Respuesta :
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.
Learn more:
1. A problem on circle https://brainly.com/question/9510228.
2. A problem on general equation of a circle https://brainly.com/question/1506955.
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.
