Solve the linear system using the Gauss-Jordan elimination method.
x+y−z= 5
3x−y+z= -9
x+4z=3
The solution of the system is x=,y=,z= (Simplify your answer. Type integers or fractions.)
Answers
The solution of the linear system using the Gauss-Jordan elimination method is x = 2, y = -3, z = -1.
To solve the system, we can start by representing the given equations in augmented matrix form:
| 1 1 -1 | 5 |
| 3 -1 1 | -9 |
| 1 0 4 | 3 |
Next, we'll perform row operations to transform the matrix into row-echelon form. The goal is to create zeros below the leading coefficient in each row. We'll achieve this by multiplying rows by appropriate factors and adding or subtracting them from other rows. The steps involved in performing the Gauss-Jordan elimination method are as follows:
Step 1: Multiply the first row by 3 and subtract it from the second row to eliminate the x-term below the leading coefficient in the second equation.
| 1 1 -1 | 5 |
| 0 -4 4 | -24 |
| 1 0 4 | 3 |
Step 2: Multiply the first row by 1 and subtract it from the third row to eliminate the x-term below the leading coefficient in the third equation.
| 1 1 -1 | 5 |
| 0 -4 4 | -24 |
| 0 -1 5 | -2 |
Step 3: Multiply the second row by -1/4 to make the leading coefficient of the second equation equal to 1.
| 1 1 -1 | 5 |
| 0 1 -1 | 6 |
| 0 -1 5 | -2 |
Step 4: Multiply the second row by 1 and add it to the first row to eliminate the y-term below the leading coefficient in the first equation.
| 1 0 -2 | 11 |
| 0 1 -1 | 6 |
| 0 -1 5 | -2 |
Step 5: Multiply the second row by 1 and add it to the third row to eliminate the y-term below the leading coefficient in the third equation.
| 1 0 -2 | 11 |
| 0 1 -1 | 6 |
| 0 0 4 | 4 |
Step 6: Multiply the third row by 1/4 to make the leading coefficient of the third equation equal to 1.
| 1 0 -2 | 11 |
| 0 1 -1 | 6 |
| 0 0 1 | 1 |
Step 7: Multiply the third row by 2 and add it to the first row to eliminate the z-term above the leading coefficient in the first equation.
| 1 0 0 | 13 |
| 0 1 -1 | 6 |
| 0 0 1 | 1 |
Step 8: Multiply the third row by -1 and add it to the second row to eliminate the z-term above the leading coefficient in the second equation.
| 1 0 0 | 13 |
| 0 1 0 | 7 |
| 0 0 1 | 1 |
The resulting matrix represents the row-echelon form of the system. Now, we'll perform back substitution to find the values of x, y, and z.
From the row-echelon form, we can deduce that x = 13, y = 7, and z = 1.
Therefore, the solution to the given system of equations is x = 2, y = -3, z = -1.
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Related Questions
A die is a cube with six sides and each side contains one to six dots. Suppose a blue die and a gray die are rolled together, and the numbers of dots that occur face up on each are recorded. The possible outcomes of the sample space S are listed as follows, where in each case the die on the left is blue and the one on the right is gray. S = {11, 12, 13, 14, 15, 16, 21, 22, 23, 24, 25, 26, 31, 32, 33, 34, 35, 36,
41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66}
Let E be the event that the sum of the numbers showing face up is at least 9. Write E as a set. (Enter your answer in set-roster notation. Enter EMPTY or ∅ for the empty set.)
E =
{24, 36, 53, 66}
What is the probability of E?
Answers
The probability of event E, which represents the sum of the numbers showing face up on the blue and gray dice being at least 9, can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this case, event E consists of the outcomes {24, 36, 53, 66}, which means there are 4 favorable outcomes. The total number of possible outcomes is 36 since there are six possible outcomes for each die roll. Therefore, the probability of event E is 4/36 or 1/9.
To calculate the probability of event E, we need to determine the number of favorable outcomes and the total number of possible outcomes. In this case, the event E represents the sum of the numbers on the blue and gray dice being at least 9. The favorable outcomes are the outcomes in the sample space S that satisfy this condition, namely {24, 36, 53, 66}. There are four favorable outcomes in this case.
