solve -3(x-3)≤ 5(1-x)
5(2x - 1) + 3(x - 3) = -(4x - 6) + 2(13 - 3x)
Answer:
The value of x is 2.
Step-by-step explanation:
5(2x -1) + 3(x -3) = -(4x - 6) + 2(13 -3x)
Expand both sides of the equation to remove parenthesis
10x - 5 + 3x - 9 = -4x + 6 + 26 - 6x
Transpose all terms with x as the coefficient on the left side of the equation and transpose the constants to the right.
10x + 3x + 4x + 6x = 6 + 26 + 5 + 9
Simplify
23x = 46
Divide both sides of the equation by 23
23x/23 = 46/23
x = 2
a blanket measures 1 1/4 yards on each side. how many square yards does the blanket cover?
Answer:
Add 1 1/4 x 1 1/4
Step-by-step explanation:
Kelly went to a store to purchase a coffee pot. She will use a coupon for 20% off. She can calculate the cost before sales tax using the following expression, where c
represents the original cost of the coffee pot.
c−0.2c
Which other expression could Kelly use to calculate her cost before sales tax?
A. 0.8c
B. 1.2c
C.1.8c
D.80c
in testing a hypothesis from a random sample involving three or more means, the appropriate test would be a one-way anova. True or false?
Using Trig to find a side.
Solve for x. Round to the nearest tenth, if necessary.
Answer:
[tex]\large\boxed{\tt x \approx 95.6}[/tex]
Step-by-step explanation:
[tex]\textsf{We are asked to solve for x by using \underline{Trigonometric Identities}.}[/tex]
[tex]\large\underline{\textsf{What are Trigonometric Identities?}}[/tex]
[tex]\boxed{\begin{minipage}{20 em} \\ \underline{\textsf{\large Trigonometric Identities;}} \\ \\ \textsf{Trigonometric Identities are trigonometric ratios determined with what's given in order to find a missing value. For a Right Triangle, the Trigonometric Identities are Sine, Cosine, and Tangent. These are used to find missing sides.} \\ \\ \tt Sine = \tt $ \tt \frac{Opposite}{Hypotenuse} \\ \\ Cosine = \frac{Adjacent}{Hypotenuse} \\ \\ Tangent = \frac{Opposite}{Adjacent} \end{minipage}}[/tex]
[tex]\textsf{We should determine whether Sine, Cosine, or Tangent will actually help us}[/tex]
[tex]\textsf{determine x. We are given a Right Triangle that has 1 15}^{\circ} \ \textsf{angle, and a side with}[/tex]
[tex]\textsf{a length of 99. Because this side is opposite of the right angle, this side is called}[/tex]
[tex]\textsf{the \underline{Hypotenuse}.}[/tex]
[tex]\textsf{The side labeled x is \underline{Adjacent}, which means that it's touching the given angle.}[/tex]
[tex]\textsf{Using what was given to us, we should use Cosine since we are asked for the}[/tex]
[tex]\textsf{Adjacent Angle when given the Hypotenuse.}[/tex]
[tex]\large\underline{\textsf{Solving;}}[/tex]
[tex]\textsf{Remember that;}[/tex]
[tex]\tt \cos(15^{\circ}) =\frac{Adjacent}{Hypotenuse}[/tex]
[tex]\textsf{We're given;}[/tex]
[tex]\tt \cos(15^{\circ}) =\frac{x}{99}[/tex]
[tex]\textsf{To find the value of x, we first should remove the fraction using cancellation.}[/tex]
[tex]\textsf{We are able to use the \underline{Multiplication Property of Equality} to prove that the}[/tex]
[tex]\textsf{equation remains equal.}[/tex]
[tex]\underline{\textsf{Multiply both expressions by 99;}}[/tex]
[tex]\tt 99 \cos(15^{\circ}) =\not{99} \frac{x}{\not{99}}[/tex]
[tex]\tt 99 \cos(15^{\circ}) =x[/tex]
[tex]\underline{\textsf{Evaluate;}}[/tex]
[tex]\tt 99 \cos(15^{\circ}) \approx \boxed{\tt 95.6}[/tex]
[tex]\large\boxed{\tt x \approx 95.6}[/tex]
Tonya is making a circle graph of the data shown in the table. How many degrees will the section for piano have?
Guitar: 1/4
Piano: 1/6
Drums 13/45
Flute 1/10
Trumpet 7/36
The section for piano will have 60 degrees in the circle graph.
