# For which c is there 1/0 solution?

• MHB
• mathmari
In summary, the conversation discusses solving a linear system of equations with a fixed value of c in the real numbers. The matrix form of the system is shown and it is concluded that there will always be one solution, as the second line cannot be a multiple of the first. Geometrically, the solution is represented as a line, meaning there are infinitely many solutions.
mathmari
Gold Member
MHB
Hey!

I am looking the following:

Solve for a fix number $c\in \mathbb{R}$ the following linear system of equations: $$\begin{cases}x_1-cx_2+(2c-1)x_3=-(c+1) \\ 3x_2+(5c+8)x_3=-(c-2)\end{cases}$$
For which values of $c$ is there one solution and for which values are there no solution? I have done the following:

First we write this in matrix form:
\begin{equation*}\begin{pmatrix}\left.\begin{matrix}1 & -c & 2c-1 \\ 0 & 3 & 5c+8 \end{matrix}\right|\begin{matrix}-(c+1) \\ -(c-2)\end{matrix}\end{pmatrix}\end{equation*}
It is already in echelon form.

This at the second line we have "$3$" which doesn't depend on $c$, then the case "No solution"doesn't occur, right?

Since the second line cannotbe a multiple of the first one,we conclude that we always have One solution.

Is that correct?

:unsure:

Hey mathmari!

From the second equation, we can write $x_2$ as a function of $x_3$, which will indeed have at least 1 solution due to the non-zero coefficient $3$.
After that we can always find $x_1$ from the first equation as a function of $x_2$ and $x_3$.
So we will always have infinitely many solutions, won't we?

Looking at it geometrically, we have the intersection of 2 planes.
Those are actual planes since the coefficients of each cannot all be zero, regardless of the value of $c$.
They can either coincide, or be parallel (no solutions), or intersect in a line.
Since the coefficients in the second line cannot be a multiple of the coefficients in the first line, we can conclude that those planes do not coincide, and they are not parallel either.
That leaves that the solution must be a line, which means we have infinitely many solutions.

Last edited:
Klaas van Aarsen said:
From the second equation, we can write $x_2$ as a function of $x_3$, which will indeed have at least 1 solution due to the non-zero coefficient $3$.
After that we can always find $x_1$ from the first equation as a function of $x_2$ and $x_3$.
So we will always have infinitely many solutions, won't we?

Looking at it geometrically, we have the intersection of 2 planes.
Those are actual planes since the coefficients of each cannot all be zero, regardless of the value of $c$.
They can either coincide, or be parallel (no solutions), or intersect in a line.
Since the coefficients in the second line cannot be a multiple of the coefficients in the first line, we can conclude that those planes do not coincide, and they are not parallel either.
That leaves that the solution must be a line, which means we have infinitely many solutions.

I see! Thank you! (Handshake)

## 1. What does "1/0 solution" mean in mathematics?

The term "1/0 solution" refers to a mathematical expression that involves dividing a number by zero. In mathematics, division by zero is considered undefined and does not have a valid solution.

## 2. Is there a value of c that would make 1/0 a valid solution?

No, there is no value of c that would make 1/0 a valid solution. Division by zero is undefined in mathematics and cannot be assigned a value.

## 3. Why is 1/0 considered undefined in mathematics?

Division by zero is considered undefined in mathematics because it leads to contradictory or inconsistent results. For example, if we divide a number by a smaller and smaller number approaching zero, the result becomes larger and larger, making it impossible to assign a specific value to the expression.

## 4. Are there any real-world applications where 1/0 can be used as a solution?

No, there are no real-world applications where 1/0 can be used as a solution. In practical situations, division by zero is not allowed and is considered an error.

## 5. Can we manipulate the expression 1/0 to make it a valid solution?

No, we cannot manipulate the expression 1/0 to make it a valid solution. Division by zero is a fundamental mathematical concept and cannot be changed or manipulated to produce a valid solution.

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