The total number of possible outcomes can be found by counting all the elements in the sample space S. Since each die has six sides and can show numbers from 1 to 6, there are 6 possible outcomes for each die roll. As there are two dice being rolled together, the total number of possible outcomes is 6 * 6 = 36.
To calculate the probability, we divide the number of favorable outcomes (4) by the total number of possible outcomes (36). Therefore, the probability of event E is 4/36, which can be simplified to 1/9.
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987654321.123456-78906543.21+23456.88+376483.7332=
Answers
Answer:
909147718.527
Step-by-step explanation:
It's called a calculator
A rose garden is formed by joining a rectangle and a semicircle, as shown below. The rectangle is 21ft long and 14ft wide. If the gardener wants to build a fence around the garden, how many feet of fence are required?
Answers
Answer: 70
Step-by-step explanation: 21+21+14+14
On a map of Arizona 2 inches represents 54 miles. The map distance from Yuma to Sentinel is 3 inches. What is the actual distance in miles between the cities?
Answers
Answer: 81 miles
Step-by-step explanation:
if 2 inches = 54miles then 1 inch is 27 miles because 54/2=27
so 3 inches means 3(27)=81 miles from yuma to sentinel
What is x^2+4x+4 simplified?
Answers
Answer:
(x+2)^2 or (x+2)(x+2)
Step-by-step explanation:
x^2+4x+4 simplified is (x+2)^2 or (x+2)(x+2)
Answer:
(x+2)^2 or (x+2)(x+2)
Step-by-step explanation:
Brainly pls :D I tried for a while just to do this
PLEASE HELP ILL GIVE BRAINLIEST
Answers
tables originally priced at $70 are now on sale for $42.What is the discount, as a percentage?
Answers
Answer:
40%
Step-by-step explanation:
you saved $28.00
find the area of the surface. the part of the hyperbolic paraboloid z = y2 − x2 that lies between the cylinders x2 y2 = 9 and x2 y2 = 16.
Answers
To find the area of the surface between the cylinders x^2 y^2 = 9 and x^2 y^2 = 16 for the hyperbolic paraboloid z = y^2 − x^2, we can set up a double integral over the region of interest.
First, let's find the limits of integration for x and y. The equation x^2 y^2 = 9 represents a hyperbola, and x^2 y^2 = 16 represents another hyperbola. We can solve for y in terms of x for both equations:
For x^2 y^2 = 9:
y^2 = 9 / (x^2)
y = ±3 / x
For x^2 y^2 = 16:
y^2 = 16 / (x^2)
y = ±4 / x
Since the hyperbolic paraboloid is symmetric about the x and y axes, we only need to consider the positive values of y. Thus, the limits for y are from 3/x to 4/x.
To find the limits for x, we can equate the two equations:
3 / x = 4 / x
3 = 4
This is not possible, so the two curves do not intersect. Therefore, the limits for x can be determined by the region bounded by the hyperbolas. We solve for x in terms of y for both equations:
For x^2 y^2 = 9:
x^2 = 9 / (y^2)
x = ±3 / y
For x^2 y^2 = 16:
x^2 = 16 / (y^2)
x = ±4 / y
Again, considering only positive values, the limits for x are from 3/y to 4/y.
Now we can set up the double integral for the area:
A = ∬ R √(1 + (∂z/∂x)^2 + (∂z/∂y)^2) dA
where R represents the region of integration and dA is the differential area element.
The integrand √(1 + (∂z/∂x)^2 + (∂z/∂y)^2) simplifies to √(1 + 4y^2 + 4x^2).
Therefore, the area A can be expressed as:
A = ∫∫ R √(1 + 4y^2 + 4x^2) dA
To evaluate this double integral, we integrate with respect to y first, and then with respect to x, using the limits determined earlier:
A = ∫[3/y, 4/y] ∫[3/x, 4/x] √(1 + 4y^2 + 4x^2) dx dy
After integrating, the resulting expression will give us the area of the surface between the two cylinders.
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16 students at a school were asked about their favorite pasta dish. A graph of the results is on the left.
Create a bar graph showing the possible results for all 400 students in the school. Be sure to number the vertical axis.
