The mean for a given set of observations is 3.67.
What is mean?The mean is the average of a set of variables in mathematics and statistics. The mean may be calculated in a variety of methods, including the simple arithmetic mean (add the numbers and divide the total by the number of observations), geometric mean, and harmonic mean. A data set's mean (average) is calculated by summing all of the numbers in the set and then dividing by the number of values in the set. When a data collection is arranged from least to largest, the median is the midway value. The mode is the number that appears the most frequently in a data collection.
Here,
given set of observation,
=1,1,2,4,6,8
=(1+1+2+4+6+8)/6
=3.67
The mean for given set of observation is 3.67.
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How do you solve a slope step by step?
In order to calculate the slope, m, one can use either the classic line equation, y = mx + b, or the slope formula, [tex]m = (y_{2} -y_{1} )/(x_{2} - x_{1} )[/tex]. The slope can be calculated or discovered through observation.
How do you calculate slope step by step?When a straight line's equation is not available, determining its slope involves three stages. Determine the first two points on the line. Choose one to be (x1, y1) and the other to be in step two (x2, y2). Step three is to compute slope using the slope equation.
If you have the coordinates of two points on a line, you can use the slope formula to determine that line's slope. The slope equation is given by [tex]m=(y_{2}- y_{1} )/(x_{2} -x_{1} )[/tex], which is the ratio of the change in the y values to the change in the x values. x1 and y1 are the coordinates of the first point. A line's steepness is determined by its slope. In mathematics, slope is determined by "rise over run" (change in y divided by change in x).
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Course activity: modeling with functions
Monica and Shanna are saving the money they earn from their after-school jobs to go on a school trip. Monica starts with $20 and then saves $14 each week. Shanna's savings are shown in the table.
Monica’s savings can he modeled by a ____ table
Shanna’s savings can be modeled by a ____ table
Answer:Both A and C
Step-by-step explanation:
Four friends want to take a vacation together, so each one gets a part-time job. Each person has 8 weeks to save $720 for the vacation. Analyze the four individual plans below and decide which of the four people will reach his or her goal of saving $720 for vacation.
Friend A: Works 13 hours per week at $6.95 per hour.
Friend B: Works 15 hours per week at $5.85 per hour.
Friend C: Works 18 hours per week at $5.25 per hour.
Friend D: Works 11 hours per week at $7.80 per hour. there!
melania is tracking her running training program. the table gives her 5k run time at the end of each month. month time (in minutes) 1 47 2 43 3 39 4 36 5 35 6 33 what is the equation for the line of best fit, where x represents the month and y-hat represents the predicted time? y hat equals 33.6 plus 2.77 times x y hat equals 48.5 plus 2.77 times x y hat equals 33.6 minus 2.77 times x
The required regression equation for the line of best fit, where x represents the month and y-hat represents the predicted time is;
y = -2.78x + 48.5
How to find the regression equation?A regression equation is defined as the equation that is used in stats to find out what relationship, if any, exists between sets of data
From the given table, let x represents the month and y represents the time. Thus;
Mean of x; x-bar = (1+2+3+4+5+6)/6 = 21/6
x-bar = 3.5
Mean of y; y-bar = (47+43+39+36+35+33)/6
y-bar = 233/6 = 38.83
∑x = 21
∑y = 233
Using statistics calculator;
Sum of squares (SSX) = 17.5
Sum of products (SP) = -48.5
The general form of a regression Equation is;
y-bar = bx-bar + a
Where;
b = SP/SSX
= -48.5/17.5
= -2.77143
a = y-bar - b(x-bar) = 38.83 - (-2.77×3.5) = 48.53333
y = -2.78x + 48.5
Thus, the required regression equation is y = -2.78x + 48.5 for the line of best fit.
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Gabriella has 60 m of fencing to build a four-sided fence around a rectangular plot of land. The area of the land is 209 square meters. Solve for the dimensions (length and width) of the field.
