Answer: B = 50 - 0.06x
Step-by-step explanation:
This equation shows that the remaining balance on the gift card (B) is equal to the initial amount of the gift card (50) minus the cost of the prints (0.06x) where x is the number of prints purchased.
What was the price of the ABC bond on the previous day?
a. 100 1/3
B. 100 3/4
C. 104 1/2
D. 104 3/4
The price of the ABC bond on the previous day is $104 3/4
How to determine the price of the bondFrom the question, we have the following parameters that can be used in our computation:
Rate = $10 per day
Current price = $114 3/4
Using the above as a guide, we have the following:
Previous price = Current price - Rate
Substitute the known values in the above equation, so, we have the following representation
Previous price = $114 3/4 = $10
Evaluate the difference
Previous price = $104 3/4
Hence, the previous price is $104 3/4
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Complete question
ABC bond increase by $10 everyday. If the price on a given is $114 3/4, what was the price of the ABC bond on the previous day?
Draw the taste of the values for the relation y = x² for the Interval -3≤x≤4.
Answer: I don't have the capability to draw images. However, I can explain how to graph the equation y = x² for the interval -3≤x≤4.
To graph the equation, we can plot a set of points by substituting values of x from the given interval into the equation and solving for y.
For example:
when x = -3, y = (-3)² = 9
when x = -2, y = (-2)² = 4
when x = -1, y = (-1)² = 1
when x = 0, y = (0)² = 0
when x = 1, y = (1)² = 1
when x = 2, y = (2)² = 4
when x = 3, y = (3)² = 9
when x = 4, y = (4)² = 16
We can plot these points on a coordinate plane, and then connect them with a smooth curve.
The graph of y = x² for the interval -3≤x≤4 will be a parabola that opens upward and its vertex is located at (0,0) point.
The x-coordinates of the vertex are in the center of the inte
Step-by-step explanation:
which number rounded to the nearest hundred is 4,700? 4,070 4,617 4,681 4,797
Answer:
Let's look at each number separately. Look in the tens place to determine which way to round.
4,070 will round up since there's a 7. However, it will only round up to 4,100 so this is not the right answer.
4,617 will round down since there's a 1. However, it will only round down to 4,600 so this is not the right answer.
4,681 will round up since there's an 8. This number will round up to 4,700 so this is the correct answer.
4,797 will round up since there's a 9. However, it will only round up to 4,800 so this is not the right answer.
The correct answer is 4,681.
Round 74,600 to the nearest thousand
Answer:
75,000
Step-by-step explanation:
We have to round up to nearest thousand ,so we need to check hundred place ...
If hundred place is equal or greater than 5 ,we need to add +1 to the number which is in thousand place , If it is less than 5 we don't need to do so ....
Determine whether the system is consistent or inconsistent and whether the equations are dependent or independent.
-3x + 3y= 21
4x + 4y= 60
The given pair of linear equations is consistent.
What are Linear Equations?
An equation is said to be linear if the maximum power of the variable is consistent. Another name for it is a one-degree equation.
Equations are declarations of equality between two mathematical expressions containing one or more variables. A linear equation is one in which the maximum power of the variable is one. a real number equation of the form ax+by+c = 0, with real numbers a, b, and c with the highest power one variable x and y.
In our question, we can see:
a1 = -3 and a2 = 4
b1 = 3 and b2 = 4
c1 = 21 and c2 = 60
a1/a2 = -¾
b1/b2 = ¾
c1/c2 = 7/20
Since a1/a2 is not equal to b1/b2, therefore our pair of liner equation is consistent.
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row-echelon forms determine if the following matrices are in row-echelon form or reduced row-echelon form. explain why the matrix is in the determined form
A matrix is in the row-echelon form if:
1. The first nonzero element of each nonzero row (called the leading entry) is to the right of the leading entry of the row above it.