Answers
Answer:
Step-by-step explanation:
to solve this question, we need to use the graph on the left to find the proportions of students who prefer each pasta dish, and then multiply those proportions by 400 to get the estimated number of students in the whole school who prefer each pasta dish. Then we need to plot those numbers on a bar graph with the pasta dishes on the horizontal axis and the number of students on the vertical axis. The graph below shows one possible way to create the bar graph:
We can see that the vertical axis is numbered from 0 to 120 in increments of 20. The bars show the estimated number of students who prefer each pasta dish, based on the sample of 16 students. For example, since 4 out of 16 students prefer spaghetti, we can estimate that 4/16 x 400 = 100 students in the whole school prefer spaghetti. Similarly, since 3 out of 16 students prefer lasagna, we can estimate that 3/16 x 400 = 75 students in the whole school prefer lasagna. We can repeat this process for the other pasta dishes and plot them on the graph.
✧☆*: .｡. Hope this helps, happy learning! (*✧×✧*) .｡.:*☆
Which statement correctly compares the areas of these two rectangles?
Answers
A statement that correctly compares the areas of these two rectangles is that the area of the yellow rectangle is greater than the area of the blue rectangle by 8 square units.
How to calculate the area of a rectangle?In Mathematics and Geometry, the area of a rectangle can be calculated by using the following mathematical equation:
A = LW
Where:
A represent the area of a rectangle.W represent the width of a rectangle.L represent the length of a rectangle.Based on the information provided about these rectangles, we have the following:
Area of blue rectangle = 10 × 4
Area of blue rectangle = 40 square units.
Area of yellow rectangle = 6 × 8
Area of yellow rectangle = 48 square units.
Difference = Area of yellow rectangle - Area of blue rectangle
Difference = 48 - 40
Difference = 8 square units.
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please help me graph this thanks :)
Answers
Answer: umm I don't now how to word a graph but I will try to help. okay so you will need to simplify it to look like y=mx+b but without the equal sign, next you have to have a solid line, that's really it besides that it is a positive line that looks like I
Step-by-step explanation:
So, we will solve for y:
Add 20 to both sides:
-5y ≤ -15x+20
Divide by -5 on both sides. NOTE: when dividing or multiplying by a negative, the inequality sign flips. So, instead of less than or equal to, it flips to greater than or equal to.
y ≥ -15/-5x+20/-5
Simplify:
y ≥ 3x-4
So, we now how the inequality in slope-intercept form, which is easier to graph than standard form.
The first step to graphing a linear inequality is to create a boundary or limit with the line, so we will graph the line as usual.
Y-intercept (b): -4
Slope (m): 3/1
So, graph the y-intercept first and then use the slope to graph the line. The y-intercept is where the graph intercepts the y axis. Remember, slope is rise over run, so from -4, the graph rises 3 units up and runs 1 unit right.
Once you have the line, understand the inequality. In a function, y is the output dependent on x (the input). Therefore, y is the total or output and x is the input.
When you graph a function, you choose values from the x-axis to plug into the equation, which then returns an output value that f graphs along the y-axis.
So, in y ≥ 3x-4, y is greater than or equal to the expression on the right. Since we know that y is the output and is the vertical value for an x value, this means that the output value based on x must be greater than the output value of the line you graphed. This is why the line acts as a boundary; y is greater than or equal to the outputs on that line.
To show this, you will shade all the space above the equation, and the linear equation should be a solid line, not dashed. Solid lines represent that y can be collinear (lie on) to that line Here’s a picture to illustrate this:
Hope this helps! Please select! Brainliest!
30 POINTS What is the answer????
Answers
Answer:
Step-by-step explanation:
the third i think
Find the volume of a cube with a side of 6 cm. It has a mass of 457 g. Find its density and label with correct units.
Answers
Answer: 8.1 g/cm
Step-by-step explanation:
Density is equal to mass divided by volume. You are given a mass of
64.7
grams. You can find the volume of the cube by doing height multiplied by width multiplied by the depth.