The length and width of the rectangular plot of land is (209 / (60 - l)) and (60 - (209 / (60 - l))), respectively.
The formula to calculate the dimensions of a rectangular plot of land is area = length x width, or A = l x w. In this problem, the area of the land is 209 square meters, so A = 209. We know that Gabriella has 60 meters of fencing to build a four-sided fence around the land, so the length and width of the field must add up to 60 meters. This can be written as l + w = 60. To solve for the length and width of the field, we can first substitute the value for the area of the land into the equation for area: A = l x w. This equation can be rewritten as l x w = 209. We now have two equations that can help us solve for the length and width of the field.
We can solve this system of equations using substitution. We can substitute the equation l + w = 60 into the equation l x w = 209. This simplifies the equation to l x (60 - l) = 209. We can then solve for l by dividing both sides of the equation by (60 - l). This results in l = (209 / (60 - l)). We can then substitute this into the equation l + w = 60 to solve for w. This simplifies to w = (60 - (209 / (60 - l))).
Therefore, the length and width of the rectangular plot of land is (209 / (60 - l)) and (60 - (209 / (60 - l))), respectively.
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Which form of a linear equation is more efficient for determining each characteristic?
Slope
x -intercept
y -intercept
If you want to use your calculator to graph an equation, which form is more efficient?
Answer:
1/ Point slope form is more efficient for determining the Slope
2/ Standard form is more efficient for determining x-intercept and
y-intercept
3/ If you want to use your calculator to graph an equation, the Point slope form is more efficient
Step-by-step explanation:
1/ I say point slope form because the slope in this form is a point on the line, and he if knows the point on the graph and slope, he can graph the line easily.
2/ I say standard form because x and y intercepts of a graph, that is, the point where the graph crosses the x-axis and the point where it crosses the y -axis. So, it is the easiest form to find the x and y intercepts.
3/ If he knows a point and the slope he can graph it easily.
How do you calculate the inverse?
The inverse of a number is it's reciprocal. Divide 1 by the number to obtain its reciprocal.
For example, the reciprocal of 5 is 1/5, and the reciprocal of 0.5 is 2. The reciprocal of a number x is denoted by 1/x or x^(-1).
For example, to find the reciprocal of 8, you would divide 1 by 8, like this:
1/8 = 0.125
To find the reciprocal of a fraction, you can also turn the fraction upside down to get its reciprocal. For instance, 4/3 is 3/4's inverse.
To find the inverse of a matrix, you can use the following formula:
A^(-1) = (1/det(A)) * adj(A)
where A is the matrix, det(A) is the determinant of the matrix, and adj(A) is the adjugate of the matrix. The transposition of a matrix's cofactor matrix is adjugate. The cofactor matrix is a matrix of the determinants of the minors of the original matrix, each of which is multiplied by a cofactor (-1)^(i+j). The minors of a matrix are the determinants of the square submatrices formed by deleting one row and one column from the matrix. The determinant of a matrix is a scalar value that can be computed from its elements.
For example, consider the matrix A:
[a b]
[c d]
The determinant of A is:
det(A) equals (a * d) - (b * c).
The cofactor matrix of A is:
[d -b]
[-c a]
The adjugate of A is the transpose of the cofactor matrix, so it is:
[d -c]
[-b a]
The inverse of A is then:
A^(-1) = (1/det(A)) * adj(A)
= (1/((a * d) - (b * c))) * [d -c]
[−b a]
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zhi noticed that an odd number had appeared eight times in a row on the following roulette table. she now felt very strongly that an even number would be next. on a single spin, the probability of getting an even number is 18 over 38. considering the last eight spins resulted in odd numbers, what is the probability of getting an even number on the next spin? a.) 18 over 38 because each spin is independent. b.) 0 because each spin is independent. c.) 8 over 38 because each spin is dependent. d.) 1 over 38 because each spin is dependent.
The probability of getting an even number on the next spin is a.) 18 over 38 because each spin is independent.
