Answer: These three planes are consistent and independent.
Explanation:
Since, if the system of planes has a solution then it is called Consistent, While, if it does not have any solution then it is called inconsistent.
Further, If the consistent system has infinite solution then it is dependent but if it has only a unique solution then it is called independent.
Here, given equations of planes are
12x+5y-3z=36 -------(1 )
x-2y+4z=3 -------(2)
9x-10y+5z=27 -------(3)
From equation (1), 3z=12x+5y-36⇒z=4x+5y/3-12 ------(4)
after putting this value in equation (2) and (3), we will get two equation in variables x and y
So, x-2y+4(4x+5y/3-12)=3 ⇒x-2y+16x+20y/3-48=3⇒3x-6y+48x+20y-144=9⇒51x+14y=153 --------(5)
And, 9x-10y+5(4x+5y/3-12)=27⇒ 9x-10y+20x+25y/3-60=27⇒ 27x-30y+60x+25y-180=81⇒87x-5y=261 --------(6)
after solving equation equation (5) and (6) we will get x=3 and y=0
substituting these values in equation (4) we will get z=0
Thus the solution of these three plane (1), (2) and (3) is x=3, y=0 and z=o
Which is the unique solution, Thus the given planes are consistent and independent.
The given system of equations is consistent and independent so, [tex]\fbox{\begin\\\ \bf option (3)\\\end{minispace}}[/tex] is correct.
Further explanation:
A system of equations is said to be a consistent system if the solution exists and if the solution does not exist then it is an inconsistent system.
If a consistent system has a unique solution then it is an independent system of equations but if the number of solutions is infinite then it is a dependent system of equations.
Label the given equations as shown below:
[tex]\boxed{x-2y+4z=3}[/tex] …… (1)
[tex]\boxed{12x+5y-3z=36}[/tex] …… (2)
[tex]\boxed{9x-10y+5z=27}[/tex] …… (3)
The augmented matrix for the above equations is, as follows:
[tex]\left[\begin{array}{ccc}1&-2&4\\12&5&-3\\9&-10&5\end{array}\Biggm\vert\begin{array}{c}3\\26\\27\end{array}\right][/tex]
Apply row transformation [tex]R_{2}\rightarrow R_{2}-12R_{1}[/tex] and [tex]R_{3}\rightarrow R_{3}-9R_{1}[/tex] as,
[tex]\left[\begin{array}{ccc}1&-2&4\\0&29&-51\\0&8&-31\end{array}\Biggm\vert\begin{array}{c}3\\0\\0\end{array}\right][/tex]
Now, apply row transformation [tex]R_{2}\rightarrow R_{2}-4R_{3}[/tex] and [tex]R_{3}\rightarrow R_{3}-3R_{2}[/tex] as,
[tex]\left[\begin{array}{ccc}1&-2&4\\0&-3&73\\0&-1&188\end{array}\Biggm\vert\begin{array}{c}3\\0\\0\end{array}\right][/tex]
Now, apply row transformations [tex]R_{2}\rightarrow -R_{2}+2R_{3}[/tex] and [tex]R_{3}\rightarrow R_{3}+R_{2}[/tex] as,
[tex]\left[\begin{array}{ccc}1&-2&4\\0&1&303\\0&0&491\end{array}\Biggm\vert\begin{array}{c}3\\0\\0\end{array}\right][/tex]
The equations obtained from the above augmented matrix are,
[tex]\begin{aligned}x-2y+4z&=3\\y+303z&=0\\491z&=0\end{aligned}[/tex]
The first equation is simplified to obtain the value of z as,
[tex]\begin{aligned}491z&=0\\z&=0\end{aligned}[/tex]
Substitute [tex]0[/tex] for [tex]z[/tex] in the equation [tex]y+303z=0[/tex] to obtain the value of [tex]y[/tex] as,
[tex]\begin{aligned}y+(303\cdot0)&=0\\y&=0\end{aligned}[/tex]
Now, substitute [tex]0[/tex] for [tex]z[/tex] and [tex]0[/tex] for [tex]y[/tex] in the equation [tex]x-2y+4z=3[/tex] to obtain the value of [tex]x[/tex] as,
[tex]\begin{aligned}x-(2\cdot0)+(4\cdot0)&=3\\x&=3\end{aligned}[/tex]
Therefore, the value of [tex]x[/tex] is [tex]3[/tex], the value of [tex]y[/tex] is [tex]0[/tex] and the value of [tex]z[/tex] is [tex]0[/tex] and the system has a unique solution.