2. Each leading entry is 1 (also called a pivot).
3. Each element above and below a pivot is 0.
A matrix is in the reduced row-echelon form if it is in row-echelon form and also:
4. Each pivot is the only nonzero element in its column.
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The numbers, symbols, or expressions in a matrix are called its entries or elements. Matrices are often used in linear algebra, a branch of mathematics that deals with vector spaces and linear transformations.
They can be used to represent systems of linear equations, perform operations such as matrix addition and multiplication, and solve systems of linear equations.
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data consist of weights in grams the unit for population standard deviation would be
Answer: s squared.s2.sigma i think
Step-by-step explanation:
complete the implementation of the function `same diff ints`, which takes in a list of integers (`ints`) and returns `true` if there exist two list elements $i$ positions apart, whose absolute difference as integers is also $i$. if there are no two elements satisfying this condition, `same diff ints` should return `false`
The complete implementation of the function 'same diff ints' is written italic below.
How to complete the implementation of the function 'same diff ints'?Below is the implementation for the function same_diff_ints in Python:
def same_diff_ints(ints):
for i in range(1, len(ints)):
for j in range(i, len(ints)):
if abs(ints[j] - ints[j-i]) == i:
return True
return False
The function takes in a list of integers as input and iterates through the list using nested for loops. The outer for loop iterates through the list starting at index 1, and the inner for loop iterates through the list starting at the outer loop's current index.
The function compares the absolute difference between the two elements at index j and j-i with the value of i. If the absolute difference between the two elements is equal to i, the function returns True. If no such pair of elements is found, the function returns False.
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lists the parts of the expression shown 8r + 5p + 7 terms variables, coefficient of p constant
Answer/Step-by-step explanation:
Starting with:
8r + 5p + 7
There are three terms. 8r is a term; 5p is a term; 7 is a term. See how the terms are separated by + or -
r and p are variables. Most often the variable is a letter, that way it can stand for an unknown value. It can vary.
5 is the coefficient of p. Its the number right next to the p. (They are multiplied together)
7 is the constant. Constant means unchanging, right? So 7 is a plain old number, it doesn't change like a variable can. A plain number with no variable is a constant.
Do these data (Mod2-2Commercials (Links to an external site.)Links to an external site.) allow us to infer that the mean number of commercials watched during the game is different from 16? Assume the level of significance is 10%.
No, the data provided in the link does not allow us to infer that the mean number of commercials watched during the game is different from 16.
To test the hypothesis that the mean number of commercials watched is different from 16, we would need to perform a statistical hypothesis test. The level of significance (10%) would be used to determine the critical value(s) for the test statistic, which would then be used to make a decision about the null hypothesis (that the mean number of commercials watched is equal to 16).
Without the results of such a test, it is not possible to make a conclusion about the difference between the mean number of commercials watched and 16.
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the distribution system for the herman company consists of three plants, two warehouses, and four customers. plant capacities and shipping costs per unit (in $) from each plant to each warehouse are as follows: customer demand and shipping costs per unit (in $) from each warehouse to each customer are as follows: choose the correct network representation of this problem. (i) (ii) (iii) (iv) formulate a linear programming model of the problem. for subtractive or negative numbers use a minus sign even if there is a sign before the blank. let xij represents relation between plants to warehouses or relation between warehouses to customers. min fill in the blank 2 x14 fill in the blank 3 x15 fill in the blank 4 x24 fill in the blank 5 x25 fill in the blank 6 x34 fill in the blank 7 x35 fill in the blank 8 x46 fill in the blank 9 x47 fill in the blank 10 x48 fill in the blank 11 x49 fill in the blank 12 x56 fill in the blank 13 x57 fill in the blank 14 x58 fill in the blank 15 x59 s.t. 1) fill in the blank 16 x14 fill in the blank 17 x15 < fill in the blank 18 2) fill in the blank 19 x24 fill in the blank 20 x25 < fill in the blank 21 3) fill in the blank 22 x34 fill in the blank 23 x35 < fill in the blank 24 4) fill in the blank 25 x46 fill in the blank 26 x47 fill in the blank 27 x48 fill in the blank 28 x49 fill in the blank 29 x14 fill in the blank 30 x24 fill in the blank 31 x34
All variables must be greater than or equal to 0, indicating that no shipping is allowed if the cost is negative.
