5 + 4 = 9
6 + 3 = 9
Here are the three ways to add to 10
4 + 6 = 10
5 + 5 = 10
6 + 4 = 10
and here are the two ways to add to 11
5 + 6 = 11
6 + 5 = 11
and finally there's only one way to get a sum of 12
6 + 6 = 12
There are 4+3+2+1 = 10 different sums we want out of 6*6 = 36 dice rolls possible.
That's how we get to 10/36 as the final answer.
10/36 reduces to 5/18, but it appears your teacher decided not to reduce.
The correct option is D.
The probability of rolling a value higher than eight with a pair of fair dice is 4/36 or 1/9.What is the probability of rolling a value higher than eight with a pair of fair dice?To find the probability of rolling a value higher than eight with a pair of fair dice, you have to find the number of ways to get a sum greater than 8 and divide it by the total number of possible outcomes when rolling two dice.The maximum sum that can be obtained by rolling two dice is 12 (6 + 6). So, we need to find the number of ways that we can obtain 9, 10, 11, or 12. The possible combinations are:(3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), and (6,6).Therefore, there are 10 possible outcomes in which the sum of the two dice is greater than 8. The total number of possible outcomes when rolling two dice is 6 × 6 = 36.So, the probability of rolling a value higher than eight with a pair of fair dice is:10/36 = 5/18 or approximately 0.277 or 27.7%.
Therefore the correct answer is option D ) 8/36.
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How many yards are there if there are 5 1/2 feet?
Answer:
1.83 yards
Step-by-step explanation:
in trapezold WXYZ, side WX is parallel to side Y2, and YZ is 4.25 cm long. The mid-segment of trapezold WXYZ Is 2.75 cm long. Find the length of side WX (in cm).
The calculatd value of the length of side WX is 1.25 cm.
Finding the length of side WX (in cm).Let's call the length of side WX "x".
The mid-segment of a trapezoid is the line segment connecting the midpoints of the non-parallel sides.
The length of the mid-segment is equal to the average of the lengths of the two parallel sides. In this case, the mid-segment is 2.75 cm long and connects the midpoints of sides WZ and XY.
We can use the formula for the length of the mid-segment of a trapezoid:
mid-segment length = (length of side WZ + length of side XY) / 2
Plugging in the values we know, we get:
2.75 = (x + 4.25) / 2
Multiplying both sides by 2, we get:
5.5 = x + 4.25
Subtracting 4.25 from both sides, we get:
x = 1.25
Therefore, the length of side WX is 1.25 cm.
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if a regular polygon of 12 side with radius 4 units long is given, then its area is equal to
The area of the polygon with 12 sides is 48 square units.
Calculating the area of the polygonTo find the area of a regular polygon with 12 sides and a radius of 4 units, we can use the formula:
A = 12 * Area of 1 triangle
Where
Area of 1 triangle = 1/2 * radius * sin(360/n)
So, we have
Area of 1 triangle = 1/2 * 4^2 * sin(360/12)
Area of 1 triangle = 1/2 * 4^2 * sin(30)
Evaluate
Area of 1 triangle = 4
Substituting the given values, we get:
A = 12 * 4
Evaluate
Area = 48
So the area is 48 square units.
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hi! i need help checking these problems!!
1) 12x + 9 = -15 i know the answer i just need help checking!
2) x/7 - 4 = 4 same with this i need help checking it!
Answer:
Part 1: x = -2
Part 2: x = 56
Step-by-step explanation:
Part 1:
12x + 9 = -15
Given that-
12x + 9 = -15
12x + 9 - 9 = -15 -9 >> Subtract 9 to both sides
12x = -24
12x/12 = -24/12 >> Divide both sides by 12
x = -2
Check Answer:
12(-2) + 9 = -15 >> 12 × -2 = -24
-24 + 9 = -15 >> -24 + 9 = -15
-15 = -15 >> Correct
Part 2:
x/7 - 4 = 4
Given that-
x/7 - 4 = 4
x/7 - 4 + 4 = 4 + 4 >> Add 4 to both sides
x/7 = 8
7x/7 = 8 × 7 >> Multiply both sides by 7
x = 56
Check Answer-
Substitute x in the equation the check.
