# How to express solution to system of equations

• dewert
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.
dewert
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:

$$2x_{1} - 2x_{2} - 3x_{3} = -2$$
$$3x_{1} - 3x_{2} - 2x_{3} + 5x_{4} = 7$$
$$x_{1} - x_{2} - 2x_{3} - x_{4} = -3$$

The attempt at a solution

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

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 $$\in \mathbb{R}$$}

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:
Are you sure you don't get something else, like, say,
$$$\left( \begin{array}{cccc|c} 2 & -2 & 0 & 3 & 5 \\ 0 & 0 & 1 & 2 & 4 \\ 0 & 0 & 0 & 0 & 0 \end{array} \right)$$$

## 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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