How to solveTo find out how many degrees the section for piano will have in the circle graph, we first need to find the fraction of the circle that the piano represents.
The total degrees in a circle is 360 degrees. We can use the given fraction of piano (1/6) and multiply it by 360 degrees to find out the degrees of the section for piano.
Degrees for piano = (Fraction of piano) × 360
Degrees for piano = (1/6) × 360
Now, multiply the fraction by 360:
Degrees for piano = 60
So, the section for piano will have 60 degrees in the circle graph.
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→ 60° ( btw, ° means degrees )
— if a WHOLE circle is 360° and the piano is 1/6 of the circle, then all you have to do is multiply 1/6 by 360° which gives you 60 OR 60°
HEY- you look hungry, eat up !! →
SALES An automobile company sold 2.3 million new cars in a year. If the average price per car was $21,000, how
much money did the company make that year? Write your answer in scientific notation.
Therefore, the company made $48.3 million (written in scientific notation as 4.83 x 10⁷) that year from selling new cars.
What is equation?An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equals sign (=). The expressions on both sides of the equals sign must have the same value for the equation to be true. Equations can involve a wide range of mathematical operations such as addition, subtraction, multiplication, division, exponents, and roots. They are used to solve problems in various fields such as physics, engineering, economics, and many others.
Here,
To find the total revenue generated by the company, we need to multiply the number of new cars sold by the average price per car. We can do this as follows:
Total revenue = number of new cars sold x average price per car
Total revenue = 2.3 million x $21,000
To multiply these two numbers, we can use the distributive property:
Total revenue = (2.3 x 10⁶) x ($21,000)
Total revenue = 2.3 x $21 x 10⁶
Multiplying 2.3 by 21 gives us 48.3, which we can write in scientific notation as 4.83 x 10¹. We can then add the exponents to get:
Total revenue = 4.83 x 10¹ x 10⁶
Total revenue = 4.83 x 10⁷
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Suppose that $18,000 is invested at 5. 2% compounded. Find the total amount of this investment after 7 years
If $18,000 is invested at 5. 2% compounded then the full sum of the investment after 7 long years is roughly $24,810.89.
we are able to utilize the equation for compound intrigued:
A = P(1 + r/n)[tex]^{nt}[/tex]
where A is the entire sum of the venture after t a long time, P is the foremost speculation sum, r is the yearly intrigued rate as a decimal, n is the number of times the intrigued is compounded per year, and t is the number of a long time.
In this case, P = $18,000, r = 0.052 (since the intrigued rate is 5.2%), n = 1 (since the intrigued is compounded every year), and t = 7 (since we need to discover the full sum after 7 a long time). Substituting these values into the equation, we get: A = 18000(1 + 0.052/1)[tex]^{1*7}[/tex]
= 18000(1.052)[tex]^{7}[/tex]
= $24,810.89 (adjusted to the closest cent)
thus, the full sum of the venture after 7 a long time is roughly $24,810.89.
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Your cousin bought a used car. He paid a total of $12,755 for the car. The total cost includes: • The list price of the car, x • A 7% sales tax on the list price . Plus $450 in additional fees.
What is the price of the car your cousin bought?
List price:
Let's use the variable x to represent the list price of the car.
The total cost your cousin paid for the car is the list price plus a 7% sales tax on the list price, plus $450 in additional fees.
We can write this as an equation:
Total cost = List price + 0.07(List price) + 450
We know that the total cost your cousin paid was $12,755, so we can substitute that into the equation:
12,755 = x + 0.07x + 450
Simplifying the right side:
12,755 = 1.07x + 450
Subtracting 450 from both sides:
12,305 = 1.07x
Dividing both sides by 1.07:
x = 11,500
Therefore, the list price of the car your cousin bought was $11,500.
The list price of the car your cousin bought was $11,500.
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example:
1 + 3x + 4y = 7 is an expression.com
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
Let's start by setting up an equation to represent the total cost of the car:
Total cost = List price + 7% of List price + $450
We know that the total cost is $12,755, so we can substitute that in:
$12,755 = List price + 0.07(List price) + $450
Simplifying this equation, we get:
$12,755 = 1.07(List price) + $450
Subtracting $450 from both sides, we get:
$12,305 = 1.07(List price)
Dividing both sides by 1.07, we get:
List price = $11,500
Therefore,
The list price of the car your cousin bought was $11,500.
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if x is a matrix of centered data with a column for each field in the data and a row for each sample, how can we use matrix operations to compute the covariance matrix of the variables in the data, up to a scalar multiple?