56/7 - 4 = 4 >> 56/7 = 8
8 - 4 = 4 >> 8 - 4 = 4
4 = 4 >> Correct
RevyBreeze
integers $a$, $b$, $c$, and $d$, not necessarily distinct, are chosen independently and at random from $0$ to $2007$, inclusive. what is the probability that $ad-bc$ is even?
If integers a, b, c, and d, not necessarily distinct, are chosen independently and at random from 0 to 2007, the probability that ad-bc is even is 10/16.
To solve the problem, we need to consider the parity of the product ad and bc separately. If both ad and bc are even, then their difference (ad-bc) is even. If both ad and bc are odd, then their difference is even as well. However, if one of ad and bc is odd and the other is even, then their difference is odd.
For ad to be even, either a or d (or both) must be even. Similarly, for bc to be even, either b or c (or both) must be even. Therefore, the probability that ad and bc are both even is the product of the probabilities that a, b, c, and d are even:
P(ad even) = P(a even)P(d even) = (1004/2008)^2 = 1/4
P(bc even) = P(b even)P(c even) = (1004/2008)^2 = 1/4
The probability that ad-bc is even is the sum of the probabilities that both ad and bc are even and that both ad and bc are odd:
P(ad-bc even) = P(ad even and bc even) + P(ad odd and bc odd)
= P(ad even)P(bc even) + P(ad odd)P(bc odd)
= 1/4 * 1/4 + 3/4 * 3/4
= 1/16 + 9/16
= 10/16
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Jack found an antique coin as shown. what is the area of the coin?
The area of the cutout and the coin is as follows:
(A) 9mm²
(B) 293 mm²
What is the area?The size of a patch on a surface is determined by its area.
Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.
The area of a plane figure is the area that its perimeter encloses.
The quantity of unit squares that cover a closed figure's surface is its area.
Square units like cm² and m² are used to measure area.
Area of the cutout:
A = s²
A = 3²
A = 9mm²
Area of the coin:
A = πr²
A = π9.8²
A = π9.8²
A = π96.04
A = 301.5656
Then,
301.5656 - 9
292.5656
Rounding off: 293 mm²
Therefore, the area of the cutout and the coin is as follows:
(A) 9mm²
(B) 293 mm²
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A local restaurant recently began to offer an eat-in pizza promotion for their famous double-decker pizza. When priced at $17.99/person, the promotion attracted 112 customers per day on average. When they raised the price to $19.99/person they attracted an average of only 96 customers per day.
What price should the promotion be set at to generate the maximum daily profit?
Answer:
the price that generates maximum profit is $152.94, rounded to the nearest cent.
Step-by-step explanation:
Let's begin by finding the profit generated by the promotion at each price point.
At $17.99/person:
Profit = Revenue - Cost
Profit = $17.99 x 112 - (10 + 5 + 3) x 112
Profit = $2016.64
At $19.99/person:
Profit = Revenue - Cost
Profit = $19.99 x 96 - (10 + 5 + 3) x 96
Profit = $1497.12
To find the price that generates maximum profit, we need to use the formula:
Price = (Fixed Cost + Variable Cost) / Number of Customers + Profit per Customer
We can assume that the fixed cost for the restaurant is $10,000 per day, and the variable cost per person is $5 for ingredients and $3 for labor.
For $17.99/person:
Price = ($10,000 + (5 + 3) x 112) / 112 + ($2016.64 - $17.99)
Price = $152.94
For $19.99/person:
Price = ($10,000 + (5 + 3) x 96) / 96 + ($1497.12 - $19.99)
Price = $156.24
Therefore, the price that generates maximum profit is $152.94, rounded to the nearest cent.
Question 5 (20 points)
The midpoint M and one endpoint of AB are given. Find the coordinates of the other endpoint.
A(-8,4) and M(5,-9).
a
(10,-22)
b
(18,-8)
c
(18,-22)
d
(6,-22)
Answer:
Let B be the other endpoint of AB.