Thus, the given system of equations is consistent and independent.
Option (1)
Here, the first option is inconsistent and dependent.
There exists a solution of the given system of equations so it is consistent.
Therefore, option (1) is incorrect.
Option (2)
Here, the second option is consistent and dependent.
There exists a solution of the given system of equations so it is consistent.
Also, the solution is unique therefore the system of equations is independent.
Therefore, option (2) is incorrect.
Option (3)
Here, the third option is consistent and independent.
There exists a solution of the given system of equations so it is consistent.
Also, the solution is unique therefore the system of equations is independent.
Therefore, option (3) is correct.
Option (4)
Here, the fourth option is inconsistent and independent.
There exists a solution of the given system of equations so it is consistent.
Therefore, option (4) is incorrect.
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Answer details
Grade: High school
Subject: Mathematics
Chapter: System of linear equations
Keywords: Equations, unique solution, independent, dependent, consistent, inconsistent, infinite solutions, homogeneous equation, non- homogeneous equation, determinant.
There are different ways to think about the meaning of speed.
Which description correctly completes this sentence?
Speed is ________________________.
A.time traveled times distance traveled
B.time traveled divided by distance traveled
C.the number of hours it takes to travel a certain distance
D.the distance traveled divided by the time it takes to travel that distance
30 POINTS
Answer:
D
Step-by-step explanation:
Find the coordinates of the point (x,y,z)(x,y,z) on the plane z=2x+3y+3z=2x+3y+3 which is closest to the origin.
To find the point closest to the origin on the plane with equation z = 2x + 3y + 3, an offset vector from the origin to a point on the plane is used. This vector's dot product with the plane's normal vector should be zero, since the shortest path to the plane will be perpendicular to it. This creates a system of equations that can be solved for x, y, and z.
Explanation:To find the closest point (x, y, z) on the given plane to the origin in three-dimensional Cartesian axes, we can start by creating an offset vector from the origin to an arbitrary point on the plane. Let's use the equation of the plane, z = 2x + 3y + 3. Let's presume a point P(x, y, z) on this plane. The vector to this point from the origin is (x, y, z - 3).
Because we want the shortest distance to the origin, the dot product of this vector and the normal vector to the plane (2, 3, -1) should be zero, because the shortest path to the plane will be perpendicular to it. So, the formula for the dot product gives us:
2*x + 3*y - (z - 3) = 0
This simplifies to:
2x + 3y - z + 3 = 0
Which combined with the equation of the plane (z = 2x + 3y + 3), provides a system of equations that can be solved using methods like Gaussian elimination or substitution.
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reuben deposits $700 into an account that pays simple interest at a rate if 5% per year. How much interest will he be paid in the first 3 years.
Reuben deposits $700 at a 5% annual interest rate for 3 years, using the simple interest formula, he will earn $105 in interest over that period.
Explanation:The question deals with calculating the simple interest Reuben will receive by depositing $700 into an account at a 5% annual interest rate over 3 years. To calculate simple interest, the formula used is Interest = Principal × Rate × Time. In this scenario, the principal (P) is $700, the rate (R) of interest per year is 5% or 0.05 when expressed as a decimal, and the time (T) is 3 years.
Substituting these values into the formula provides us with: Interest = $700 × 0.05 × 3. Simplifying this gives us: Interest = $105.
Therefore, Reuben will be paid $105 in interest over the first 3 years.
Determine whether it is possible to circumscribe a circle about the quadrilateral.
80 POINTS
Point E is located at (2, −3) and point F is located at (−2, −1). Find the y value for the point that is 3 over 4 the distance from point E to point F.