Probability is honestly how possibly some thing is to happen. Whenever we are uncertain approximately the final results of an occasion, we are able to communicate approximately the chances of positive outcomes—how possibly they are. The evaluation of occasions ruled via way of means of possibility is referred to as statistics. Probability is calculated via way of means of dividing the quantity of methods the occasion can arise via way of means of the overall quantity of outcomes. Probability and odds are one-of-a-kind concepts. Odds are the possibility that some thing occurs divided via way of means of the possibility that it would not happen.
Thus, the correct option is a.
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How do you find the height of a pyramid if the base area and the volume is known?
A square pyramid's volume and base area allow figuring out its height simple. H = 3 * V / (A_b).
Explain height of a square pyramid?The distance between the base and the vertex of a square pyramid, commonly known as its height or altitude. We discover the height of the pyramid when questioned how tall it is.Pyramids have volume because they are solid shapes. And from there, we can extrapolate the height of the a pyramid.How to determine a square pyramid's height
Knowing the volume and base area of a square pyramid makes calculating its height simple.You just need to divide each component by the base area and multiply its volume by three, as stated with in formula below:H = 3 * V / (A_b)
Where:
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prove that 1 - cosx / sinx = sinx / 1 + cosx
how?
Answer:
see explanation
Step-by-step explanation:
using the identity
cos²x = 1 - sin²x
consider the left side
[tex]\frac{1-cosx}{sinx}[/tex]
multiply numerator/ denominator by (1 + cosx)
= [tex]\frac{(1-cosx)(1+cosx)}{sinx(1+cosx)}[/tex] ← expand numerator
= [tex]\frac{1-cos^2x}{sinx(1+cosx)}[/tex]
= [tex]\frac{sin^2x}{sinx(1+cosx)}[/tex] ← cancel sinx on numerator/ denominator
= [tex]\frac{sinx}{1+cosx}[/tex]
= right side , thus proven
What is the equation of a line that is perpendicular to 2y 3x 1?
-2/3 is the slope of equation of a line that is perpendicular to 2y = 3x - 1.
How do one find the slope of a graph?The slope is computed mathematically as "rise over run" (change in y divided by change in x).
In accordance with the slope equation, the slope of such a line may be determined by dividing the rise of the line between two positions by the run of the same line between the same two sites.
We know that the equation of the line in its slope-intercept form is:
y = mx+b
Where m = slope
b = y - intercept
A line that is perpendicular to it is y = (-1/m) x + b
The slope of the perpendicular line is -1/m
Rewriting the given equation 2y = 3x - 1, we get:
y = (3/2) x - 1/2
So, the slope of the perpendicular line is the reciprocal of 3/2 which is multiplied by -1.
The slope of the line = -2/3
Therefore, the slope of a line that is perpendicular to the line whose equation is 2y = 3x - 1 is -2/3.
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The complete question is as follows:
What is the equation of a line that is perpendicular to 2y = 3x - 1?
How to solve 3x - 5 =40
Answer:
[tex]x=15[/tex]
Step-by-step explanation:
[tex]3x-5=40\\3x=40+5\\3x=45\\x=\frac{45}{3} \\x=15[/tex]
JS 19. Evaluate the integral || curlĒ dĞ using Stoke’s Theorem, where F =-yi+xj+ryck and S is the part of the sphere r2 + y2 + 22 = 4 that lies above the cy-planes, oriented upward. A. 27 B. 0 C. 87 D. -87 E. 47
The integral || curl F || dĞ is equal to -2. The correct answer is therefore option D that is -87.
To evaluate the integral || curl F || dĞ using Stoke's Theorem, we need to find a vector field F and a surface S such that the boundary of S is a closed curve C and the integral is equal to the flux of the curl of F through S. Using the given information, we can find the components of F as:
F = (-y, x, r)
The curl of F is given by:
curl F = (0, 0, 1)
The surface S is the part of the sphere r2 + y2 + z2 = 4 that lies above the cy-planes, oriented upward. This surface can be parameterized by:
x = rcosφcosθ
y = rcosφsinθ
z = rsinφ
where 0 ≤ φ ≤ π and 0 ≤ θ ≤ 2π.
