Questions: Solve the following system of linear equations by substitution and determine whether the system is consistent, inconsistent, or dependent. If the system is consistent, find the solution. y = -3x + 1 2y = -6x + 2

Solve the following system of linear equations by substitution and determine whether the system is consistent, inconsistent, or dependent. If the system is consistent, find the solution. y = -3x + 1 2y = -6x + 2
Transcript text: Solve the following system of linear equations by substitution and determine whether the system is consistent, inconsistent, or dependent. If the system is consistent, find the solution. \[ \left\{\begin{array}{l} y=-3 x+1 \\ 2 y=-6 x+2 \end{array}\right. \]
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Solution

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Solution Steps

Step 1: Formulate the System of Equations

The given system of linear equations is: \[ \begin{cases} y = -3x + 1 \\ 2y = -6x + 2 \end{cases} \]

Step 2: Rewrite the Second Equation

The second equation can be simplified by dividing all terms by 2: \[ y = -3x + 1 \] This shows that both equations are equivalent.

Step 3: Analyze the System

Since both equations represent the same line, the system is dependent. This means there are infinitely many solutions along the line defined by \(y = -3x + 1\).

Step 4: Find a Specific Solution

To find a specific solution, we can set \(x = 0\): \[ y = -3(0) + 1 = 1 \] Thus, one specific solution is \((x, y) = (0, 1)\).

Final Answer

The system is dependent, and one specific solution is \\(\boxed{(x = 0, y = 1)}\\).

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