We will see that f'(x) > 0, which means that f(x) is an increasing function.
How to prove that the function is increasing?
For any function f(x), if f'(x) > 0, then f(x) is increasing for any value of x.
Here we have the cubic function:
f(x) = x³ + 4x
If we differentiate this, we get:
f'(x) = df(x)/dx = 3x² + 4.
And notice that x² is always positive, then f'(x) > 0, which means that f(x) is an increasing function.
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Pls help me with my math
Answer:
The definition for the given piecewise-defined function is: [tex]\boxed{\displaystyle\sf\ Option\:D:\:\: f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex].
Step-by-step explanation:
General Concepts:Piecewise-defined functions.Interval notations.What is a piecewise-defined function?
A piecewise-defined function represents specific rules over different intervals of the domain.
Symbols used in expressing interval notations:Open interval: This means that the endpoint is not included in the interval.
We can use the following symbols to indicate the exclusion of endpoints in the interval:
Left or right parenthesis, "( )" (or both). Greater than (>) or less than (<) symbols. Open dot "[tex]\circ[/tex]" is another way of expressing the exclusion of an endpoint in the graph of a piecewise-defined function.Closed interval: This implies the inclusion of endpoints in the interval.
We can use the following symbols to indicate the inclusion of endpoints in the interval:
Open- or closed brackets (or both), "[ ]." Greater than or equal to (≥) or less than or equal to (≤) symbols. Closed circle or dot, "•" is another way of expressing the inclusion of the endpoint in the graph of a piecewise-defined function. Determine the appropriate function rule that defines different parts of the domain.The best way to determine which piecewise-defined function represents the graph is by observing the endpoints and orientation of both partial lines.
Open circle on (-1, 2): The graph shows that one of the partial lines has an excluded endpoint of (-1, 2) extending towards the right. This implies that its domain values are defined when x > -1. Closed circle on (-1, 1): The graph shows that one of the partial lines has an included endpoint of (-1, 1) extended towards the left. Hence, its domain values are defined when x ≤ -1.Based on our observations from the previous step, we can infer that x > -1 or x ≤ -1 apply to piecewise-defined functions A or D. However, only one of those two options represent the graph.
Solution:a) Test option A:[tex]\boxed{\displaystyle\sf Option\:A)\:\:\:f(x) = \begin{cases}\displaystyle\sf\ 2x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex]
Piece 1: If x ≤ -1, then it is defined by f(x) = 2x + 2.We must choose a domain value that falls within the interval of x ≤ -1 whose output is included is included in the graph of the partial line with a closed dot.
Substitute x = -2 into f(x) = 2x + 2:
f(x) = 2x + 2f(-2) = 2(-2) + 2f(-2) = -4 + 2f(-2) = -2 ⇒ False statement.⇒ The output value of f(-2) = -2 is not included in the graph of the partial line whose endpoint is at (-1, 1).
Piece 2: If x > -1, then it is defined by f(x) = x + 4.
We must choose a domain value that falls within the interval of x > -1 whose output is included in the graph of the partial line with an open dot.
Substitute x = 0 into f(x) = x + 4:
f(x) = x + 4f(0) = (0) + 4f(0) = 4 ⇒ True statement.⇒ The output value of f(0) = 4 is included in the graph of the partial line whose endpoint is at (-1, 2).
Conclusion for Option A:
Option A is not the correct piecewise-defined function because one of the pieces, f(x) = 2x + 2, does not specify the interval (-∞, -1].
b) Test option D:[tex]\boxed{\displaystyle\sf Option\:D)\:\:\:f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex]
Piece 1: If x ≤ -1, then it is defined by f(x) = x + 2.We must choose a domain value that falls within the interval of x ≤ -1 whose output is included is included in the graph of the partial line with a closed dot.
Substitute x = -2 into f(x) = x + 2:
f(x) = x + 2f(-2) = (-2) + 2f(-2) = 0 ⇒ True statement.⇒ The output value of f(-2) = 0 is included the graph of the partial line whose endpoint is at (-1, 1).