The network representation of the Herman Company distribution system is as follows:
Plants (P1, P2, P3) -> Warehouses (W1, W2) -> Customers (C1, C2, C3, C4).
A linear programming model of the problem can be formulated as:
Minimize Z = 2x14 + 3x15 + 4x24 + 5x25 + 6x34 + 7x35 + 8x46 + 9x47 + 10x48 + 11x49 + 12x56 + 13x57 + 14x58 + 15x59
Subject to:
x14 + x15 ≤ 2
x24 + x25 ≤ 3
x34 + x35 ≤ 4
x46 + x47 + x48 + x49 ≤ 5
x14 + x24 + x34 ≤ 6
x15 + x25 + x35 ≤ 7
x46 + x56 + x66 ≤ 8
x47 + x57 + x67 ≤ 9
x48 + x58 + x68 ≤ 10
x49 + x59 + x69 ≤ 11
All variables ≥ 0
The objective of this problem is to minimize the total shipping cost from plants to warehouses and warehouses to customers. The constraints represent the limited capacities of each plant, warehouse and customer and the maximum number of units that can be shipped from each plant to each warehouse or each warehouse to each customer. All variables must be greater than or equal to 0, indicating that no shipping is allowed if the cost is negative.
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Below is the data for the unemployment rates from September 2017 issue of the Economist from 10 different countries. Country Unemployment Rates US 4.4 China 4 France 9.8 Spain 17.1 India 5 Greece 21.2 Chile 6.9Mexico 3.2 Egypt 12 Turkey 10.2 You can enter the data into R using this line of code below. GlobalRates<-c(4.4.4,9.8.17.1,5,21.2,6.9.3.2,12,10.2) Constuct a box and whisker plot and deterimine if there are any outliers in the Global Unemployment Rates. O There is one outlier, Greece, O There are two outliers, Greece and Spain.O There are three outliers, Spain, Greece and Turkey. O There are no outliers.
The answer is -There are no outliers.
According to the question, there are no outliers in the Global Unemployment Rates. From the given data, no outliers are observed.
An outlier is an observation that lies an abnormal distance from other values in a sample from a population. In a sense, this explanation leaves it up to the analyst to decide what will be considered abnormal.
An outlier is a single data point that goes far outside the average value of a set of statistics. Outliers may be exceptions that stand outside individual samples of populations.
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is the formula z =
adratic inequality
-6± √b²
2a
4ac
used to find the solutions to a quadratic equation of the form az² + bz+c= 0.
Yes, the formula z = -b± √(b²-4ac)/2a is used to find the solutions (or roots) of a quadratic equation of the form az² + bz + c = 0. This is known as the Quadratic Formula.
What is the Quadratic Equation?
A quadratic equation is a type of polynomial equation of degree 2 in one variable (x), which can be written in the form of ax² + bx + c = 0, where a, b, and c are real numbers, and a is not equal to zero. The solutions or roots of this equation can be found by using the Quadratic Formula which is z = -b ± √(b²-4ac) / 2a.
The Quadratic formula is valid for any quadratic equation in standard form (ax² + bx + c = 0), where a, b, and c are real numbers, and a is not equal to zero. The two solutions are given by the formula are the roots of the equation, and they are always real and distinct, unless the discriminant b² - 4ac is negative, in which case the roots are complex numbers.
Hence, Yes, the formula z = -b± √(b²-4ac)/2a is used to find the solutions (or roots) of a quadratic equation of the form az² + bz + c = 0. This is known as the Quadratic Formula.
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While you could use the associative property on all the following expressions, select two of the following expressions that using the associative property to write an equivalent expression that would help you calculate the value of the expression using mental math.