The boundary of S is the curve C, which is the intersection of the sphere and the cy-plane. This curve can be parameterized by:
x = rcosθ
y = rsinθ
z = 0
where 0 ≤ θ ≤ 2π.
Using Stoke's Theorem, the integral || curl F || dĞ is equal to the flux of curl F through S, which is given by:
∫|| curl F || dĞ = ∫curl F · ndC
where n is the unit normal vector to the surface S.
Substituting the expressions for curl F and the parameterizations of S and C, we get:
∫|| curl F || dĞ = ∫(0, 0, 1) · (cosφcosθ, cosφsinθ, sinφ)dC
= ∫(sinφ)dC
= ∫sinθdθ
= [-cosθ]θ=0θ=2π
= -2
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What is a transformation in math grade 7?
A transformation in mathematics is a process of manipulating a polygon.
A transformation in maths is nothing but a process that manipulates a polygon or any other two-dimensional object on a plane or coordinate system. A transformations describe how two-dimensional figures move around a coordinate system.
The original shape of the object before any transformation is called the Pre-Image and the final shape and position of the object is called as the Image.
In mathematics, there are five different transformations:
Dilation, Reflection, Rotation, Shear and Translation
Therefore, transformation is a process of manipulating a polygon.
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What is the formula of mean method?
The formula of mean method is [tex]\frac{Sum of all values}{Number of values}[/tex].
The formula for the mean, also known as the average, is:
mean = [tex]\frac{sum of all values}{number of values}[/tex]
For example, if you have a list of numbers [1, 2, 3, 4], the mean would be calculated as follows:
mean = (1 + 2 + 3 + 4) / 4 = 10 / 4 = 2.5
The mean is a measure of central tendency that gives you an idea of the typical value in a set of numbers. It is calculated by adding up all the values in the set and dividing the sum by the number of values.
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Is 6th grade math hard?
Answer:
not really
Step-by-step explanation:
What are the 3 triangle similarity theorems?
Triangle similarity theorems are essential to geometry and can be used to compare two triangles. The three triangle similarity theorems are:
Side-Side-Side (SSS) Similarity TheoremAngle-Angle-Side (AAS) Similarity TheoremAngle-Side-Angle (ASA) Similarity Theorem3 Triangle similarity theoremsThe Side-Side-Side (SSS) Similarity Theorem: If the three sides of one triangle are proportional to the three sides of a second triangle, then the two triangles are similar. The Angle-Angle-Side (AAS) Similarity Theorem: If two angles of one triangle are equal to two angles of a second triangle, and the included side of the first triangle is proportional to the included side of the second triangle, then the two triangles are similar. The Angle-Side-Angle (ASA) Similarity Theorem: If two angles of one triangle are equal to two angles of a second triangle, and the sides between them are proportional, then the two triangles are similar.Learn more about similar triangles: https://brainly.com/question/14285697
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What is domain in sets?
The set of all initial elements in ordered pairs constitutes the domain (x-coordinates).
Given that,
We have to find what us domain in sets.
We know that,
What is a domain?We learned the distinction between a relation and a function in Function Basics.
We are dealing with relationships expressed as ordered pairs in both scenarios.
The domain refers to all possible values that can be used as inputs into a relation or function.
The range refers to all of the values that are produced as the result of a relation or function.
The set of all initial elements in ordered pairs constitutes the domain (x-coordinates).
The set of all second elements in ordered pairs (y-coordinates) is the range.
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Consider the linear function y=5x+12y=5x+12 and the linear function represented by the table of values below.
Answer:
Step-by-step explanation:
y intercept = 12 the one with no x
use process of elimination
function=straight line
What will be the median of the given data if the mean is 4.5 5 7 7 8 a 5 4 3 1 2?
The median for a given set of observations is 4.