Piece 2: If x > -1, then it is defined by f(x) = 2x + 4.
We must choose a domain value that falls within the interval of x > -1 whose output is included is included in the graph of the partial line with an open dot.
Substitute x = 0 into f(x) = 2x + 4:
f(x) = 2x + 4f(0) = 2(0) + 4f(0) = 0 + 4 = 0 ⇒ True statement.⇒ The output value of f(0) = 4 is included in the graph of the partial line whose endpoint is at (-1, 2).
Final Answer:
We can infer that the piecewise-defined function that represents the graph is:
[tex]\boxed{\displaystyle\sf\ Option\:D:\:\: f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex].
________________________________________
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The half-life of cobalt-60 (used in
radiation therapy) is 5.26 years (actual
data). How much of a 200 g sample of
cobalt-60 will remain after 26.3 years?
Answer:
6.25 gm left
Step-by-step explanation:
Find the number of half lives in 26.3 years
26.3 / 5.26 = 5 half lives
(1/2) ^5 = 1/32 nd of the original will be left
1/32 * 200g = 6.25 gm left
Ian buys 4 bananas and 2 apples. Each banana has a mass of 120 grams. Each apple has a mass of 180 grams.
Which is the total mass of the fruit that Ian buys?
A.
300 grams
B.
580 grams
C.
840 grams
D.
960 grams
Answer:
THE ANSWER IS A CORRECT ME IF I WRONG
Identify the composition that is represented by: r (90, 0) T(−2,4)
A translation left 2, up 4 and then a reflection of 90°
A rotation of 90° and then a translation left 2, up 4.
A translation of left 2, up 4 and then a rotation of 90°.
A reflection of 90° and then a translation left 2, up 4.
The composition of the transformation is (b) A rotation of 90° and then a translation left 2, up 4.
How to determine the transformation?The transformation rule is given as:
r (90, 0) T(−2,4)
The r(90,0) represents a rotation of 90 degrees.
The other part of the transformation rule can be rewritten as:
T(-2,4) => (x - 2,y + 4)
This means a translation right by 2 units and up by 4 units
Hence, the composition of the transformation is (b)
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Answer: left 2
Step-by-step explanation:
1. Find the equation of a normal to the curve y= 2² - 2x +3 at the point (3,0)
I think you meant to say the equation is
y = 2x² - 2x + 3
Differentiate both sides with respect to x :
dy/dx = 4x - 2
At the point (3, 0), the slope of the tangent line is dy/dx(3) = 4•3 - 2 = 10. Then the normal line to the curve at (3, 0) has slope -1/10.
Using the point-slope formula, the equation of the normal line is
y - 0 = -1/10 (x - 3) ⇒ y = (3 - x)/10
Determine if the function is an even function, an odd function or neither. > = −6x² – 5x² −2 even neither odd
Answer: even
Step-by-step explanation:
A function is even if f(x)=f(-x), and odd if f(x)=-f(-x). If a function satisfies neither of these, it is neither even nor odd.
[tex]f(x)_=-6x^{2}-5x^{2}-2=-11x^{2}-2\\\\f(-x)=-11(-x)^{2}-2=-11x^{2}-2\\\\\therefore f(x)=f(-x)[/tex]
Therefore, the function is even.
Goal
Your task is to create a monthly budget given certain circumstances.
Role
You are a single person who just received a full-time job in a factory (40 hours). You will make $12.00 an hour.
Audience
You need to convince your parents that you can maintain a budget to live in a one-bedroom apartment that costs $300 for rent a month.
Situation
The challenge involves dealing with finding average heating, water, sewage bills in your area so that you can include those in your budget.
Product, Performance, and Purpose
You need to develop a monthly budget so that you can show your parents that you can afford the expenses.
Please upload your completed assignment here. Be sure you have included your name at the top of your document and as part of the file name.
The development of the personal budget that convinces parents that one can maintain an independent living is detailed as follows using the 50: 30: 20 rule:
50% for necessities (housing, food, transportation, utilities) $740
30% for luxuries, savings, vacations, entertainment, etc. $461
20% for Emergency Funds and Retirement $335
What is a personal budget?A personal budget is the household budget for a single person for a period.