Group of answer choices
Answer:
A, C
Step-by-step explanation:
A)
(120 ∙ 8) ∙ 5 = 120 ∙ (8 ∙ 5)
Multiply 8 and 5 first to get 40.
Then 40 × 120 = 4800
C)
(17 + 75) + 25 = 17 + (75 + 25)
Add 75 and 25 first to get 100.
then 17 + 100 = 117
use ratio test to see if this series converges
This given series [tex]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n!}[/tex] diverges.
What is the ratio?The ratio can be defined as the number that can be used to represent one quantity as a percentage of another. Only when the two numbers in a ratio have the same unit can they be compared. Ratios are used to compare two objects.
Given a series, [tex]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n!}[/tex] we need to see if the series converges or not.
the ratio test will fail if [tex]\displaystyle \lim_{n\to \infty} \left |\frac{a_{n+1}}{a_n} \right | =1[/tex]. Otherwise, the series converges absolutely if [tex]\displaystyle \lim_{n\to \infty} \left |\frac{a_{n+1}}{a_n} \right | < 1[/tex] and diverges if [tex]\displaystyle \lim_{n\to \infty} \left |\frac{a_{n+1}}{a_n} \right | > 1[/tex]
Testing for the series [tex]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n!}\\[/tex],
=> [tex]\displaystyle \lim_{n\to \infty} \left |\frac{a_{n+1}}{a_n} \right |[/tex]
=> [tex]\displaystyle \lim_{n\to \infty} \left |\frac{(n+1)!}{n!} \right | \\[/tex]
=> [tex]\lim_{n \to \infty} (n + 1)[/tex]
This value will always be greater than 1 for any positive value of n.
Therefore, This given series [tex]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n!}[/tex] does not converge.
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3. If AB = 26 and BD = 20, find AE.
In a parallelogram, opposite sides are equal in length, which can be used to calculate the length of AE. AB is equal to 26 and BD is equal to 20, so AE is equal to 6.
Is a parallelogram a rhombus?In contrast to a parallelogram, which has equal sides on both sides and diagonals that bisect each other, a rhombus has equal sides on all four sides and diagonals that cross at 90 degrees.
The length of each side is the same. A rhombus has equal opposite angles. At a 90° angle, the diagonals split in half.
In a parallelogram, opposite sides are equal in length. Therefore, we can use this fact to calculate the length of AE in the given parallelogram.
Since AB is equal to 26 and BD is equal to 20, we can set up the equation:
26 = AE + 20
Subtract 20 from both sides of the equation to solve for the length of AE:
26 - 20 = AE
6 = AE
Therefore, AE is equal to 6.
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giving a test to a group of students, the grades by the two different classes are summarized below. a a b b c c total morning class 4 4 20 20 8 8 32 32 afternoon class 13 13 5 5 7 7 25 25 total 17 17 25 25 15 15 57 57 enter your answer as a fraction or as a calculation involving fractions. if one student is chosen at random, find the probability that the student got a a a given that the student was in the afternoon class.
The probability that the student got a a a given that the student was in the afternoon class is 25/57
In math the term probability is defined as based on the possible chances of something to happen
Here we have given that a test to a group of students, the grades by the two different classes are summarized.
and it can be written like the following;
a b c Total
Morning class 4 20 8 32
Evening Class 13 5 7 25
Total 17 25 15 57
Here we know that the total number of students is obtained as
=> 57
And the number of students who have been taking the afternoon class is calculated as,
=> 25
Then the probability the student got a a a given that the student was in the afternoon class is written as,
=> 25/57
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3. Find (a) the net cost equivalent and (b) the single discount equivalent for a
series discount of 5/10/12.