Because all of the sides of a cube are the same length, the volume just ends up being
(2 cm) = 8 cm
Now plug both of those numbers into your density equation
p= m/v = 64.7 g/8 cm = 8.1 g/cm
~Hope this helps ;0~
Answer:
\(2.11 \: \: gram \: per \: cm {}^{3}\)
The density is 2.11 g/cm^3
Step-by-step explanation:
Density is the ratio of the mass of a body per volume. To find density we need mass and volume.
side given (l) =6 cm
Volume of the cube = l^3
= 6^3
= 6 * 6 * 6
= 216 cm^3
we have mass = 457 g
Now,
density= mass/volume
= 457/216
=2.11
Soo, the density is 2.11 g/cm^3
if you want to round a number within an arithmetic expression, which function should you use?
Answers
If you want to round a number within an arithmetic expression, you should use the ROUND function.
The ROUND function allows you to specify the number of decimal places to which you want to round a given number. It is commonly used in programming languages and spreadsheet software.
The syntax for the ROUND function typically involves specifying the number or expression you want to round and the number of decimal places to round to. For example, if you want to round a number, let's say 3.14159, to two decimal places, you would use the ROUND function like this: ROUND(3.14159, 2), which would result in 3.14.
Using the ROUND function ensures that the rounded number is calculated within the arithmetic expression, providing the desired level of precision in the calculation.
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Write the equation of the line (in y = ax+b form) in the
graph below. do not use any spaces when typing your
answer
Answers
Answer:
y=3x-8
Step-by-step explanation:
what the root of this question?
Answers
Answer:
\( \sqrt{125 {p}^{2} } = p \sqrt{25} \sqrt{5} = 5p \sqrt{5} \)
D is the correct answer.
How does adding salt to water affect the boiling point?
Answers
When salt is added to water, the boiling point of water is increased. The boiling point elevation depends on the concentration of salt added. The addition of salt increases the boiling point of water.
The extent of the increase in boiling point is directly proportional to the amount of salt added. Boiling point is defined as the temperature at which the vapor pressure of a liquid is equal to the pressure exerted on it by the surrounding atmosphere.
In layman's terms, this means that a liquid boils when its vapor pressure equals atmospheric pressure. Boiling point of a liquid can be increased or decreased by the addition of solutes such as salt.When a solute such as salt is added to water, the boiling point of water increases due to the phenomenon of boiling point elevation.
This is because the solute particles in the water interfere with the formation of water vapor molecules at the surface of the liquid. As a result, more energy is required to boil the water, which means that the boiling point is elevated.
The amount of elevation is directly proportional to the concentration of salt added. In short, the addition of salt to water increases the boiling point of water.
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A particle starts from rest at a fixed point A and moves in a straight line with an acceleration which, t seconds after leaving A, is given by a = 4t. After 2 seconds the particle reaches a point B and the acceleration then ceases. Find:
i) the velocity when the particle reaches B
ii) the distance AB
The particle moves on immediately with acceleration given by -3t, where t seconds is the time after the particle leaves A, until it comes to rest at a point C. Find:
iii) the value of t when the particle reaches C
(iv) the distance AC
Answers
(a) The velocity when the particle reaches B is 8 m/s.
(b) The distance between point A and B is 5.33 m.
(c) The value of t when the particle reaches C is 1.63 seconds.
(d) The distance AC is 8.7 m.
Velocity when the particle reaches BThe velocity when the particle reaches B is calculated as follows;
v = ∫a. dt
where;
v is velocitya is acceleration of the particlev = ∫(4t . dt)
v = 4t²/2
v = 2t²
v(2) = 2(2)²
v(2) = 8 m/s
Thus, the velocity when the particle reaches B is 8 m/s.
Distance ABx = ∫v
where;
x is the distance between A and Bx = ∫2t². dx
x = ²/₃t³
x(2) = ²/₃(2)³
x(2) = 5.33 m
Thus, the distance between point A and B is 5.33 m.
value of t when the particle reaches Cwhen the particle reaches point C, final velocity, vf = 0
vf = v + at
where;
v is the velocity at point0 = 8 - 3t(t)
0 = 8 - 3t²
3t² = 8
t² = 8/3
t² = 2.67
t = √(2.67)
t = 1.63 seconds
Distance between A and Cx = ∫vf
x = ∫(8 - 3t²)
x = 8t - t³
x(1.63) = 8(1.63) - (1.63)³
x(1.63) = 8.7 m
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Question 10
When choosing a loan, which of the following scenarios is best?