The personal budget shows an estimate of the person's revenue and expenses over the period.
Data and Calculations:Hourly rate of earnings = $12
Working hours per week = 40 hours
Working hours per month = 160 hours (40 x 4 weeks)
Total monthly earnings = $1,920 ($12 x 160)
Assumed tax rate = 20%
After-tax take-home pay = $1,536 ($1,920 x 1 - 20%)
Monthly Necessities:Rent of apartment = $300
Heating = $40
Water cost = $20
Sewage bill = $10
Food = $300
Transportation = $50
Other costs = $20
Total cost for necessities = $740
Thus, the development of the personal budget shows that one can maintain an independent living from the parents as a single person.
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How do i solve this one
Answer:
900π
Step-by-step explanation:
Hello!
RadiusThe radius is represented by the function [tex]r(t) = 15\sqrt{t +2}[/tex].
The radius of the ripple can be found when [tex]t=2[/tex]
Solve:
[tex]r(t) = 15\sqrt{t + 2[/tex][tex]r(2) = 15\sqrt{2 +2}[/tex][tex]r(2) = 15\sqrt4[/tex][tex]r(2) = 15*2[/tex][tex]r(2) = 30[/tex]The radius of the ripple is 30 inches
AreaThe area of a circle is calculated by the formula [tex]A = \pi r ^2[/tex]
Given the radius is 30, we can find the area of the circle.
Solve:
[tex]A = \pi r^2[/tex][tex]A = \pi (30^2)[/tex][tex]A = 900\pi\\[/tex]Denise bikes 3 miles to her friend's house, and then she bikes home. The average rate biking to her friend's house is twice the average rate coming home. Write and simplify an expression for the time it takes Denise to make a round-trip in terms of the average rate coming home x.
Hint : Use d = rt.
The total time for the trip is:
T = 3mi*( 3/2x).
Where x is the rate at which she comes home.
How to find the time for the total trip?
Remember the relation:
distance = rate*time.
We know that the distance is 3 miles (done twice).
First, the rate is 2x and then the rate is x, then the time it took the first half is:
t = 3mi/2x
And for the coming back:
t = 3mi/x
Then the total time for the trip is:
T = 3mi/2x + 3mi/x = 3mi( 1/2x + 1/x) = 3mi*( 3/2x).
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The length of a new rectangular playing field is 4 yards longer than quadruple the width. If the perimeter of the rectangular playing field is 558 yards, what are its dimensions?
The width is yards.
Answer:
55 x 224 yards
Step-by-step explanation:
w + w + 2 * ( 4 + 4w) = 558 (given)
10 w + 8 =558
w = 55 then L = 4 + 4w = 224 yards
Find the values of a and b that make the quadrilateral a parallelogram.
Step-by-step explanation:
both halves of a diagonal must be equal.
so,
8b - 30 = 6b + 6
2b = 36
b = 18
and
9a - 13 = 5a + 5
4a = 18
a = 18/4 = 4.5
The members of a club decide to sell hats to raise money. They originally
planned to charge $12 for a hat, but they reduced that price by $2. They sold
41 hats at the reduced price.
Select the expression representing the amount of money earned.
OA. 41(12-2)
OB. 41(12+2)
OC. 41(12) +2
OD. 41(12) - 2
Answer:
A. 41 (12 - 2)
Step-by-step explanation:
If the club decided to charge $12 for a hat, but reduced the price by $2, the reduced price of the hat is (12 - 2). The amount of money earned is based upon the price of the hat multiplied by the number of hats sold. Therefore, the answer is 41 (12 - 2).
Hope this helped :)
The shorter leg of a right triangle is 5 inches shorter than the longer leg. The hypotenuse is 5 inches longer than the longer leg. Find the side lengths of the triangle.
Answer:
longer leg: 20 in
shorter leg: 15 in
hypotenuse: 25 in
Formulas:
Pythagorean Theorem
[tex]c^2 = a^2 + b^2[/tex]
c ... hypotenuse
a ... one leg
b ... another leg
Pythagorean theorem is used in right triangles (triangles in which one angle is 90°).
