Questions: Homework Use the echelon method to solve the following system of 6 x-5 y=-7 -12 x+10 y=14

Homework Use the echelon method to solve the following system of 6 x-5 y=-7 -12 x+10 y=14
Transcript text: Homework Use the echelon method to solve the following system of \[ \begin{array}{r} 6 x-5 y=-7 \\ -12 x+10 y=14 \end{array} \]
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Solution

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Solution Steps

To solve the system of equations using the echelon method, we first convert the system into an augmented matrix. Then, we perform row operations to transform the matrix into row-echelon form. Finally, we use back substitution to find the values of \(x\) and \(y\).

Step 1: Set Up the Augmented Matrix

Convert the system of equations into an augmented matrix: \[ \begin{bmatrix} 6 & -5 & \vert & -7 \\ -12 & 10 & \vert & 14 \end{bmatrix} \]

Step 2: Perform Row Operations

To achieve row-echelon form, perform the following operation:

  • Multiply the first row by 2 and add it to the second row: \[ \begin{bmatrix} 6 & -5 & \vert & -7 \\ 0 & 0 & \vert & 0 \end{bmatrix} \]
Step 3: Analyze the Row-Echelon Form

The second row is all zeros, indicating that the system is dependent and has infinitely many solutions. The first row gives: \[ 6x - 5y = -7 \]

Step 4: Express \(y\) in Terms of \(x\)

Solve the equation from the first row for \(y\): \[ 6x - 5y = -7 \implies y = \frac{6}{5}x + \frac{7}{5} \]

Final Answer

The system has infinitely many solutions, expressed as: \[ \boxed{y = \frac{6}{5}x + \frac{7}{5}} \]

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