Answer:(a) To find the net cost equivalent, we can use the formula:
Net Cost Equivalent = Original Cost / (1 - (Discount Rate / 100))
Using this formula, the net cost equivalent for a series discount of 5/10/12 is:
Net Cost Equivalent = Original Cost / (1 - (5/100)) / (1 - (10/100)) / (1 - (12/100))
(b) To find the single discount equivalent for a series discount of 5/10/12, we can use the formula:
Single Discount Equivalent = (1 - (1 - (Discount Rate1 / 100)) x (1 - (Discount Rate2 / 100)) x ... x (1 - (Discount RateN / 100))) x 100
Using this formula, the single discount equivalent for a series discount of 5/10/12 is:
Single Discount Equivalent = (1 - (1 - (5/100)) x (1 - (10/100)) x (1 - (12/100))) x 100
What is the 18th term of the arithmetic sequence where a3 = −6 and a12 = −3?
Group of answer choices
0
1
−3
−1
Step-by-Step Explanation:
3rd term (a3) = -6
12th term (a12) = -3
We know, nth term of an AP = a1 + (n - 1)d, where d = common difference
a3 = a + (3-1)d = a + 2d = -6...(I)
a12 = a + (12-1)d = a + 11d = -3..(II)
Subtracting (II) from (I), we have
2d - 11d = -6 + 3
=> -9d = -3
=> d = 1/3
Therefore, a + 2×1/3 = -6
=> a + 2/3 = -6
=> a = -6-2/3 = -20/3
Therefore, 18th term
= a + (18-1)d
= -20/3 + 17/3
= -3/3
= -1
Answer:
-1
Hope it helps.
If you have any query, feel free to ask.
What is the Domain:x > 3 , -3 ≤ x ≤ 3, all real numbers
The domain and the range of y = 12x - 36 are the set of all real numbers
How to determine the domain and the range?From the question, we have the following parameters that can be used in our computation:
y = 12x - 36
There is no restriction on the input values, x
So, the domain of y = 12x - 35 is all real numbers, since there is no restriction on the input (x).
The range of y = 12x - 35 is also all real numbers, since the output (y) can be any real value, depending on the input (x).
This can be seen by considering that for any real number x, the output y = 12x - 35 will also be a real number.
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Complete question
What is the Domain of y = 12x - 36
x > 3 ,
-3 ≤ x ≤ 3,
all real numbers
Also, determine the range
based on the responses, which of the following is a 95 percent confidence interval for the proportion of all city residents who would respond very supportive or somewhat supportive of the proposal?
As per the given confidence interval, the proportion of all city residents who would respond very supportive or somewhat supportive of the proposal is 336 ± 0.197
The term confidence interval is defined as the statistical measure used to indicate the range of estimates within which an unknown statistical parameter is likely to fall.
Here we have given that the random sample of 1,018 city residents were asked to rate their level of support for a proposal being considered by the city council.
Here we have also know that the percent of confidence interval is 95%.
Here we also identified the following values,
Total population = 1,018
Very supportive population = 336
Somewhat supportive population = 387
Not supportive population= 295
Then the value of confidence interval is calculated as,
Proportion of positive results = P = x/N = 0.3301
Lower bound = 0.3012
Upper bound = 0.3599
Therefore, it can be written as,
=> 336 ± 0.197
Complete Question:
A random sample of 1,018 city residents were asked to rate their level of support for a proposal being considered by the city council. Based on the responses, which of the following is a 95% confidence interval for the proportion of all city residents who would respond very supportive of somewhat supportive of the proposal?
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HURRY HAVE A TIMER In the winter fuel line antifreeze must be added at a rate of one can per 8 gal of fuel.
How many cans should be added to an 18-gallon tank
The number of cans to be added to 18 gallon tank is given by equation
A = 2.25 cans
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the number of cans for 18 gallon tank be A
Now , the equation will be
The number of cans per 8 gallons = 1 can
So , the number of cans for 1 gallon = number of cans per 8 gallons / 8
On simplifying the equation , we get
The number of cans for 1 gallon = 1/8
The number of cans for 1 gallon = 0.125 can
And ,
The number of cans for 18 gallons A = 18 x number of cans for 1 gallon
Substituting the values in the equation , we get
The number of cans for 18 gallons A = 18 x 0.125
The number of cans for 18 gallons A = 2.25 cans
Hence , the number of cans is 2.25 cans
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Two communications companies offer calling plans. With Company X, it costs 30c to connect and then 4c for each minute. With Company Y, it costs 15c to connect and then 3c for each minute. Write and simplify an expression that represents how much more Company X charges, in cents, for n minutes.