Step-by-step explanation:
longer leg: x
shorter leg: x - 5
hypotenuse: x + 5
To find x (longer leg), let's use Pythagorean theorem.
[tex]c^2 = a^2 + b^2\\(x+5)^2 = x^2 + (x-5)^2\\x^2 + 10x + 25= x^2 + x^2 -10x + 25\\x^2 + 10x = 2x^2 - 10x\\0 = x^2 - 20x[/tex]
Now let's factorize to get solutions for x.
[tex]0 = x(x-20)[/tex]
First solution:
[tex]x = 0[/tex]
Second solution:
[tex]x - 20 = 0\\x = 20[/tex]
Since a side of a triangle has to be a positive number, x is equal to 20.
Now let's just substitute x back to get side lengths. From the question the lengths are in inches.
longer leg: x = 20 in
shorter leg: x - 5 = 20 - 5 = 15 in
hypotenuse: x + 5 = 20 + 5 = 25 in
someone please help ASAP will give brainliest
The scale factor have been categorized into Expansion and Contraction.
What is a Scale Factor ?It is the measure by which two figures are same in appearance but different in size.
It can be positive or negative as the figure is getting bigger or smaller.
Different scale factors are given and it has been asked they describe which type of dilation.
1. -2/3 This will contract the figure
2. 3.25 This shows expansion
3. 2/3 This will contract the figure
4. -3 This will expand but the image will be form on the other side of Center of Enlargement.
Therefore the scale factor have been categorized into Expansion and Contraction.
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HELP PLEAAASEEE Ireland opened a bag or skittles and recorded the colors and their frequencies.
What is the RELATIVE FREQUENCY, rounded to the NEAREST HUNDREDTH, for
the number of PURPLE candies?
=================================================
Explanation:
We want the relative frequency of the purple candies.
There are 16 purple out of 18+20+10+16 = 64 total
Divide the values to get 16/64 = 0.25
Exactly 25% of the candies are purple. This converts to the fraction 1/4.
The correct answer is 25%
What is Relative Frequency?The number of times an event occurs divided by the total number of events occurring in a given data. How to solve the problem?This problem can be solved by following steps.The data of color and frequency is given in table.We need find the relative frequency for PurpleFirst calculate the the total number of frequency
18+20+10+16
= 64
The total number of frequency is 64
Hence the relative frequency of purple is 16/64
Therefore , relative frequency of purple is 1/4 = 0.25
Therefore 25% of purple candies are there
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Can you help me please!!
The ABEF is a parallelogram because DC is extended to point F; AB ≅ FE and FA ≅ BE.
What is parallelogram?In two-dimensional geometry, it is a plane shape having four sides, in which two pairs of sides are parallel to each other and equal in length. The sum of all angles in a parallelogram is 360°.
We have ABCD is a parallelogram.
BE is perpendicular to FC
FA is perpendicular to FC
As we know in the parallelogram ABCD:
AB ≅ DC
AD ≅ CB
As the DC is extended to F
So, AB ≅ FE
And BE is perpendicular to FC
FA is perpendicular to FC (given)
∴ FA ≅ BE
∴ ABEF is a parallelogram.
Thus, the ABEF is a parallelogram because DC is extended to point F; AB ≅ FE and FA ≅ BE.
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The physician prescribed penicillin 250 mg. The bottle from the supply cabinet is labeled, "Penicillin 500 mg per cc." The correct amount to administer would be CC.
Select one:
A. 0.5
B. 5.0
C. 0.25
D. 0.75
Using proportions, it is found that the correct amount to administer of CC would be of:
A. 0.5.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
In this problem, 500 mg are equivalent to one cc. How many cc are equivalent to 250 mg? Hence the rule of three is given by:
500 mg - 1 cc
250 mg - x cc
Applying cross multiplication:
500x = 250
c = 0.5.
Hence option A is correct.
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A recipe for 8 servings lists: 6 cups spaghetti sauce 4 cups ricotta cheese and 4 cups mozzarella cheese. Determine the amount of ingredients necessary to serve 5 people.