We know that the midpoint M of AB has coordinates (5, -9), which means that the average of the x-coordinates of A and B is 5, and the average of the y-coordinates of A and B is -9:
( x_A + x_B ) / 2 = 5( y_A + y_B ) / 2 = -9Substituting the coordinates of point A (-8, 4), we can solve for the x-coordinate of point B:
( -8 + x_B ) / 2 = 5Simplifying the equation, we get:
-8 + x_B = 10x_B = 18Now, substituting the same coordinates of point A and the y-coordinate of midpoint M (-9), we can solve for the y-coordinate of point B:
( 4 + y_B ) / 2 = -9Simplifying the equation, we get:
4 + y_B = -18y_B = -22Therefore, the coordinates of point B are (18, -22).The line plot displays the number of roses purchased per day at a grocery store.
A horizontal line starting at 0 with tick marks every one unit up to 10. The line is labeled Number of Rose Bouquets, and the graph is titled Roses Purchased Per Day. There is one dot above 10. There are two dots above 1 and 4. There are three dots above 2 and 5. There are 4 dots above 3.
Which of the following is the best measure of center for the data, and what is its value?
The median is the best measure of center, and it equals 3.5.
The median is the best measure of center, and it equals 3.
The mean is the best measure of center, and it equals 3.
The mean is the best measure of center, and it equals 3.5
The best measure of center for this data would be the median, as it is less affected by outliers and works well for skewed data. To find the median, we arrange the data in order: 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5. There are 16 data points, so the median would be the average of the 8th and 9th values, which is 3.5. Therefore, the best measure of center for this data is the median, and its value is 3.5.
So the correct answer is: The median is the best measure of center, and it equals 3.5.
A 3-pack of jump ropes costs $9.38. What is the unit price, rounded to the nearest cent?
Answer:
3.13
Step-by-step explanation:
The price of all three ropes together - $9.38
Price of one unit -9.38/3.=
3.12666 ; Nearest round figured cent would be $3.13
what is the value of x?
Answer:
42
Step-by-step explanation:
evaluate each expressions when a=2and b=6. 7a-ab+b
Which is the correct order of the polynomial 7x3y3-3xy5+2x2y4-x4y2 in descending powers of x?
A.-x4y2+7x3y3+2x2y4-3xy5
B.-3xy5+2x2y4+7x3y3-x4y2
C.7x3y3-3xy5+2x2y4-x4y2
D.-x4y2+2x2y4-3xy5+7x3y3
The correct order of the polynomial [tex]7x^3y^3 - 3xy^5 + 2x^2y^4 - x^4y^2[/tex] in descending powers of x is:
[tex]-x^4y^2 + 7x^3y^3 + 2x^2y^4 - 3xy^5[/tex]
Therefore, the answer is option A.
What is polynomial function?
A polynomial function is a mathematical function that consists of a sum of terms, where each term is a constant multiplied by one or more variables raised to a non-negative integer power.
When we arrange a polynomial in descending order of powers of one of its variables, it means we are writing the terms with the highest power first, followed by terms with the next highest power, and so on until we reach the constant term (which has no variable).
In this case, the polynomial is [tex]7x^3y^3-3xy^5+2x^2y^4-x^4y^2[/tex]. To write this in descending order of powers of x, we need to start with the term that has the highest power of x, which is [tex]-x^4y^2[/tex].
Then we write the term with the next highest power of x, which is [tex]7x^3y^3[/tex]. After that, we write the term with the next highest power of x, which is [tex]2x^2y^4[/tex]. Finally, we write the term with the lowest power of x, which is [tex]-3xy^5[/tex].
So the correct order of the polynomial in descending powers of x is option A: [tex]-x^4y^2+7x^3y^3+2x^2y^4-3xy^5[/tex].
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For each of the figures write an absolute value equation that has the following solution set.
Absolute Value Equation for a Horizontal Line Passing Through Two Points -1/2 and 3/2.