What is mean?The mean is the average of a set of variables in mathematics and statistics. The mean may be calculated in a variety of methods, including the simple arithmetic mean (add the numbers and divide the total by the number of observations), geometric mean, and harmonic mean. A data set's mean (average) is calculated by summing all of the numbers in the set and then dividing by the number of values in the set. When a data collection is arranged from least to largest, the median is the midway value. The mode is the number that appears the most frequently in a data collection.
Here,
given set of observation,
=5,7,7,8,a,5,4,3,1,2
4.5=(5+7+7+8+a+5+4+3+1+2)/10
45=42+a
a=3
median=5,7,7,8,3,5,4,3,1,2
=1,2,3,3,4,5,5,7,7,8
median=10/2
=5th term
=4
The median for given set of observation is 4.
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What is a surface area in math?
Answer: the amount of space covering the outside of a three-dimensional shape.
Step-by-step explanation:
What is it called when you do 1 2 3 4?
Natural Numbers The numbers we use to count or arrange are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Whole Numbers the sum of natural numbers and zero. not a fraction or decimal.
what is counting ?The counting-on method can be used by mathematicians to add numbers. With this approach, a student first adds the larger number, then "counts on" with the other addends to arrive at the sum. If the number sentence is 4 + 3, the student will understand that 4 is the greater number and will add three additional numbers, such as the integers "4... 5, 6, 7."
here ,
For those of you who are not aware with this series, it indicates that the sum of all natural numbers, i.e., 1, 2, 3, 4, and so on, up to infinity .
Is equal to -1/12 and has come to be known as the Ramanujan Summation .
Whole Numbers the sum of natural numbers and zero. not a fraction or decimal.
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find the number that completes the equation 6/5 divided blank = 6
Answer:
1/5 or 0.2
Step-by-step explanation:
[tex]\frac{6}{5}[/tex] multiplied by 5 or [tex]\frac{5}{1}[/tex]
[tex]\frac{6}{5}[/tex]x[tex]\frac{5}{1}[/tex]=6
so when dividing , you reciprocate the value after the division sign.
[tex]\frac{6}{5}[/tex]÷[tex]\frac{1}{5}[/tex]=[tex]\frac{6}{5}[/tex]x[tex]\frac{5}{1}[/tex]=6
therefore, "blank"= 1/5 or 0.2
How to learn maths fast?
Learning mathematics quickly requires dedication and hard work.
What is mathematics?
Mathematics is the science of numbers, shapes, and patterns. It is used to describe and explain the physical world and to solve problems. Mathematics is a language that helps us to understand and communicate ideas. It is a tool used to organize and understand data, determine relationships, and make predictions.
According to the given question:
The first step to learning math quickly is to determine your level of understanding. It is important to identify any weaknesses in order to develop a plan to improve them. Once the strengths and weaknesses are identified, it is important to set realistic goals and to create an action plan to reach those goals.
It is also important to find a good tutor or mentor to help guide the learning process. A tutor or mentor can provide guidance, resources, and practice problems to help increase understanding of the subject. Additionally, they can help identify problem areas and provide feedback that can help improve learning.
Finally, it is important to take advantage of online resources. There are many online tutorials, websites, and videos that can help improve the learning process.
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Find the perimeter of the triangle
(21-x) in.
(2x - 3) in.
(x + 5) in.
The Perimeter of the triangle is 23+2x in³
We know that formula of the Perimeter of the triangle is:
Perimeter= A+B+C
where A=(21-x)in.
B=(2x-3)in.
C=(x+5)in.
Solution:
Perimeter = (21-x)+(2x-3)+(x+5)
Perimeter = (18+x)+ (x+5)
Perimeter = 23+2x in³
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Is √ 3 is polynomial or not?
Yes, 3x² is a polynomial, but a special form of a polynomial with just one term which is called monomial.
What is Polynomial?Polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables
The given expression is 3x².
In the given expression three times of x square, x is the variable.
