2. The purpose of this problem is to practice manipulating mathematical objects
ID: 2267474 • Letter: 2
Question
2. The purpose of this problem is to practice manipulating mathematical objects using only their definitions and properties. I have broken the problem up into parts to help guide your response. (a) What is the definition of a linear system with n unknowns and m equations (b) Suppose that first equation in part (a) is replaced by the first equation minus two times the second equation. Express this new system using the notation from part (c) Show that a solution to the system in part (a) is also a solution to the system in part (b). [SUGGESTION: Create a new symbol (or new symbols) to represent a solution to the system from part (a).] (d) Show that a solution to the system in part (b) is also a solution to the system in part (a). [SUGGESTION: Create a new symbol (or new symbols) to represent a solution to the system from part (b).] (e) What is the definition of equivalent systems, and what can you now conclude about the systems from part (a) and part (b)?Explanation / Answer
a)A linear system can be defined as A system of equations is a set or collection of equations that you deal with all together at once
A linear equation in n unknowns x1; x2;... ; xn is an equation of the form a1x1 +a2x2 +....+anxn = b; where a1; a2;....; an; b are given real numbers.
Ex:lets consider x and y instead of x1 and x2,the linear equation 5x+8y = 6 describes the line passing through the points (5; 0) and (0; 8).
A system of m linear equations in n unknowns x1, x2,..., xn is a family of linear equations:-
a11x1 + a12x2 + · · · + a1nxn = b1 ; a21x1 + a22x2 + · · · + a2nxn = b2 ; ... am1x1 + am2x2 + · · · + amnxn = bm.
Such a linear system is called a homogeneous linear system if b1 = b2 = · · · = bm = 0.
consider two variable system
1)2x + y = 3 ; x 9y = 8 .Here,x = 1, y = 1 is the only solution to this system.Geometrically ,Points where graphs of the two equations meet.
2)system 2x + y = 3; 2x + y = 7 does not have any solution. Such system would be called as an inconsistent system. Geometrically, these equations in the system represent two parallel lines i.e: they never meet.
If such a system has a solution i.e if there exists numbers x1,x2,..,xn.which satisfies the equation.
System is consitent if it has a solution,otherwise system is called inconsistent.
Geometrically ,solving a system of linear equations is determining whether the family of lines (or planes) has a common point of intersection or not.
In three variables, two equation system is given by : 2x + y + 2z = 3 ; x 9y + 2z = 8.Here, x = 1, y = 1, z = 0 is a solution to this system.
This system has many more solutions. x = 11, y = 0, z = 19/2 is also a solution of this system.
This specifies: Geometrically, solution given by precisely the points where the graphs (two planes) in 3-space of these two equationns meet.
consider part a system is 2x + y = 3 ; 3x 9y = 8
b)a11x1 + a12x2 + · · · + a1nxn - 2(a21x1 + a22x2 + · · · + a2nxn) = b1-2b2
Let's consider equation 2x+y=3 and 2nd equation 3x-9y=-8 .The resultant equation is represented as 2x+y-6x+18y=3+16 => -4x+18y=19
System in partb is -4x+19y=19;3x-9y=-8
c)Solution for system in part a is x=19/21,y=25/21 which when substituted in system in part b equations gives the solution to the part b system.
d) The system in part b also have same solution as in part a system which will give solution for the system in part a.
e)Equivalent systems are those systems which have the same solution set.Here as per the example considered it's concluded that the systems in part a and part b are equilavalent with a solution set (19/21,25/21)
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.