The amount of ingredients necessary to serve 5 people is 3.75 cups spaghetti sauce, 2.5 cups ricotta cheese, and 2.5 cups mozzarella cheese.
What is the ratio?It is described as the comparison of two quantities to determine how many times one obtains the other. The proportion can be expressed as a fraction or as a sign: between two integers.
We have:
A recipe for 8 servings lists: 6 cups spaghetti sauce 4 cups ricotta cheese and 4 cups mozzarella cheese.
First we take the ratio of ingredients for 8 servings:
spaghetti sauce : ricotta cheese : mozzarella cheese = 6:4:4
The ratio of ingredients for 1 servings:
spaghetti sauce : ricotta cheese : mozzarella cheese = 6/8 : 4/8 : 4/8
The ratio of ingredients for 5 servings:
spaghetti sauce : ricotta cheese : mozzarella cheese
= 5×0.75 : 5×0.5 : 5×0.5
= 3.75 : 2.5 : 2.5
The amount of ingredients necessary to serve 5 people:
3.75 cups spaghetti sauce 2.5 cups ricotta cheese2.5 cups mozzarella cheese.Thus, the amount of ingredients necessary to serve 5 people is 3.75 cups spaghetti sauce, 2.5 cups ricotta cheese, and 2.5 cups mozzarella cheese.
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If-5x + 4y = -3 is a true equation, what would be the value of
-6(-5x+4y)?
Answer:
Step-by-step explanation:
-5x+4y=-3
-6(-5x+4y)=-6×(-3)=18
please help! acellus
find the missing side of this right triangle.
30
3
x
x= [?]
Answer:
The number that belongs in the green box is equal to 909.
General Formulas and Concepts:
Algebra I
Equality Properties
Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of EqualityTrigonometry
[Right Triangles Only] Pythagorean Theorem:
[tex]\displaystyle a^2 + b^2 = c^2[/tex]
Step-by-step explanation:
Step 1: Define
Identify given variables.
a = 30
b = 3
c = x
Step 2: Find x
Let's solve for the general equation that allows us to find the hypotenuse:
[Pythagorean Theorem] Square root both sides [Equality Property]:Now that we have the formula to solve for the hypotenuse, let's figure out what x is equal to:
[Equation] Substitute in variables:∴ the hypotenuse length x is equal to √909 and the number under the square root, our answer, is equal to 909.
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Topic: Trigonometry
Substratum is an open-source network that allows anyone to allocate spare computing resources to support the foundation of the decentralized web. SUB is the cryptocurrency that supports this network. On the 10th of January 2018, the price of SUB was $2.88 and on the 5th of April 2019 the price was $0.02. When rounded to one decimal place, which of the below gives the percentage decrease of the price of SUB from the 10th of Jan 2018 to the 5th of April 2019?
The percentage decrease will be equal to 99.30 %.
What are percentages?The Percentage is defined as representing any number with respect to the 100. It is denoted by the sign %.
Given that:-
SUB is the cryptocurrency that supports this network. On the 10th of January 2018, the price of SUB was $2.88 and on the 5th of April 2019 the price was $0.02The percentage decrease will be calculated as:-
Percentage decrease = [tex]\dfrac{2.88 - 0.02}{2.88}[/tex]
Percentage decrease = 0.9930 = 99.30%
Therefore the percentage decrease will be equal to 99.30 %.
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writing equations from graphs pixel art graph
Answer:
y= -2/3x - 4
Step-by-step explanation:
the 2/3 comes from going up 2 spaces and over left 3 spaces. the b is negative 4 because of where it falls. the line is negative because of the direction it is in.
Select the correct answer from each drop-down menu. y = f(x) = (1/2) ^ 5 Consider the function The value of f(- 7) is The value of f(5) rounded to the nearest ten thousandth, is
The values of the functions are:
1. y = f(-7) = 128
2. y = f(5) = 0.0313
How to Find the Value of a Function?We can evaluate a function by substituting the given value into the equation of the function, then simplify.
