Question

Rewrite the system of equations in matrix form. Find the solution to the linear system by simultaneously manipulating the equations and the matrix. $$ \begin{array}{l} -2 x+3 y=2 \\ -x+y=1 \end{array} $$

Short answer

The solution to the system is \(x = -1\) and \(y = 0\).

Step-by-step solution

Write System of Equations in Standard Form

First, ensure the equations are written in a standard linear form: \(-2x + 3y = 2\) and \(-x + y = 1\).

Express in Matrix Form

The linear system can be rewritten in matrix form as follows: \[\begin{bmatrix} -2 & 3 \-1 & 1 \end{bmatrix} \begin{bmatrix} x \y \end{bmatrix} = \begin{bmatrix} 2 \1 \end{bmatrix}\]

Row Reduce the Augmented Matrix

Combine the coefficient matrix and the constants column to form an augmented matrix: \[\begin{bmatrix} -2 & 3 & | & 2 \-1 & 1 & | & 1 \end{bmatrix}\]Perform row operations to simplify this matrix. Multiply the second row by 2 and add to the first row to get:\[\begin{bmatrix} 0 & 1 & | & 0 \-1 & 1 & | & 1 \end{bmatrix}\]Swap the rows to have:\[\begin{bmatrix} 1 & -1 & | & -1 \0 & 1 & | & 0 \end{bmatrix}\]

Solve for Variables

From the reduced matrix, directly read off the equations: \(x - y = -1\) and \(y = 0\). Substituting \(y = 0\) into the first equation gives \(x = -1\).

Verify the Solution

Substitute \(x = -1\) and \(y = 0\) back into the original equations to verify the solution:\(-2(-1) + 3(0) = 2\) and \(-(-1) + 0 = 1\). Both equations are satisfied with this solution.

Key concepts

Row Reduction

Row reduction is a systematic method for solving systems of linear equations by transforming the augmented matrix to a simpler form. This form is usually either row-echelon or reduced row-echelon form.
The goal is to make the matrix easy to work with, essentially resembling upper triangular form for quick equation derivation.
During row reduction, we use three types of row operations:

• Swapping two rows, used to position a row of lower rank on top.
• Multiplying a row by a non-zero scalar, useful for creating leading ones in the diagonal.
• Adding or subtracting the multiple of one row to another row, often employed for introducing zeros below and above the leading ones.

These operations are repeated in a systematic way to lead the matrix to its desired reduced row-echelon form, making it easier to solve for variables step-by-step.

Augmented Matrix

An augmented matrix combines both the coefficients of a system of linear equations and the constant terms into a single matrix. This format allows for efficient use of matrix operations to solve systems of equations.
For example, a system:\[-2x + 3y = 2\] \[-x + y = 1\]is represented in matrix form as:\[\begin{bmatrix} -2 & 3 & | & 2 \-1 & 1 & | & 1 \end{bmatrix}\]Here, the vertical line separates the coefficient matrix on the left from the constant column on the right.
Using augmented matrices streamlines computations and is particularly favorable in terms of applying row operations that keep track of changes to both variables and constants simultaneously.
This approach helps in efficiently implementing methods like Gaussian elimination for solving systems of equations.

Solution Verification

Solution verification is a crucial step in solving linear equations, ensuring the solution obtained is indeed correct.
After deriving values for the variables, we substitute them back into the original equations to check for consistency.
Let's use the solution from the example: \(x = -1\) and \(y = 0\).Perform the check:

• Substituting into \(-2x + 3y = 2\): \(-2(-1) + 3(0) = 2\), which simplifies to \(2 = 2\), a true statement.
• For the second equation \(-x + y = 1\), substitute: \(-(-1) + 0 = 1\), which simplifies to \(1 = 1\), also true.

Both equations hold true, confirming that the solution is correct. This process minimizes errors and solidifies confidence in the calculated solution.