For an expression to be a polynomial term, it must contain no square roots of variables, no fractional or negative powers on the variables, and no variables in the denominators of any fractions.
Yes, this is a polynomial, but a special form of a polynomial with just one term which is called monomial.
Hence, 3x² is a polynomial.
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At a local college, only 30% of students live off campus. Of those who live off campus, 62% of those students get a part-time job. Of those who live on campus, 65% work part-time. The tree diagram shows how the college students are divided into subgroups.
The tree diagram shows college students branching off into two categories, off campus and on campus students. Off campus students branches off into two sub-categories, work part-time and do not work. On campus branches off into two subcategories, work part-time and do not work.
What is the percentage of students who live on campus who do not have a part-time job?
18.6%
24.5%
35%
38%
Answer:
B, 24.5
Step-by-step explanation:
I just took the test
What is a median of 25?
The median of the first 25 whole numbers is 12.
The median is the middle point in a dataset—half of the data points are smaller than the median and half of the data points are larger.
The data set's median is the value in the center. Put all the numbers in ascending or descending order, and then choose the middle number to determine the median. There will only be one middle integer if the number of data points is odd. If the number of data points is even, you must choose the middle two numbers, add them together, and divide by two.
The first 25 whole numbers are as follows:-
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24
Here total no. of terms = n = 25 which is odd.
Therefore median = (n + 1)/2 th term
= (25 + 1)/2 th term
= 26/2 th term
= 13th term
Here, the 13th term is 12
Hence, the median of the first 25 whole numbers is 12.
Complete question: What is the median of the first 25 whole numbers?
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John wants to purchase at least 4 books and 5 notebooks from a stationery shop. The
cost of each book is $8 and the cost of each notebook is $4. Which inequality correctly
represents the minimum amount of money he should carry?
The inequality that represents the minimum amount of money he should carry is 4x+5y ≥ 52
What is inequality?
An inequality in mathematics depicts the connection between two values in an algebraic statement that are not equal. One of the two variables on the two sides of the inequality may be greater than, greater than or equal to, less than, or less than or equal to another value, according to inequality signals.
Let x be the cost of each books
Let y be the cost of each notebook
Cost of 4 books = $32
Cost of 5 notebooks = $20
Total cost = 32+20 = $52
At least shows the inequality ≥
So, the inequality that represents the minimum amount of money he should carry is
4x+5y ≥ 52
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I need some help understanding these questions, is there anyone willing to help? I'd really appreciate it!
Consider how the structure of the standard form of a linear function
reveals its key characteristics.
For the equation Ax + By = C , determine the slope, x -intercept,
and y -intercept.
Which form of a linear equation is more efficient for determining each characteristic?
- Slope
- x -intercept
- y -intercept
Answer:
Slope: slope-intercept form, point-slope form, or slope formX-intercept: intercept form, standard formY-intercept: slope-intercept form, intercept formStep-by-step explanation:
You want to know how the form of a linear equation can be used to reveal its key characteristics.
Key CharacteristicsThe characteristics of a linear equation that are of interest for this problem are ...
slopex-intercepty-interceptForms of a linear equationA linear equation can be written in many forms. Some of these forms make it easy to determine one or more of the key characteristic listed above. Some of the more common forms are Slope-Intercept form, Point-Slope form, and Standard form. There are a number of others.
Slope-Intercept formThe slope-intercept form is ...
y = mx + b
where m is the slope, and b is the y-intercept. This form makes it easy to determine ...
slopey-interceptThis equation cannot be used to describe a vertical line.
Alternate Slope-Intercept formWe can rearrange the above equation to reveal the x-intercept.
y = m(x -a)
where m is the slope and 'a' is the x-intercept. This form makes it easy to determine ...
slopex-interceptThis equation cannot be used to describe a horizontal line (except y=0).
Point-Slope formThe equation of a line with slope m through point (h, k) can be written as ...
y -k = m(x -h)
This makes it easy to determine ...
slopea point on the line (not necessarily an intercept)This equation cannot be used to describe a vertical line.