To compute the covariance matrix of the variables in the data, the "matrix-operation" which should be used is ([tex]X^{t}[/tex] × X)/n.
The "Covariance" matrix is defined as a symmetric and positive semi-definite, with the entries representing the covariance between pairs of variables in the data.
The "diagonal-entries" represent the variances of individual variables, and the off-diagonal entries represent the covariances between pairs of variables.
Step(1) : Compute the transpose of the centered data matrix X, denoted as [tex]X^{t}[/tex]. The "transpose" of a matrix is found by inter-changing its rows and columns.
Step(2) : Compute the "dot-product" of [tex]X^{t}[/tex] with itself, denoted as [tex]X^{t}[/tex] × X.
The dot product of two matrices is computed by multiplying corresponding entries of the matrices and summing them up.
Step(3) : Divide the result obtained in step(2) by the number of samples in the data, denoted as "n", to get the covariance matrix.
This step scales the sum of the products by 1/n, which is equivalent to taking the average.
So, the covariance matrix "C" of variables in "centered-data" matrix X can be expressed as: C = ([tex]X^{t}[/tex] × X)/n.
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The given question is incomplete, the complete question is
Let X be a matrix of centered data with a column for each field in the data and a row for each sample. Then, not including a scalar multiple, how can we use matrix operations to compute the covariance matrix of the variables in the data?
Find all of the cube roots of 216i and write the answers in rectangular (standard) form.
The cube roots of 216 written in the rectangular (standard) form are 3 + 3√3, -3+3√3, and 6.
What is a cube root?In mathematics, the cube root formula is used to represent any number as its cube root, for example, any number x will have the cube root 3x = x1/3. For instance, 5 is the cube root of 125 as 5 5 5 equals 125.
3√216 = 3√(2x2x2)x(3x3x3)
= 2 x 3 = 6
the prime factors are represented as cubes by grouping them into pairs of three. As a result, the necessary number, which is 216's cube root, is 6.
Therefore, the cube roots of 216 are 3 + 3√3, -3+3√3, and 6.
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what is the 4th term/number of (a+b)^9, pascal’s triangle?
Step-by-step explanation:
hope this will help you Thanks
Find the length of the side labeled x. Explain.
The value of the length marked x is 61.5
What is trigonometrical ratio?Trigonometric ratios in trigonometry relate the ratio of sides of a right triangle to the respective angle. The basic trigonometric ratios are sine, cosine, and tangent ratios. The other important trig ratios, cosec, sec, and cot can be derived using the sin, cos and tan respectively.
First using left hand side
Cos = Adj/Hypo
Cos22 = Adj/50
Adj = 50 * Cos 22
The adj = 50*0.9272
The adj = 46.4
The to find x, using
Cos 41 = 46.4/x
xCos41 =46.4
x = 46.4/Cos41
x = 46.4/.0755
Therefore the value of x = 61.5
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Fill in the blanks with the appropriate justifications (reasons) for the steps used in solving the equation. Statements
statements:
1. x/2-9=-4 given
2. x/2=-13
3.x=-26
what are the justifications for 2 and 3
The justifications for 2 and 3 are;
Addition property of equality and
Multiplication property of equality
What are the justifications for the steps used in solving the equation?1. x/2 - 9 = -4 given
Add 9 to both sides
2. x/2 = -13 (Addition property of equality)
cross product
3.x = -26 (Multiplication property of equality)
Therefore, x/2 - 9 = -4 where x Is -26 is justified by addition and multiplication property of equality.
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brainlist
show all steps nd i will make u brainlist
Step-by-step explanation:
Again, using similar triangle ratios
7.2 m is to 2.4 m
as AB is to 12.0 m
7.2 / 2.4 = AB/12.0 Multiply both sides of the equation by 12
12 * 7.2 / 2.4 = AB = 36.0 meters
1. Find the square root of each of the following numbers: (i) 152.7696
Ariel wants to join the volleyball team in the fall, so she went to an overnight volleyball camp last weekend to practice her skills. Before she left, she packed a shower caddy shaped like a rectangular prism with a volume of 420 cubic inches. The caddy is 10
1
2
inches long and 5 inches tall.
How wide is the shower caddy?
The width of the shower caddy is 8 inches.
What is Volume?
Volume is a measure of the total amount of material that an object contains. In mathematical terms, volume is calculated by multiplying the length, width, and height of an object.
