Question

By Gaues elimination method, solve \(x+y=2\) and \(2 x+3 y=5\).

Short answer

The solution is \(x = 1\) and \(y = 1\).

Step-by-step solution

Represent the System as a Matrix

First, we need to represent the given system of equations \(x + y = 2\) and \(2x + 3y = 5\) as an augmented matrix. The augmented matrix is: \[\begin{bmatrix}1 & 1 & | & 2 \2 & 3 & | & 5\end{bmatrix}\]

Transform to Upper Triangular Matrix

We need to make the element below the first pivot (1 in the first column) equal to zero. We do this by replacing the second row with the second row minus 2 times the first row: \[\begin{bmatrix}1 & 1 & | & 2 \0 & 1 & | & 1\end{bmatrix}\]

Solve for the Variables

Now, solve the equations starting from the bottom up. From the second equation, \(0 \cdot x + 1 \cdot y = 1\), we find \(y = 1\).Substitute \(y = 1\) into the first equation: \(x + 1 = 2\).Solve for \(x\):\[x = 2 - 1 = 1\]

Verify the Solution

Substitute \(x = 1\) and \(y = 1\) back into the original equations to ensure they are satisfied. For \(x + y = 2\):\[1 + 1 = 2\] For \(2x + 3y = 5\):\[2(1) + 3(1) = 2 + 3 = 5\]Both equations are satisfied, confirming our solution is correct.

Key concepts

Augmented Matrix

An augmented matrix is a convenient way to represent a system of linear equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. This method simplifies the process of performing matrix operations. In the given exercise, the equations \(x + y = 2\) and \(2x + 3y = 5\) are converted into the augmented matrix:\[\begin{bmatrix}1 & 1 & | & 2 \2 & 3 & | & 5\end{bmatrix}\]Here, the vertical bar separates the coefficients of the variables from the constant terms. The augmented matrix is useful for applying Gaussian elimination to systematically solve the system and find the values of the variables.

Upper Triangular Matrix

Transforming an augmented matrix into an upper triangular form is a key step in Gaussian elimination. An upper triangular matrix is a type of square matrix where all the entries below the main diagonal are zeros. This form aids in simplifying calculations to find the solutions to linear equations.To achieve this form in the step-by-step solution given, the second row was modified by subtracting twice the first row from it. This step transformed the matrix to:\[\begin{bmatrix}1 & 1 & | & 2 \0 & 1 & | & 1\end{bmatrix}\]In this form, the first row remains unchanged while the second row is adjusted to have zero below the leading coefficient of the first row. This modification helps in solving the equations easily by back substitution.

Linear Equations

Linear equations like \(x+y=2\) and \(2x+3y=5\) are fundamental in mathematics. They represent straight lines in a two-dimensional plane. Solving a system of linear equations means finding a common solution or point that satisfies all equations simultaneously.In the context of Gaussian elimination, these equations are expressed in matrix form, which acts as a shortcut to finding the solution. The ultimate goal is to discover values of the variables that make all the equations true. When represented in matrix format, these solutions are derived by performing operations to simplify the system, such as turning the matrix into triangular form.

Matrix Operations

Matrix operations are mathematical procedures used to manipulate matrices to solve linear equations. These operations include row switching, row multiplication, and row addition or subtraction. In Gaussian elimination, these operations are applied to convert an augmented matrix into an easier form for solving, such as the upper triangular matrix. For example, in the exercise, the operation of subtracting twice the first row from the second row was essential to simplify and solve the matrix. Understanding matrix operations is crucial as they provide a powerful and organized approach to solving complex systems of equations efficiently. By mastering these steps, learners can tackle larger systems with ease.