1Points
A
lower APR, shorter loan, and extra principal payments
B
lower APR, longer loan, and extra interest payments
C
higher APR, shorter loan, and make interest payments
Answers
When choosing a loan, the best scenario is lower APR, shorter loan, and extra principal payments.
How to illustrate the information?A loan is a type of debt that a person or other entity incurs. The lender advances the borrower a certain amount of money, typically on behalf of a business, financial institution, or government. The borrower accepts a specific set of terms in return, which may include any financial costs, interest, a repayment schedule, and other requirements.
APR represents the total yearly cost of a loan to the borrower, including all fees. The APR is stated as a percentage, just like an interest rate. It does, however, contain other costs or fees such as mortgage insurance, the majority of closing costs, discount points, and loan origination fees, unlike an interest rate. It should be noted that the APR simply means the annual percentage rate which is the interest that's generated on the amount borrowed.
Therefore, it should be noted that when choosing a loan, the best scenario is a lower APR, shorter loan, and extra principal payments
This is important in order to reduce the value that the person will pay and make it easier to return the loan.
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I'm confused, please help. :/
Answers
uh i did this wrong so uh just ignore what i said
Answer:
82°
Step-by-step explanation:
Angle C and B equal the measure of angle A so,
143 - 61 = 82
Angle B is 82°
Solve the inequality:
14+4t>t+8≥8+4t
Answers
After solving the given inequality 14+4t>t+8≥8+4t the solution is given by 0 ≤ t < -2.
As given in the question,
Given inequality is :
14+4t>t+8≥8+4t
Solve the inequality to get the solution :
14+4t>t+8≥8+4t
Subtract 4t from the inequality we get,
14+4t -4t > t+8 -4t ≥ 8+4t -4t
⇒ 14 > 8- 3t ≥ 8
Subtract 8 from the inequality we have,
14 -8 > -3t ≥ 8 -8
⇒ 6 > -3t ≥ 0
Divide by -3 we get,
0 ≤ -3t /-3 < 6/-3
⇒ 0 ≤ t < -2
Therefore, after solving the given inequality 14+4t>t+8≥8+4t the solution is given by 0 ≤ t < -2.
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Trapezoid ABCD is reflected over the line y - x. What rule shows the input and output of the reflection, and what is the new coordinate of A (1 point)
B
0
A
(X,Y)--(-x): A is at (1,5)
(X,Y) (YX); A' is at (1-5)
(x,y)--(-x,y); A' is at (5, 1)
(x,y)--(-X, "): A' is at (5, -1)
Answers
Answer:
(x,y)--(-X, "): A' is at (5, -1)
Step-by-step explanat
Answer:
(x,y)→(−x,−y); A' is at (5, −1)
Step-by-step explanation:
a ratio of the vertical change to the horizontal change of two points is called:_____.
Answers
Answer:
Step-by-step explanation:
Slope of the line.
Find the perimeter. Simplify your answer.
Answers
Answer:
The perimeter of the Parallelogram is 20z + 16
Step-by-step explanation:
For a Parallelogram
Perimeter = sum of all sides
P = (7z + 3) + (7z + 3) + (3z + 5) + (3z + 5)
P = 7z + 7z + 3z + 3z + 3 + 3 + 5 + 5
P = 20z + 16
Thus, The perimeter of the Parallelogram is 20z + 16
-TheUnknownScientist 72
A 7 oz contains approximately 210 pieces. write a unit ratio for the number of pieces to ounces
Answers
A unit ratio for the number of pieces to ounces is 30.
What is the unit rate?In Mathematics, the unit rate is sometimes referred to as unit price or unit ratio and it can be defined as the price that is being charged by a seller for the sale of a single unit of product or quantity.
How to determine the unit rate?In order to determine the unit price, we would use the following mathematical expression;
Unit rate = Number of pieces/Number of ounces
Unit rate = 210/7
Unit rate = 30.
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How much pure acid should be mixed with gallons 5 of a 30% acid solution in order to get a 50% acid solution?