Given the function, y = f(x) = (1/2) ^ x
1. Find the value of f(-7):
y = f(-7) = (1/2)^-7
y = f(-7) = 2^7
y = f(-7) = 128
2. 1. Find the value of f(5):
y = f(5) = (1/2)^5
y = f(5) = 0.0313
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find the slope of the lined graph
Answer:
1
Step-by-step explanation:
The points go rise:1 up:1 so the slope is 1.
Answer:
hey!! we can taIk here
Step-by-step explanation:
The number of compact discs N purchased each year, in millions, can be
approximated by the function
N(t) = 384(1.13)
where t is the number of years since 1991.
After what amount of time will two billion compact discs be sold?
What is the doubling time for the number of compact discs sold in one year?
Answer:
126.5 years
Step-by-step explanation:
N(t) = 384(1.13)^t
2,000,000,000 = 384(1.13)^t
(1.13)^t = 2,000,000,000/384
log [(1.13)^t) = log (2,000,000,000/384)
t × log 1.13 = log (2,000,000,000/384)
t = [log (2,000,000,000/384)]/(log 1.13)
t = 126.5
what is the least common multiple of 6 and 2
1cm(6,2) =
Please help! I don’t think my answers are correct. Photo is attached.
Answer:
Step-by-step explanation:
(a). In set notation:
B. { x | x < 2 }
(b).
( - ∞ , 2 )
(c). See attachment.
Which one is the right conversion?
Answer:
1, 4, 5, 6
Step-by-step explanation:
to convert a rational to an exponential it's the index (number left of radical) over the power (number on the right).
you can also double check by plugging both into a calculator and see if they equal the same number.
Solve 8x+5y=2 and y-2=0.Also verify thmm....
Answer:
Solution: Here, the given equations are,
[tex]: \implies8x+5y=2...(i)\:and\:y-2=0...(ii)[/tex]
From equation,(ii); y-2=0
[tex] \therefore \: y = 2.........(iii)[/tex]
Substituting the value of y from equation (iii) to equation (i),we get
[tex]: \implies{8x + 5y = 2}[/tex]
[tex]: \implies{8x + 5(2) = 2}[/tex]
[tex]: \implies{8x = 2 - 10}[/tex]
[tex]: \implies{8x = - 8}[/tex]
[tex]: \implies{x = - 1}[/tex]
[tex] \therefore \: x = - 1[/tex]
Thus, x= -1 and y= 2 is the solution.
Checking at (-1,2)8x+5y=2...(i) y-2=0...(ii)
8(-1)+5(2)=2 2-2=0
2=2(True) 0=0(True)
Consider the given system of linear equations. 2x+y-z=8 x+4y+z=7 -3x+2y-3z=21
The system of equations has the solution:
x = 111/42
y = 27/14
z = -15/6
How to solve the system of linear equations?We start with 3 linear equations:
2x+y-z=8
x+4y+z=7
-3x+2y-3z=21
To solve this, first we need to isolate one of the variables. I will isolate x on the second one:
x = 7 - 4y - z
Now we can replace that in the other two:
2*( 7 - 4y - z) + y - z = 8
-3*(7 - 4y - z) + 2y - 3z = 21
Now we need to isolate other variable, let's isolate y on the above one:
14 - 8y - 2z + y - z = 8
-7y - 3z = 8 - 14 = -6
z = (-6 + 7y)/-3 = 2 - (7/3)*y
Now we replace this on the last equation:
-21 + 12y -3z + 2y - 3z = 21
14y - 6z = 42
14y - 6*(2 - (7/3)*y) = 42
14y - 12 + 14y = 42
28y = 42 + 12 = 54
y = 54/28 = 27/14
Now that we know the value of y, we can find the value of z:
z = 2 - (7/3)*y = 2 - (7/3)*27/14 = 2 - 27/6 = -15/6
And the value of x:
x = 7 - 4y - z = 7 - 4*(27/14) + 15/6 = 7 - 48/7 + 15/6 = 111/42
So the solution is:
x = 111/42
y = 27/14
z = -15/6
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