We can use the formula for the volume of a rectangular prism, which is V = lwh, where l is the length, w is the width, and h is the height.
We know that the volume of the caddy is 420 cubic inches, the length is 10.5 inches, and the height is 5 inches. Let's substitute these values into the formula and solve for the width:
420 = 10.5w × 5
Divide both sides by 52.5:
8 = w
Therefore, the width of the shower caddy is 8 inches.
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need help in a test Write the decimal as a percent 9.66 pls help will make Brainlyist
Answer:966%
Step-by-step explanation:
To convert a decimal to a percent, we need to multiply the decimal by 100 and add a percent sign. So, to convert 9.66 to a percent, we do the following:
9.66 × 100% = 966%
Therefore, 9.66 is equivalent to 966% as a percent.
Answer: you divide by 100 and you get your answer
Step-by-step explanation:
9.66 ÷ 100 = 0.0966
Craig gets a bonus with his club for every frisbee golf hole on which he makes a score of 3. He played last week and scored a total of 3 on 5 holes. Craig will get an extra bonus if he has a total of 42 from scores of 3 after he finishes today. On how many holes does he need to score a 3 today?
The total number of holes Craigs need to have a score of 3 today is equal to 37.
On every every frisbee golf hole having a score 3 = one bonus.
Total scored while playing last week = 3 on 5 holes
Total extra bonus scored by Craig = 5
Getting extra bonus on total = 42 from score of 3
let us consider Craigs need 'x' holes on a score of 3 today
Required equation is,
x + 5 = 42
Subtract 5 from both the side of the equation we get,
⇒ x = 42 - 5
⇒ x = 37 holes
Therefore, Craigs need to have 37 holes to score a 3 today.
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hat is the maximum speed of a point on the outside of the wheel, 15 cm from the axle?
It depends on the rotational speed of the wheel. To calculate this speed, we need to know the angular velocity of the wheel.
The maximum speed of a point on the outside of the wheel, 15 cm from the axle, if we assume that the wheel is rotating at a constant rate, we can use the formula v = rω, where v is the speed of the point on the outside of the wheel, r is the radius of the wheel (15 cm in this case), and ω is the angular velocity of the wheel. Therefore, the maximum speed of a point on the outside of the wheel would be directly proportional to the angular velocity of the wheel.
The formula to calculate the maximum linear speed (v) is:
v = ω × r
where v is the linear speed, ω is the angular velocity in radians per second, and r is the distance from the axle (15 cm, or 0.15 meters in this case).
Once you have the angular velocity (ω) of the wheel, you can plug it into the formula and find the maximum speed of a point on the outside of the wheel.
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3x - 1 <_ 2x what is the answer
Answer:
x<-1/5
Step-by-step explanation:
Imma assume the _ is a - soo
3x-1<-2x
5x<-1
x<-1/5
Find the missing side.
The measure of the unknown side from the given triangle is 14.48.
Solving trigonometry identityThe given triangle is a right triangle with the following sides;
Hypotenuse = 15
Adjacent = x
Acute angle = 52 degrees
We are to determine the measure of the unknown side using trigonometry identity
Cos 15 = Adjacent/Hypotenuse
Cos 15 = x/15
x = 15cos15
x = 15(0.9659)
x = 14.48
Hence the measure of the unknown side is 14.48
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Theorem: A line parallel to one side of a triangle divides the other two proportionately.
In the figure below, segment DE is parallel to segment BC and segment EF is parallel to AB:
Which statement can be proved true using the given theorem?
Group of answer choices
Segment BD = 12
Segment BD = 4
Segment BF = 16
Segment BF = 9
Since BC = 12 and AB = 16, segment BF must be 9, which is one-half of 16.
What is triangle?A triangle is a three-sided polygon that is one of the basic shapes in geometry. It is defined by three points that are connected by three line segments. Triangles have three angles, which add up to 180 degrees, and three sides, which add up to the sum of the lengths of the other two sides. The three sides of a triangle are typically referred to as the base, the height, and the hypotenuse. Triangles come in a variety of forms, from the equilateral triangle, which has three sides of equal length, to the isosceles triangle, which has two sides of equal length, to the scalene triangle, which has three sides of different lengths.
The correct answer is: Segment BF = 9. This can be proved true using the given theorem, since segment DE is parallel to segment BC and segment EF is parallel to AB. Therefore, since BC = 12 and AB = 16, segment BF must be 9, which is one-half of 16.