A. -2.5
B. -4.5
C. -3.5
D. -1.5
How many extraneous solutions does the equation below have? 9 over n squared +1 =n+3 over 4? 0 1 2 3
Answer:
Option 1 - There is no extraneous solution i.e. 0.
Step-by-step explanation:
Given : Expression [tex]\frac{9}{n^2+1}=\frac{n+3}{4}[/tex]
To find : How many extraneous solutions does the equation have ?
Solution :
First we solve to expression to determine the extraneous solution,
[tex]\frac{9}{n^2+1}=\frac{n+3}{4}[/tex]
Cross multiply,
[tex]9\times 4=(n+3)(n^2+1)[/tex]
[tex]36=n^3+n+3n^2+3[/tex]
[tex]n^3+3n^2+n-33=0[/tex]
An extraneous solution is defined as a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem.
The equation form is a cubic function so it has 3 solutions.
Therefore, There is no extraneous solution.
A group of 30 bikers went on a trip. some rode bicycles and the others rode tandems. if the total number of bicycles and tandems was 23, how many tandems were used?
plez help What is the value of y in the equation 4 + y = −3?
7
1
−1
−7
Draw a number line and show all numbers that are between 1 and 20 and have exactly 4 factors
(or visualize, make it somewhat close to being a number line)
(work with a number line all positives, 1 being smallest, 20 being biggest.
Numbers which have exactly 4 factors between 1 and 20 are 6, 8, 10, 14, 15 represented on number line.
Define factor ?
" In mathematics factor is a number by which when we divides another number remainder is zero."
According to the question,
Factors of all the numbers between 1 and 20
2, 3, 5, 7, 11, 13, 17, 19 are prime number , so they have only two factors.
4 = 1, 2, 4
6 = 1, 2, 3, 6
8 = 1, 2, 4, 8
9 = 1, 3, 9
10 = 1, 2, 5, 10
12 = 1, 2, 3, 4, 6, 12
14 = 1, 2, 7, 14
15 = 1, 3, 5, 15
16 = 1, 2, 4, 8, 16
18 = 1, 2, 3, 6, 9, 18
Hence, we conclude that numbers which have exactly 4 factors between 1 and 20 are 6, 8, 10, 14, 15.
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A math teacher assigns 3 projects. She gives the students 5 project options to choose from. How many different groups of projects can the students choose from?
The population of a particular town is given by the function P(t) = 920(1.06)2t, where t is the time in years and P(t) is the population after t years. What is the current population, the percentage growth rate, and the population size (rounded to the nearest whole person) in 4 years?
A: 920, 6% annually, and 1,161
B:920, 6% semiannually, and 1,466
C: 920, 0.6% per year, and 7,801
D:920, 60% semiannually, and 4,135
Last week,you finished level 2 of a video gamr in 32 minutes. Today, you finish level 2 in 28 minutes. What is your percent change
PLEASE HELP WILL GIVE BRAINLIEST!!! What is the y-intercept of f(x)=(1/2)^x
Try this explanation:
'y-interception' means x=0.
If x=0, then y=1.
answer: (0;1)
What is the y-value when x equals -11?
y = -200 – 12(x).
Real Question:
You Make A Table To Use AS A Quick Reference Guide. [tex]\left[\begin{array}{ccc}y = -200 -12 (x)\end{array}\right][/tex]
Use The Table Tool To Answer The Question:
[tex]\left[\begin{array}{ccc}-8&-104\\-9&-92\\-10&-80\\-11&-68\\-12&-56\end{array}\right][/tex]
What is the y-value when x equals -11?
[tex]\left[\begin{array}{ccc}y = -200 -12 (x)\end{array}\right][/tex]
Step-By-Step Explanation:
Plug In The X-Value [tex]\left[\begin{array}{ccc}y=-200-12(x)\\y=-200-12(-11)\end{array}\right][/tex]
Then You'll Multiply [tex]\left[\begin{array}{ccc}-12(x) Or -12(-11)\\132\end{array}\right][/tex]
To Find The Y-Value Simplify [tex]\left[\begin{array}{ccc}y=-200+132\\y=-68\end{array}\right][/tex]
Answer:
[tex]\left[\begin{array}{ccc}y=-200-12(x)\\y=-200-12(-11)\\y=-200+132\\ y=-68\end{array}\right][/tex]
A random sample of size n = 64 is taken from a population with mean μ = −12.2 and standard deviation σ = 5.
a. calculate the expected value and the standard error for the sampling distribution of the sample mean. (negative values should be indicated by a minus sign. round "expected value" to 1 decimal place and "standard error" to 4 decimal places.)