Answer:
30 + 4n
Step-by-step explanation:
Company X = 30 + 4n
Please help.
Find the percent of increase from 5.5 feet to 6 feet. Round the percent to the nearest tenth if necessary.
The percent of increase from 5.5 feet to 6 feet is 9.1%
What is percentage and its calculation for given terms?Percentage is a way to express a number as a fraction of 100.
It is often used to express a proportion or rate in a convenient and understandable way.
To calculate percentage, you can use the formula: (part/whole) x 100.
For example, if you scored 80 out of 100 on a test, your percentage score would be 80%.
Percentage is often used in many fields such as finance, statistics, and education.
It can be used to express the increase or decrease between two values or to compare the parts of a whole.
You can use the following calculation to determine the percentage increase from 5.5 feet to 6 feet:
Old Value * 100 / (New Value - Old Value) = Percent Increase
In this case:
(6 - 5.5) / 5.5 * 100 = 9.090909090909091%
You can round this to the nearest tenth, which would be 9.1%.
Therefore, the percent of increase from 5.5 feet to 6 feet is 9.1%.
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❗️PLEASEE HELPPPPPP ILL GIVE YOU BRAINLIESTTTTTT ❗️
Find the value of v that makes quadrilateral PQRS a parallelogram.
P v+38°
s 8v-56°
R 6v–72°
Q v+98°
PLEASEE HELPPPPPP ILL GIVE YOU BRAINLIESTTTTTT
Answer:
there is no value of v that makes quadrilateral PQRS a parallelogram.
Step-by-step explanation:
Hopefully this helps, may be wrong so...
A quadrilateral is a parallelogram if and only if opposite angles are equal. Therefore, in order for quadrilateral PQRS to be a parallelogram, angle P must be equal to angle S, and angle Q must be equal to angle R. From the given information, we know that:
angle P = v + 38°
angle S = 8v - 56°
angle Q = v + 98°
angle R = 6v - 72°
To find the value of v that makes quadrilateral PQRS a parallelogram, we must set angle P equal to angle S and angle Q equal to angle R:
v + 38° = 8v - 56°
v + 98° = 6v - 72°
Solving for v in the first equation:
v + 38° = 8v - 56°
-7v = -94°
v = 13.428571428571429
Solving for v in the second equation:
v + 98° = 6v - 72°
-5v = -170°
v = 34
We can see that both values of v do not match, therefore there is no value of v that makes quadrilateral PQRS a parallelogram.
A quadrilateral is a parallelogram if and only if the opposite angles are equal. In this case, the opposite angles are P and S, and Q and R.
Therefore, for the quadrilateral PQRS to be a parallelogram, we must have:
v+38° = 8v-56° and v+98° = 6v–72°
Solving the first equation for v:
v+38° = 8v-56°
38 = 7v
v = 38/7
Now we use this value of v to check if the second equation is true:
v+98° = 6v–72°
(38/7) +98 = 6(38/7) - 72
98 = -34
As we can see, the second equation is not true, so the value of v = 38/7 doesn't make quadrilateral PQRS a parallelogram.
No value of v exists that would make quadrilateral PQRS a parallelogram, as the opposite angles are not equal in this case.
ladder is leaning against a wall so that it makes an angle of 35 degrees with the wall.
If the base of the ladder is 4 feet from the base of the wall, how long is the ladder?