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the coefficient of determination resulting from a particular regression analysis was 0.85. what was the slope of the regression line?
The slope of the regression line is 0.922. Option C is correct.
The coefficient of determination (R-squared) is defined as the proportion of the variance in the dependent variable that is explained by the independent variable(s) in a linear regression model. It can be calculated as the square of the correlation coefficient (r) between the dependent and independent variables.
If we assume a simple linear regression model with a single independent variable (x) and a single dependent variable (y), the slope of the regression line (b) can be calculated as:
b = r * (Sy / Sx)
where r is the correlation coefficient between x and y, Sy is the standard deviation of y, and Sx is the standard deviation of x.
If the coefficient of determination is 0.85, then the correlation coefficient is the square root of 0.85, which is approximately 0.922. Assuming the standard deviations of x and y are known, we could use the formula above to calculate the slope of the regression line as:
b = 0.922 * (Sy / Sx)
Therefore, 0.922, could be the slope of the regression line in this case, if the assumptions of a simple linear regression model with known standard deviations are met. Option C is correct.
The complete question is
The coefficient of determination resulting from a particular regression analysis was 0.85. What was the slope of the regression line?
a. 0.85
b. -0.85
c. 0.922
d. There is insufficient information to answer the question.
e. None of the above
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Solve for y in the two equations below using substitution.
3x - 9y = 9
-2x - 2y = 8
Answer:
C
Step-by-step explanation:
3x - 9y = 9 → (1)
- 2x +2y = 8 ( subtract 2y from both sides )
- 2x = - 2y + 8 ( divide through by - 2 )
x = y - 4
substitute x = y - 4 into (1)
3(y - 4) - 9y = 9
3y - 12 - 9y = 9
- 6y - 12 = 9 ( add 12 to both sides )
- 6y = 21 ( divide both sides by - 6 )
y = [tex]\frac{21}{-6}[/tex] = - [tex]\frac{7}{2}[/tex]
John buys new baseball equipment for $2000. The purchase made is with a credit card that has a 19% APR. John makes a $150 payment monthly. How many months will it take John to pay off the balance?
It will take John approximately 17 months to pay off the balance.
What is simple interest?
Simple Interest (S.I.) is the method of calculating the interest amount for a particular principal amount of money at some rate of interest.
Assuming that John does not use the credit card for any other purchases and that the credit card company uses a simple interest calculation method, we can use the following steps to calculate the number of months it will take John to pay off the balance:
Calculate the monthly interest rate by dividing the annual percentage rate (APR) by 12:
Monthly interest rate = 19% / 12 = 0.01583
Calculate the monthly finance charge by multiplying the outstanding balance by the monthly interest rate:
Monthly finance charge = $2000 x 0.01583 = $31.66
Subtract the monthly payment from the monthly finance charge to get the amount that will be applied to the outstanding balance:
Payment applied to balance = $150 - $31.66 = $118.34
Divide the outstanding balance by the payment applied to balance to get the number of months it will take to pay off the balance:
Number of months to pay off balance = $2000 / $118.34 = 16.9
(rounded up to the nearest whole number)
Therefore, it will take John approximately 17 months to pay off the balance.
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A cone with a radius of 3 inches and a height of 18 inches is shown
What is the volume, in a cubic inches, of the cone. Round your answer to the nearest integer
Answer: about 169.65
Step-by-step explanation:
You do pi r squared multiplied by height over 3.
28.27 is the area of the circle and you multiply is by six because 18/3=6. 28.27x6 = about 169.65
Further statistical computation will be needed
mean
mode
median
By performing these statistical computations, you can analyze and interpret the central tendency of your dataset, which helps in understanding the overall pattern and distribution of the data.
It looks like you're seeking information on further statistical computation related to mean, mode, and median.
To calculate the mean, mode, and median of a dataset, follow these steps:
1. Mean: The mean is the average of all data points in a dataset.
- Step 1: Add up all the data points.
- Step 2: Divide the sum by the total number of data points.
2. Mode: The mode is the data point that occurs most frequently in a dataset.
- Step 1: Count the frequency of each data point.
- Step 2: Identify the data point(s) with the highest frequency.
3. Median: The median is the middle value in a dataset when the data points are arranged in ascending order.
- Step 1: Arrange the data points in ascending order.
- Step 2: If there is an odd number of data points, the median is the middle value. If there is an even number of data points, the median is the average of the two middle values.
By performing these statistical computations, you can analyze and interpret the central tendency of your dataset, which helps in understanding the overall pattern and distribution of the data.