The expected value for the sampling distribution of the sample mean is -12.2. The standard error for the sampling distribution of the sample mean is 0.6250.
Explanation:The expected value for the sampling distribution of the sample mean can be calculated using the formula µx = μ = -12.2. So, the expected value is -12.2.
The standard error for the sampling distribution of the sample mean can be calculated using the formula σx = σ/√n, where σ is the population standard deviation and n is the sample size.
Substituting the given values, we get σx = 5/√64 = 0.6250 (rounded to four decimal places).
The expected value and the standard error for the sampling distribution of the sample mean is 0.6250.
Expected Value: -12.2
Standard Error: 0.6250
1. The expected value (mean) of the sampling distribution of the sample mean is equal to the population mean, which is given as -12.2. Therefore, the expected value of the sampling distribution is also -12.2.
2. The standard error (SE) of the sampling distribution of the sample mean is calculated using the formula:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]
where [tex]\sigma[/tex] is the population standard deviation and n is the sample size.
Given [tex]\sigma[/tex] = 5 and n = 64, we can calculate the standard error as follows:
[tex]\[ SE = \frac{5}{\sqrt{64}} = \frac{5}{8} = 0.625 \][/tex]
Rounded to four decimal places, the standard error is 0.6250.
The third side of an isosceles triangle is 2 inches shorter than each of the other two sides. if the perimeter of this triangle is 67 inches find the length of each side.
what statement about this figure is true-- i know its one of the last two but they both seem right to me
I need help ASAP Please and thank you
What are the slope and the y intercept of the graph of the liner function shown on the grid
Find the average rate of change of the function
What is the answer to my math problem from my school for homework today?
About 12% of men and 10% of women are left-handed. if we select a person at random, are the event that the person is male and the event that the person is left-handed independt or associated?
The correct option is A. Gender and handedness are associated because men are more likely to be left-handed than women. This conclusion is drawn from the different probabilities of being left-handed based on gender, indicating an association rather than independence.
A. Gender and handedness are associated because men are more likely to be left-handed than women - - This statement is correct based on the given information. Men have a [tex]12\%[/tex] likelihood of being left-handed compared to women who have a [tex]10\%[/tex] likelihood. This difference shows an association between gender and handedness.
The other incorrect options are:
B. Gender and handedness are independent because all of the possible combinations of each exist - This option is incorrect because independence between two events means that the occurrence of one event does not affect the occurrence of the other. In this case, the probability of being left-handed depends on whether the person is male or female ([tex]12\%[/tex] for men and [tex]10\%[/tex] for women), indicating an association rather than independence.
C. Gender and handedness are associated because the percentage is an estimate - This option is incorrect because the percentages given ([tex]12\%[/tex]for men and [tex]10\%[/tex] for women) are specific probabilities based on data, not estimates. The difference in these probabilities reflects an association between gender and handedness.
D. Gender and handedness are independent because they are both comprised of mutually exclusive events - This option is incorrect because being male or female (gender) and being left-handed (handedness) are not mutually exclusive events. One can be male or female and still be left-handed or right-handed. Independence would require that the probability of being left-handed is the same regardless of gender, which is not the case here.
The complete Question is
About 12% of men and 10% of women are left-handed. If a person is selected at random, are the event that the person is male and the event that the person is handed independent or associated?
Choose the correct answer below.
A. Gender and handedness are associated because men are more likely to be left handed than women
B. Gender and handedness are independent because all of the possible combinations of each exist
C. Gender and handedness are associated because the percentage is an estimate
D. Gender and handedness are independent because they are both comprised of mutually exclusive events
Katie has 16 3/4 hours to finish 3 school projects. How much time may sge spend in each project if she plans to spend the same amount of time on each.