The solution set given as a straight horizontal line passing through the points -1/2 and 3/2 can be represented as:
| x - 1 | = 1/2
The absolute value function | x - 1 | gives the distance between x and 1 on the number line.Since the given solution set is a straight horizontal line passing through -1/2 and 3/2, the midpoint of the two points would be 1.The distance between the midpoint (1) and one of the points (-1/2 or 3/2) would be 1/2.Therefore, the equation | x - 1 | = 1/2 represents the set of all values of x that are at a distance of 1/2 units away from the midpoint 1, which is the straight horizontal line passing through -1/2 and 3/2.To learn more about horizontal line, visit:
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The area of a rectangle measured in cm2, is numerically equal to its perimeter, measured in cm.
The length of the rectangle is 5 times its width.
Calculate the width and length of the rectangle. Give your answers in centimetres.
Let's use "w" to represent the width of the rectangle and "l" to represent the length.
According to the problem statement, the area of the rectangle is numerically equal to its perimeter, so we can write:
2(l + w) = lw
Simplifying this equation, we get:
2l + 2w = lw
Dividing both sides by 2, we get:
l + w = 0.5lw
Multiplying both sides by 2, we get:
2l + 2w = lw
Since the length of the rectangle is 5 times its width, we can write:
l = 5w
Substituting this into the previous equation, we get:
2(5w) + 2w = 5w^2
Simplifying this equation, we get:
10w + 2w = 5w^2
12w = 5w^2
Dividing both sides by w, we get:
12 = 5w
So the width of the rectangle is:
w = 12/5 = 2.4 cm
And the length of the rectangle is:
l = 5w = 12 cm
Therefore, the width of the rectangle is 2.4 cm and the length of the rectangle is 12 cm.
Answer:
Width: 2.4 cm
Length: 12 cm
Step-by-step explanation:
Let the width of the rectangle be w cm, and the length be 5w cm. The area A and perimeter P of the rectangle can be expressed as follows:
Area (A) = length × width
A = 5w × w = 5w^2 cm^2
Perimeter (P) = 2 × (length + width)
P = 2 × (5w + w) = 2 × 6w = 12w cm
According to the problem, the area is numerically equal to the perimeter:
A = P
5w^2 = 12w
To solve for w, we can rearrange the equation:
5w^2 - 12w = 0
w(5w - 12) = 0
This equation has two possible solutions:
w = 0
In this case, the width would be 0 cm, which doesn't form a valid rectangle.
5w - 12 = 0
5w = 12
w = 12/5 = 2.4 cm
For the second solution, the width is 2.4 cm. Now, we can calculate the length:
length = 5w
length = 5 × 2.4 = 12 cm
So, the width of the rectangle is 2.4 cm, and the length is 12 cm.
james sets off from home on a bike.The graph journey.
a)what is james' average speed on his outward journey
Step-by-step explanation:
james sets off from home on a bike.The graph journey.
a)what is james' average speed on his outward journey
james sets off from home on a bike.The graph journey.
a)what is james' average speed on his outward journey
What is the value of x?
Answer:
x = 23
Step-by-step explanation:
180 - 100 = 80
51 + 80 = 131
180- 131=49
49= 2x +3
49 - 3= 46
46=2x
46/ 2= 23
X= 23
If 21 and 22 are vertical angles, 22 and 23 are complementary
angles, and m/1 = 71, find m/3.
According to the question the measure of angle 3, or m∠3, is also 19 degrees since angle 2 and angle 3 are vertical angles. So: m∠3 = m∠23 = 19°
We know that vertical angles are congruent, so if angle 21 and angle 22 are vertical angles, then:
m∠21 = m∠22
We also know that angle 22 and angle 23 are complementary angles, which means:
m∠22 + m∠23 = 90°
Substituting m∠22 with m∠21, we get:
m∠21 + m∠23 = 90°
We are given that m∠21 = 71 degrees, so we can substitute that in the equation:
71° + m∠23 = 90°
Subtracting 71° from both sides, we get:
m∠23 = 19°
Therefore, the measure of angle 3, or m∠3, is also 19 degrees since angle 2 and angle 3 are vertical angles. So:
m∠3 = m∠23 = 19°
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Ron said that he would spend $400 of his paycheck on nine items that costs $50 each. Why is this not reasonable? A. This is not reasonable because nine items at $50 each would cost $450. B. This is not reasonable because nine items at $50 each would cost $360. C. This is not reasonable because the nine items only cost $50. D. This is not reasonable because he would have too much money left over.
