# Number of solutions for system of equations

• MHB
• bargaj
In summary, the system of equations has either 0 or 1 solution for different values of c, except for $$c = -3, 1, 2$$ where it has infinitely many solutions. For c such that the determinant of coefficients is not 0, there will be exactly 1 solution, and for c such that the determinant of coefficients is 0 but the right hand side is not, there will be no solution.
bargaj
Hello!

I have a simple question about solutions, better said number of solutions for this system of equations.

$\begin{cases} x_{1 } − x_{2 } + 3x_{3 } − 2x_{4 } = 1\\ −2x_{1 } + 2cx_{2 } − 4x_{3 } + 2x_{4 } = −7\\ − 2x_{3 } + (−c + 6)x_{4 } = 2c + 15\\ − 2x_{3 } + c^{2 }x_{4 } = c^{2 }\end{cases}$

I know it's only possible that this system has either 0, 1 or $$\infty$$ number of solutions, for different values of c:

$c = -3 \rightarrow \infty\\ c = 1 \rightarrow \infty\\ c = 2 \rightarrow 0 \\ c \in ℝ \setminus \{-3, 1, 2\} \rightarrow 1$

My question is: for which c has this system at the utmost 2 solutions? Should it be only for when the whole system has only one solution or also when it has none? Thank you for your help!

YOU just said that such a system has either 0 or 1 or infinitely many solutions. So you know it is impossible to have 2 solutions. "At the utmost 2 solutions" must mean no solutions or 1 solution. There will be exactly one solution if c is such that the determinant of coefficients is NOT 0. There will be no solution if c is such that the determinant of coefficients is 0 but the right hand side is not.

## What is a system of equations?

A system of equations is a set of two or more equations that contain two or more variables. The solutions to the system are the values of the variables that make all of the equations true.

## How many solutions can a system of equations have?

A system of equations can have zero, one, or infinitely many solutions. The number of solutions depends on the number of equations and the relationships between them.

## What does it mean when a system of equations has no solutions?

If a system of equations has no solutions, it means that there is no combination of values for the variables that can make all of the equations true at the same time. This could happen if the equations are contradictory or if they represent parallel lines.

## What does it mean when a system of equations has one solution?

If a system of equations has one solution, it means that there is exactly one combination of values for the variables that makes all of the equations true at the same time. This is the point where all of the equations intersect on a graph.

## What does it mean when a system of equations has infinitely many solutions?

If a system of equations has infinitely many solutions, it means that there are an infinite number of combinations of values for the variables that make all of the equations true at the same time. This could happen if the equations are equivalent or if they represent the same line.

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