Answers
Answer: acid + acid = acid
x+%2B++.30%2A5gal+=+.50%285%2Bx%29
+.5x+=+.20%2A5
x = 2 gal
Note: 2gal + 1.5gal = 3.5gal
Set Up on ALL mixture problems is to have the same amount of the key ingredient on both sides of the EQ
Step-by-step explanation:
A cone with a radius of 6 meters has a volume of 542.87m^3. Find the slant height of the cone.
Answers
Answer:
slant height of a cone = 15.6 mStep-by-step explanation:
V = π r² h / 3
where volume V = 542.87 m³
radius r = 6 m
find height h
solution:
V = π r² h / 3
542.87 = π (6)² h
3
π (6)² h = 542.87 (3)
h = 542.87 (3)
π (6)²
h = 1628.6 / 113.1
h = 14.4 m
now solve for slant height of a cone using Pythagorean
slant height² = h² + r²
= √(14.4)² + (6)²
= √243.4
= 15.6 m
therefore, slant height of a cone = 15.6 m
This table gives information about the colors and engine types of motorcycles sold at a showroom in a month.
150 cc 180 cc
Black 18 24
Blue 15 17
Red 26 21
Identify the tables that represent the relative frequencies of this data either by row or by column. Round your answers to the nearest hundredth
Answers
The relative frequencies of this data by row is
150 cc 180 cc
Black 0.43 0.57
Blue 0.47 0.53
Red 0.55 0.45
How to determine the relative frequency table?The table of values is given as:
150 cc 180 cc
Black 18 24
Blue 15 17
Red 26 21
Calculate the row total.
150 cc 180 cc Total
Black 18 24 42
Blue 15 17 32
Red 26 21 47
Divide each element by the row total
150 cc 180 cc
Black 0.43 0.57
Blue 0.47 0.53
Red 0.55 0.45
Hence, the relative frequencies of this data by row is
150 cc 180 cc
Black 0.43 0.57
Blue 0.47 0.53
Red 0.55 0.45
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Tai’s baby brother weighs 3,745 grams. What is his weight in kilograms? A. 0.3745 kg B. 3.745 kg C. 37.45 kg D. 374.5 kg
Answers
Answer:
My answer to the question is 3.745kg.
Option B
The required weight of Tai's baby brother weighs 3,745 grams in kilograms is 3.745 kg. Option B is correct.
Given that,
Tai’s baby brother weighs 3,745 grams. What is his weight in kilograms is to be determined.
A kilogram is the standard unit of measurement of weight.
What is arithmetic?In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication, and division,
Here,
1 kilogram = 1000 gram
1 gram = 1 / 1000 kilogram
Multiply both side by 3,745
3,745 gram = 3,745 / 1000 kilogram
3,745 gram = 3.745 kilogram
Thus, the required weight of Tai's baby brother weighs 3,745 grams in kilograms is 3.745 kg. Option B is correct.
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the physical plant at the main campus of a large state university recieves daily requests to replace florecent lightbulbs. the distribution of the number of daily requests is bell-shaped and has a mean of 40 and a standard deviation of 7. using the empirical rule (as presented in the book), what is the approximate percentage of lightbulb replacement requests numbering between 19 and 40?
Answers
By using the empirical rule, the approximate percentage of lightbulb replacement requests numbering between 19 and 40 is 99.3%.
How to calculate percentageThe empirical rule is a statistical guideline which relates to bell-shaped distributions.
According to the rule, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations of the mean.
We know that mean is 40 and a standard deviation is 7.
To find the approximate percentage of lightbulb replacement requests numbering between 19 and 40
z₁ = (19 - 40) / 7 ≈ -3.00
z₂ = (40 - 40) / 7 = 0.00
Here, z₁ is the number of standard deviations that 19 is below the mean, and z₂ is the number of standard deviations that 40 is above the mean.
According to the empirical rule, approximately 99.7% of the data falls within three standard deviations of the mean.
Therefore, the approximate percentage of lightbulb replacement requests numbering between 19 and 40 is
percentage ≈ 99.7% * (1 - 0.00135) ≈ 99.3%
Note that, we subtracted the area under the normal curve beyond three standard deviations, which is approximately 0.15%, from 100% to get the percentage of data within three standard deviations.
Therefore, approximately 99.3% of the daily requests to replace fluorescent lightbulbs fall between 19 and 40.
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