The length of the ladder used in the question is; 6.974 ft
How to use trigonometric ratios?The 3 main trigonometric ratios are;
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
Now, since the ladder makes an angle of 35 degrees with the wall, it means the ladder will be the hypotenuse of the right angle triangle and the distance from the base of the wall to the base of the ladder will be labelled as h = 4 will be the opposite side of the triangle. Thus;
4/hypotenuse = sin 35
hypotenuse = 4/sin 35
hypotenuse = 4/0.5736
Hypotenuse = 6.974 ft
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Use the distributive property to write each expression as an equivalent
algebraic expression.
12(y+14)
Answer: 12y+168
Step-by-step explanation:
Basically, multiply each of the numbers that are in the parenthesis by the number outside the parenthesis (12). So 12·y=12y and 12·14=168. Therfore the answer is 12y+168 .
First solve the system for x and y.
-3y+8x=18
y=6x-6
Step-by-step explanation:
given = y = 6x-6 .
therefore ,
=) -3(6x-6)+8x=18 - ( from given ) .
=) -18x+18+8x=18 .
=) -10x=18-18 .
=) -10x = 0 .
=) x = 0/10 = 0.
therefore ,
y=6(0)-6 .
= -6 .
therefore , x = 0 & y = -6 .
[ it takes so much time to write and solve so plzzz => ]
MARK ME AS BRAINLIEST ...Determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span.
S = {(-1, 2), (2, -1), (1, 1)}
A. S spans R2.
B. S does not span R2. S spans a line in R2.
C. S does not span R2. S spans a point in R2.
A set S spans R^2 if any element of R^2 can be written as linear combination of elements of S the set S spans R2.
(a) Given that S={ (1,-1),(2,1) }
Let (x,y) be the any element in R^2.
(x,y) = C1(1,-1)+C2(2,1)
(x,y) = (c1+2c2,-c1+c2)
x = c1+2c2
y = -c1+c2
x+y = 3c2
c2= {x+y}/{3}
c1 = c2-y
= {x+y}/{3}-y
={x-2y}/{3}
So for any element (x,y) in R^{2} we have the constants c_1,c_2 so that (x,y) can be written as,
(x,y)= {(x-2y)}/{3}(1,-1) + {(x+y)}/{3}(2,1)
So S= { (1,-1),(2,1) } spans R^2.
(b) Given that S = { (1,1) }
Let (x,y) be the any element in R^2.
(x,y)=c_1(1,1)
(x,y)=(c_1,c_1)
x=c_1,y=c_1
If x,y are distinct then do not get a value of c_1 so that (x,y) = c_1(1,1).
So set S= { (1,1) } does not span R^2.
(c) Given that S= { (0,2),(1,4) }
Let (x,y) be the any element in R^2.
(x,y)=c_1(0,2)+c_2(1,4)
(x,y)=(c_2,2c_1+4c_2)
x=c_2
y=2c_1+4c_2
c_1={y-4c_2} / {2}
={y-4x} / {2}
So (x,y)= {(y-4x)}/{2}(0,2)+x(1,4)
So any element of R^2 can be written as written as linear combination of elements of Set S.
Hence set S={ (0,2),(1,4) } spans R^2.
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Wenxuan, Charlene and Siti shared a box of stickers. Wenxuan took 8 more than of the total number of stickers. Charlene took 7 more than of the remaining 1 1 3 2 stickers. Siti took the last 23 stickers. How many stickers were there in the box at first?
Answer:
We can start solving the problem by using algebraic equations. Let x be the total number of stickers in the box at first.
According to the problem, Wenxuan took 8 more than of the total number of stickers, so he took x + 8 stickers.
The remaining stickers after Wenxuan took his share is x - (x+8) = x - x - 8 = -8 stickers.
Charlene took 7 more than of the remaining stickers, which is -8 + 7 = -1 stickers.
So, Charlene took x - (x+8) + 7 = -1 stickers.
Siti took the last 23 stickers, so the total number of stickers remaining is -1 - 23 = -24.
Now we can use this information to set up the equation:
x + (x+8) + (x-x-8+7) + (-1-23) = x
Solving for x we get x = 32
Therefore, there were 32 stickers in the box at first.