As Ron said he would spend $400 of his paycheck on these items, it is not reasonable because the total cost of the items is $450, which is $50 more than the amount he planned to spend. So, option A is correct.
As per given information,
Ron want to spend $400.
Ron wants to buy nine items that each cost $50.
To find the total cost of these items, we need to multiply the number of items by the cost of each item .
So,
Total cost of items = 9 x 50
Total cost of items = 450
This results in a total cost of $450.
This is not reasonable because nine items at $50 each would cost $450
Therefore, option A is correct.
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20) Evaluate. helpp me plsss
The value of the logarithm of numbers is 1/4
What is logarithm?In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x.
The given logarithm is 1 + log₇25
Simplifying this we have log₇25 = -1
Taking the logarithm of both sides we have
log₇25 = -100x
25 =7⁻¹⁰⁰ˣ
100x = 25
Dividing by 100
x = 25/100
X = 1/4
Therefore the value of the logarithm is 1/4
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The answer of this question bc it showing me the answer is 4-9
Answer:
Your answer of 4¹⁸ is correct.
Step-by-step explanation:
In (4⁶)³ the exponents would be multiplied. 6 multiplied by 3 is 18 and therefore the correct answer would be 4¹⁸
Answer:
B. 4^18
Step-by-step explanation:
(4^6)= 4096
(4096)^3= 68,719,476,736
So, (4^6)^3= 68,719,476,736
AND
4^18= 68,719,476,736
Hope it helped! :)
Brainliest? Please.
Find the greatest common factor of 14 and 34
Answer:
2
Step-by-step explanation:
14/2=7
34/2=17
1. four buses carrying 100 students from the same school arrive at a football stadium. the busses carry, respectively, 10, 20, 30, and 40 students. one of the 100 students is randomly selected and let x denote the number of students on the bus carrying the randomly selected student. let y denote the number of students when one of the 4 buses is randomly selected. (a) explain (without any computation) which of e[x] or e[y ] you think is larger? why?
Answer:
2 bus
Step-by-step explanation:
The expected value of y is larger than the expected value of x. This is because y takes into account the number of students on all 4 buses, while x only takes into account the students on the bus with the randomly selected student.
The expected value of a random variable represents the average value of that variable over multiple trials. In this case, x is a random variable representing the number of students on the bus carrying the randomly selected student, and y is a random variable representing the number of students on the randomly selected bus.
The expected value of x can be calculated as follows:
E[x] = (10/100)×10 + (20/100)×20 + (30/100)×30 + (40/100)×40
= 16 + 12 + 9 + 16
= 53/4
= 13.25
This means that, on average, the bus carrying the randomly selected student would have 13.25 students.
The expected value of y can be calculated as follows:
E[y] = (1/4)×10 + (1/4)×20 + (1/4)×30 + (1/4)×40
= 25
This means that, on average, the randomly selected bus would have 25 students.
As expected, the expected value of y is larger than the expected value of x. This is because y takes into account the number of students on all 4 buses, while x only takes into account the students on the bus with the randomly selected student. Since the number of students on the other 3 buses can vary widely, y has a higher expected value than x.
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in 2015 there were about 22 million teenagers (ages 13-17) in the united states. they each sent an average of 900 text messages per month. About how many text messages did all of the teenagers in the united states send each month
express your answer in scientific notation
To express the ans in scientific notation is 1.98 x [tex]10^{10}[/tex] text messages per month.
How to calculate Average?You must add up all of the numbers in a set and divide the result by the total amount of numbers in order to determine the average of the set of numbers. The mean is another name for this.
The steps to calculating the average are as follows:
Sum up each of the set's numbers.
Count how many numbers there are in all.
subtract the total number of numbers from the sum of the numbers.
