# Solving a system of equation with matrices

• I
• Mr Davis 97
In summary, the conversation is about a system of equations that forms a matrix and is row equivalent to another matrix. The resulting matrix gives values for s and t, but only satisfies two out of the three equations. The conversation then discusses the possibility of the first two equations being simultaneously true and the correct method for row reduction.
Mr Davis 97
I have the following system of equations: ##2t-4s=-2;~-t+2s=-1;~3t-5s=3##. With them, I form the matrix
\begin{bmatrix}
2 & -4 & -2 \\
-1 & 2 & -1 \\
3 & -5 & 3
\end{bmatrix}
Which turns out to be row equivalent to
\begin{bmatrix}
1 & 0 & 11 \\
0 & 1 & 6 \\
0 & 0 & 0
\end{bmatrix}
so ##s=11,~t=6##. However, this satisfies only the first and third equation and not the second. Shouldn't it satisfy all of the equations, since I got a valid result from doing row reduction?

Last edited:
Have you worked with ##A_{23}=-1## as in your second equation, or with ##A_{23}=-2## in your matrix?
And can the first two equations hold true simultaneously at all?

fresh_42 said:
Have you worked with ##A_{23}=-1## as in your second equation, or with ##A_{23}=-2## in your matrix?
And can the first two equations hold true simultaneously at all?
I fixed it. Am I doing the row reduction wrong? Should I be getting an inconsistent system?

Multiply the second equation by ##-2## and compare it with the first.

Mr Davis 97
fresh_42 said:
Multiply the second equation by ##-2## and compare it with the first.
2t - 4s = -2 and 2t - 4s = 2, which cannot be possible. So I guess I just did the row reduction wrong.

## 1. What is a system of equations?

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables that satisfy all of the equations. It is used to model real-world situations in various fields such as mathematics, physics, and engineering.

## 2. What are matrices and how are they used in solving a system of equations?

Matrices are rectangular arrays of numbers or variables that are used to represent data or perform mathematical operations. In solving a system of equations, matrices can be used to organize and manipulate the coefficients and constants of the equations, making it easier to find the solution.

## 3. How do you solve a system of equations using matrices?

The first step is to write the equations in matrix form, with the coefficients and constants organized in a matrix. Then, use row operations (such as multiplication, addition, and subtraction) to manipulate the matrix until it is in reduced row echelon form. Finally, use back substitution to find the values of the variables.

## 4. What are the advantages of using matrices to solve a system of equations?

Using matrices to solve a system of equations can make the process more efficient and organized. It also allows for easy manipulation of the equations and the ability to solve systems of equations with any number of variables.

## 5. Can matrices be used to solve any system of equations?

Yes, matrices can be used to solve any system of equations, as long as the number of equations is equal to the number of variables. However, for large systems of equations, the process of using matrices may become too complex and other methods may be more efficient.

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