Solve Linear System Ax=b

For example, if x2= 0, x4= 1, and x5= 0, then x1= 2- 2(0)- 3(1)- 0= -1 and x3= 5- 2(1)- 4(0)= 3. That gives the solution (-1, 0, 3, 1, 0). If x2= 0, x4= 0, and x5= 1, then x1= 2- 2(0)- 3(0)- 1= 1 and x3= 5- 2(0)- 4(1)= 1.
  • #1
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Let Ax = b be a linear system whose augmented matrix (A|b) has reduced row echelon form
1 2 0 3 1 ||-2
0 0 1 2 4 || 5
0 0 0 0 0 || 0
0 0 0 0 0 || 0a) Find all solutions to the system

b)If


a1= (1,1,3,4) and a3= (2,-1,1,3)
determine b.
I got part a to be
x1=-2-2r-3t-w
x2=5-2t-4w
x3=r
x4=t
x5=w

I'm having difficulty getting part b. I know a1 and a3 are column vectors and I know Ax=x1a1+x2a2+...+xnan. The textbook solution says its b=(8,-7,-1,7)^T
 
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  • #2
I have no idea what you mean by "a1" and "a2"! If we call the variables x1, x2, x3, x4, and x5 (I thought at first the you meant "a1" and "a2" to be the first of those but those are, of course, numbers, not vectors) then the two equations become x1+ 2x2+ 3x4+ x5= 2 and x3+ 2x4+ 4x5= 5. From the first, x1= 2- 2x2- 3x4- x5 and from the second, x3= 5- 2x4- 4x5. So you can choose x2, x4, and x5 to be anything you want, then calculate x1 and x3.
 

What does it mean to solve a linear system Ax=b?

Solving a linear system Ax=b means finding the values of the variables in the system that satisfy the equation. In other words, it is finding the solution to the system of linear equations.

What is the importance of solving a linear system Ax=b?

Solving a linear system Ax=b is important in many areas of science and engineering. It allows us to model and analyze real-world situations, make predictions, and solve optimization problems.

What are the methods for solving a linear system Ax=b?

There are several methods for solving a linear system Ax=b, including Gaussian elimination, Cramer's rule, inverse matrix method, and iterative methods such as Jacobi and Gauss-Seidel.

How do you know if a linear system Ax=b has a unique solution?

A linear system Ax=b has a unique solution if the system is consistent (has at least one solution) and the number of equations is equal to the number of variables. This is known as the fundamental theorem of linear algebra.

What are some common applications of solving a linear system Ax=b?

Solving a linear system Ax=b is commonly used in many fields, such as engineering, physics, economics, and computer science. It can be used to solve problems involving electric circuits, chemical reactions, population dynamics, and more.

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