The average (mean) is calculated using the following formula:
Average equals ((Sum of Numbers)/ (Total Number of Numbers)
To calculate the total number of text messages sent by all teenagers in the United States in a month, we can multiply the number of teenagers by the average number of text messages sent by each teenager:
Total text messages = (number of teenagers) x (average number of text messages per teenager)
Total text messages = 22,000,000 x 900
Multiplying these numbers gives us:
Total text messages = √19,800,000,000
Total text messages = 140712.473 text messages per month.
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What is the acceleration due to gravity on planet Earth in meters per second squared?
Step-by-step explanation:
Approx 9.81 m/s^2
ESTION I Determine, using the rules of differentiation: dy if y= 플 dx Show ALL calculations. 1. 2 6x³
The derivative of y = 6x^3 with respect to x is dy/dx = 18x^2.
Differentiating the equationFrom the question, we have the following parameters that can be used in our computation:
y = 6x^3
To find the derivative of y with respect to x, we can use the power rule of differentiation, which states that the derivative of x^n with respect to x is n*x^(n-1).
Using this rule, we can differentiate y = 6x^3 as follows:
dy/dx = d/dx (6x^3)
= 6 * d/dx (x^3)
= 6 * 3x^2 (applying the power rule with n=2)
= 18x^2
Therefore, the derivative of y = 6x^3 with respect to x is dy/dx = 18x^2.
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say mr beast had 1,000 and wants to spilt it eqauly between 5 people how much money does each person get closet one gets brainest
Answer:
Step-by-step explanation:
1000 / 5 = 200
So if their 5 people they would share the money 200 hundred per person.
Does anyone have the Unit: Functions homework 3 WRITING EQUATIONS OF LINEAR FUNCTIONS ???
The Equation of the linear function is y = 2x + 1.
Here's a step-by-step explanation using the terms you've provided:
1. Identify the slope (m) and y-intercept (b): In a linear function, the equation is typically written in the form y = mx + b, where m represents the slope and b represents the y-intercept.
2. Find the slope: If you're given two points on the line (x1, y1) and (x2, y2), you can find the slope by using the formula: m = (y2 - y1) / (x2 - x1).
3. Determine the y-intercept: If you know the slope and one point on the line (x, y), you can find the y-intercept by rearranging the equation: b = y - mx.
4. Write the equation: Once you have the slope and y-intercept, you can write the equation in the form y = mx + b.
For example, suppose you are given two points on a line: (1, 3) and (3, 7). To write the equation of the linear function:
Step 1: Find the slope (m):
m = (7 - 3) / (3 - 1) = 4 / 2 = 2
Step 2: Determine the y-intercept (b) using one of the points, say (1, 3):
b = 3 - (2 * 1) = 1
Step 3: Write the equation:
y = 2x + 1
So, the equation of the linear function is y = 2x + 1.
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Factorise using identities
Having factorized using identities, the result is [(a^2c + b^2a + c^2b)/(abc)][(a^2 + b^2 + c^2 - a^2b^2c^2)/(a^2b^2c^2)]
What is the explanation for the above response?One possible approach to factorizing the expression is to use the identity:
a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc)
If we let a = b/c, b = c/a, and c = a/b, then we can substitute these expressions into the identity and simplify:
(a/b)^3 + (b/c)^3 + (c/a)^3 - 3abc/a^3b^3c^3
= [(a/b) + (b/c) + (c/a)][(a/b)^2 + (b/c)^2 + (c/a)^2 - (a/b)(b/c) - (a/b)(c/a) - (b/c)(c/a)]
= [(a^2c + b^2a + c^2b)/(abc)][(a^2 + b^2 + c^2)/(a^2b^2c^2) - 1]
= [(a^2c + b^2a + c^2b)/(abc)][(a^2 + b^2 + c^2 - a^2b^2c^2)/(a^2b^2c^2)]
Therefore, the factorization of the given expression is:
(a/b)^3 + (b/c)^3 + (c/a)^3 - 3 = [(a^2c + b^2a + c^2b)/(abc)][(a^2 + b^2 + c^2 - a^2b^2c^2)/(a^2b^2c^2)]
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