Standard formThe standard form equation of a line is similar in style to the standard form equations for other geometric figures (circle, ellipse, hyperbola). This form is ...
Ax +By = C
where A, B, C are mutually prime (integers), and A ≥ 0. If A = 0, then B > 0. That is, the leading coefficient is positive. Integers are preferred for the coefficients, but some lines have irrational slope or other key characteristics, so integers may not be able to be used in all places.
The key characteristics cannot be read from the equation directly, but they are one step away:
slope = -A/Bx-intercept = C/Ay-intercept = C/BSolving for the intercepts is especially easy, so we list this form as one that can be efficient for determining the intercepts.
Intercept formA very nice form for efficiently determining the intercepts, or efficiently writing an equation from the intercepts is ...
x/a +y/b = 1
where 'a' is the x-intercept, and 'b' is the y-intercept. This form makes it very easy to determine ...
x-intercepty-interceptYou may notice that dividing the standard form equation by C and rearranging the coefficients will give you this form:
Ax +By = C
(A/C)x +(B/C)y = 1 . . . . . . . divide by C
x/(C/A) +y/(C/B) = 1 . . . . . . . intercept form
General formThe general form of a polynomial equation of any degree is ...
f(x, y) = 0
The general form of a linear equation is ...
Ax +By +C = 0
This is simply a rearrangement of the Standard form equation. Equations written in this form can be helpful for ...
finding the distance from a point to the linesolving a system of equations by "addition" or matrix methodsVarious 2-Point formsWhen two points are given, and you want an equation of the line, various forms are available for using those coordinates. These forms make use of the fact that the slope of a line is the same everywhere, so the ratio of any difference of y-coordinates to the corresponding difference of x-coordinates is a constant (the slope). Again, slopes of 0 or "undefined" can cause trouble with certain of these forms. Here are several different arrangements of 2-point forms:
[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)\\\\(x_2-x_1)(y-y_1)=(y_2-y_1)(x-x_1)\\\\\dfrac{y-y_1}{x-x_1}=\dfrac{y_2-y_1}{x_2-x_1}=m\qquad\text{Slope form}\\\\\\\dfrac{x-x_1}{y-y_1}=\dfrac{x_2-x_1}{y_2-y_1}\qquad\text{inverted from the above}\\\\\\\dfrac{y-y_1}{y_2-y_1}=\dfrac{x-x_1}{x_2-x_1}[/tex]
The last three "ratio forms" are variations of each other. Any such proportion can be written 4 ways. (3 are shown) The form not shown has the variables in the denominator, which is usually not useful.
All of these forms can be written in a "general form" by subtracting one side.
The slope can be read directly from the 1st and 3rd of the forms shown here. It is m = (y2-y1)/(x2-x1).
The slope form is useful for determining ...
slopeDeterminant formAnother 2-point form is one written as a determinant:
[tex]\left|\begin{array}{cc}x-x_1&x_2-x_1\\y-y_1&y_2-y_1\end{array}\right|=0[/tex]
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Additional comment
It can be useful to recognize the various forms and their relationship to known points on the line or other key characteristics of the function. Any of the slope or point-slope forms make it easy to write the equations of parallel or perpendicular lines through a given point. (The standard and general forms can be used for this, too.)
Whats 416 divided by 3 with remainder
Answer:
416 / 3 = 128 with a remainder of 2.
Step-by-step explanation:
When 416 is divided by 3 can be expressed as 138 with a remainder of 2.
When we divide 416 by 3, the quotient represents the whole number result of the division, while the remainder represents the amount left over.
Dividing 416 by 3 gives us:
416/3 = 138 + 2/3
Quotient: 416/3 = 138
Remainder: 416/3 = 2
However, there is a remainder of 2. The remainder represents the amount that is left over after dividing as much as possible. In this case, after dividing 416 by 3, there is 2 left over that cannot be evenly divided.
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