How to express solution to system of equations

In summary, the conversation is about solving a system of equations. The given matrix has no unique solution, but the answer in the back of the book includes a solution involving variables r and s. The question is how to obtain this solution from the given matrix.
  • #1
dewert
3
0
Sorry if this is too basic for this forum, but it IS from a 2nd-year linear algebra course. I'm probably just being stupid, and missing something obvious, but here goes:

Homework Statement

Solve the following system of equations:

[tex]2x_{1} - 2x_{2} - 3x_{3} = -2[/tex]
[tex]3x_{1} - 3x_{2} - 2x_{3} + 5x_{4} = 7[/tex]
[tex]x_{1} - x_{2} - 2x_{3} - x_{4} = -3[/tex]

The attempt at a solution

So, clearly there won't be a unique solution. I do the work and get this matrix:
[tex]
\[ \left( \begin{array}{cccc|c}
1 & -1 & 0 & 0 & 5 \\
0 & 0 & 1 & 0 & 4 \\
0 & 0 & 0 & 1 & 0 \end{array} \right)\]
[/tex]

However, the answer in the back of the book is
{r(1,1,0,0) + s(-3,0,-2,1) + (5,0,4,0) | r,s [tex]\in \mathbb{R}[/tex]}

My question is how to get this from the solution matrix. Regardless of whether the solution is right or wrong, I don't know where this is coming from. I can conceivably see that (5,0,4,0) can be obtained by choosing x_2 = 0, but that's about it.

Thanks!
 
Last edited:
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  • #2
Are you sure you don't get something else, like, say,
[tex]
\[ \left( \begin{array}{cccc|c}
2 & -2 & 0 & 3 & 5 \\
0 & 0 & 1 & 2 & 4 \\
0 & 0 & 0 & 0 & 0 \end{array} \right)\]
[/tex]
 

Related to How to express solution to system of equations

1. What is the general method for expressing solutions to a system of equations?

The general method for expressing solutions to a system of equations is by using the substitution or elimination method. These methods involve substituting one variable into another equation or eliminating a variable by combining equations, respectively.

2. How do I know if a system of equations has a solution?

A system of equations will have a solution if the equations have at least one point in common, meaning they intersect at one point. This point is the solution to the system of equations.

3. What does it mean if a system of equations has no solution?

If a system of equations has no solution, it means that the equations do not intersect at any point. This could be because the equations are parallel or the lines are coinciding.

4. Can a system of equations have more than one solution?

Yes, a system of equations can have more than one solution. This occurs when the equations represent parallel lines that intersect at multiple points.

5. How do I express the solution to a system of equations in a written form?

The solution to a system of equations can be expressed in a written form as an ordered pair (x,y) where x represents the value of one variable and y represents the value of the other variable. This point is the intersection of the two lines and satisfies both equations in the system.

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