Questions: x+5y=36 5x+3y=26

Transcript text: \[ \left\{\begin{array}{r} x+5 y=36 \\ 5 x+3 y=26 \end{array}\right. \]

Solution

Solution Steps

To solve the given system of linear equations, we can use matrix notation and apply the Gauss-Jordan elimination method. This involves representing the system as an augmented matrix, performing row operations to reduce it to row-echelon form, and then further reducing it to reduced row-echelon form. From the reduced matrix, we can determine if there is a unique solution, no solution, or infinitely many solutions.

Step 1: Set Up the System of Equations

The given system of equations is: \[ \begin{align_} x + 5y &= 36 \\ 5x + 3y &= 26 \end{align_} \]

Step 2: Represent the System as an Augmented Matrix

We represent the system of equations as an augmented matrix: \[ \begin{bmatrix} 1 & 5 & | & 36 \\ 5 & 3 & | & 26 \end{bmatrix} \]

Step 3: Apply Gauss-Jordan Elimination

Perform row operations to reduce the matrix to reduced row-echelon form: \[ \begin{bmatrix} 1 & 0 & | & 1 \\ 0 & 1 & | & 7 \end{bmatrix} \]

Step 4: Interpret the Reduced Matrix

The reduced matrix corresponds to the following system of equations: \[ \begin{align_} x &= 1 \\ y &= 7 \end{align_} \]

Final Answer

The solution to the system of equations is: \[ \boxed{x = 1, \, y = 7} \]

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