Jake ate 1/6 of the cookies at the valentines party. If he ate 5 cookies, how many cookies were there in all?
lars has been approved for$420,000, 20-year mortgage with an APR of 5.125%.What is his monthly payment ? How much interest would he expect to pay on the loan in one month?How much of the monthly payment will go towards the principal?
Answer:
Monthly Payment = $2,286.85
Interest on first month payment = $1,793.75
Principal value on first month payment = $493.10
Step-by-step explanation:
lars has been approved for $420,000 for loan.
Time for loan = 20-years
APR (Annual Rate of interest ) = 5.125%
We need to find monthly payment. First month interest and First month principle.
Formula for loan:
[tex]p=\dfrac{PV\cdot \frac{r}{n}}{1-(1+\frac{r}{n})^{-nt}}[/tex]
Where,
p is monthly payment.
PV = Loan Amount (PV=$420,000)
R = Rate of interest (R=0.05125)
t = Loan period ( t=30)
n = Mode of payment (n=12)
Substitute the value into formula and solve for p
[tex]p=\dfrac{420000\cdot \frac{0.05125}{12}}{1-(1+\frac{0.05125}{12})^{-12\cdot 30}}[/tex]
p=$2,286.85
Thus, Monthly Payment =$2,286.85
Interest for first month [tex]=420000\times \dfrac{0.05125}{12}=1793.75[/tex]
Interest pay on the loan in on month = $1,793.75
Principle value on first month payment = 2286.85 - 1793.75 = $493.10
The monthly payment will go towards the principal is $493.10
8 students can ride in each van. How many vans are needed for 29 students?
There is the need for 4 vans.
To determine how many vans are needed for 29 students, with each van able to transport 8 students, we can use division and then round up, since you can't have a fraction of a van. We divide 29 students by 8 students per van, which gives us 3.625. Since we can't have 0.625 of a van, we need to round up to the next whole number to ensure all students have a seat. That means we need 4 vans to transport 29 students.
Divide the total number of students by the number of students each van can hold: 29 / 8 = 3.625.
Round this number up to the nearest whole number, because you need whole vans: This gives us 4.
Thus, you need 4 vans to transport 29 students.
Which is the standard form of the equation of the parabola that has a vertex of (–4, –6) and a directrix of y = 3?
Answer:
The required equation is [tex](x+4)^2=-36(y+6)[/tex].
Step-by-step explanation:
The standard form of the equation of the parabola is
[tex](x-h)^2=4p(y-k)[/tex] .... (1)
where, (h, k) is vertex and y = k - p is directrix.
It is given that vertex of parabola is (–4, –6) and the directrix is y = 3.
[tex](-4,-6)=(h,k)[/tex]
On comparing both the sides, we get
[tex]h=-4,k=-6[/tex]
Directrix of the parabola is
[tex]y=k-p[/tex]
Put y=3 and k=-6 in the above equation.
[tex]3=-6-p[/tex]
[tex]3+6=-p[/tex]
[tex]9=-p[/tex]
[tex]-9=p[/tex]
Substitute h=-4,p=-9 and k=-6 in equation (1).
[tex](x-(-4))^2=4(-9)(y-(-6))[/tex]
[tex](x+4)^2=-36(y+6)[/tex]
Therefore the required equation is [tex](x+4)^2=-36(y+6)[/tex].
Kylie writes an equation to solve for the diameter, d, of the circle shown. Solve Kylie’s equation to find the diameter of the circle. 4 (d + 4) = 72 What is the diameter? 7.5 in. 8.25 in. 11.25 in. 13.25 in.
Answer:
8.25 in.
Step-by-step explanation:
In order to solve this you just have to clear the equation that Kylie created to calcuate the diameter of the circle for diameter, in this case, you just need to do the next steps:
[tex]4(d+4)=7^{2} \\4(d+4)=49\\d+4=\frac{49}{4} \\d+4=12.25\\d=12.25-4\\d=8.25[/tex]
Now we know that the diameter of the circle is 8.25 inches.
what are the roots of the equation?
x^2+24=14x
enter answers in the boxes
Write an equation or inequality that best represents each statement below
6 times a number, x, is at least 22