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a. The most typical case is desired: Mode. The mode is a useful measure of central tendency when the most typical or common value is of relevance since it denotes the value or category that occurs most frequently in a data collection.
b. The distribution is open-ended: Median. The median is the middle value in a data set when arranged in ascending or descending order. It is a suitable measure of central tendency when the distribution is open-ended or skewed, as it is less affected by extreme values compared to the mean.
c. The data collection has an extreme value: the median. The median is less sensitive to extreme values compared to the mean, making it a better measure of central tendency in data sets with extreme values or outliers.
d. The data are categorical: Mode. The mode is appropriate for categorical data, as it represents the most frequently occurring category or value in the data set.
e. Further statistical computations will be needed: This statement does not indicate a specific measure of central tendency. Further statistical computations may be needed to determine the appropriate measure of central tendency depending on the characteristics of the data and the specific objectives of the analysis.
f. The numbers should be split into two roughly equal groups, one of which should contain the higher values and the other should contain the smaller values: Median. The median is the value that separates a data set into two equal halves, making it suitable for dividing data into two approximately equal groups based on their values.
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COMPLETE QUESTION-
For these situations, state which measure of central tendency - mean, median, or mode-should be used.
a. The most typical case is desired.
b. The distribution is open-ended.
c. There is an extreme value in the data set.
d. The data are categorical.
e. Further statistical computations will be needed.
f. The values are to be divided into two approximately equal groups, one group containing the larger values and one containing the smaller values.
Let s be the set of all orderd pairs of real numbers. Define scalar multiplication and addition on s by
In the 8 Axioms, 4 and 6 axioms fails to holds and S is not a vector space. Rest of the axioms try to hold the vector space.
To demonstrate that S is not a vector space, we must demonstrate that at least one of the eight vector space axioms fails to hold. Let us examine each axiom in turn:
Closure under addition: For any (x₁, x₂) and (y₁, y₂) in S, their sum (x₁ + y₁, 0) is also in S. This axiom holds.Commutativity of addition: For any (x₁, x₂) and (y₁, y₂) in S, (x₁ + y₁, 0) = (y₁ + x₁, 0). This axiom holds.Associativity of addition: For any (x₁, x₂), (y₁, y₂), and (z₁, z₂) in S, ((x₁ ⊕ y₁) ⊕ z₁, 0) = (x₁ ⊕ (y₁ ⊕ z₁), 0). This axiom holds.The Identity element of addition: There exists an element (0, 0) in S such that for any (x₁, x₂) in S, (x₁, x₂) ⊕ (0, 0) = (x₁, x₂). This axiom fails because (x₁, x₂) ⊕ (0, 0) = (x₁, 0) ≠ (x₁, x₂) unless x₂ = 0.Closure under scalar multiplication: For any α in the field of real numbers and (x₁, x₂) in S, α(x₁, x₂) = (αx₁, αx₂) is also in S. This axiom holds.Inverse elements of addition: For any (x₁, x₂) in S, there exists an element (-x₁, 0) in S such that (x₁, x₂) ⊕ (-x₁, 0) = (0, 0). This axiom fails because (-x₁, 0) is not well-defined as the inverse of (x₁, x₂) because (x₁, x₂) ⊕ (-x₁, 0) = (0, 0) holds only if x₂=0.Distributivity of scalar multiplication over vector addition: For any α in the field of real numbers and (x₁, x₂), (y₁, y₂) in S, α ((x₁, x₂) ⊕ (y₁, y₂)) = α(x₁ + y₁, 0) = (αx₁ + αy₁, 0) = α(x₁, x₂) ⊕ α(y₁, y₂). This axiom holds.Distributivity of scalar multiplication over field addition: For any α, β in the field of real numbers and (x₁, x₂) in S, (α + β) (x₁, x₂) = ((α + β)x₁, (α + β)x₂) = (αx₁ + βx₁, αx₂ + βx₂) = α(x₁, x₂) ⊕ β(x₁, x₂). This axiom holds.Therefore, axioms 4 and 6 fail to hold, and S is not a vector space.
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The correct question:
Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by α(x₁, x₂) = (αx₁, αx₂); (x₁, x₂) ⊕ (y₁, y₂) = (x₁ + y₁, 0). We use the symbol ⊕ to denote the addition operation for this system in order to avoid confusion with the usual addition x + y of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation ⊕, is not a vector space. Which of the eight axioms fail to hold?