Questions: 2x + 5y = 1 -4x - 10y = -2

2x + 5y = 1 -4x - 10y = -2
Transcript text: \[ \left\{\begin{array}{c} 2 x+5 y=1 \\ -4 x-10 y=-2 \end{array}\right\} \]
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Solution

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Solution Steps

Step 1: Analyze the System of Equations

The given system of equations is: \[ \begin{cases} 2x + 5y = 1 \quad \text{(Equation 1)} \\ -4x - 10y = -2 \quad \text{(Equation 2)} \end{cases} \]

Step 2: Check for Proportionality

Notice that Equation 2 is a multiple of Equation 1. Specifically, Equation 2 is \(-2\) times Equation 1: \[ -2(2x + 5y) = -2(1) \implies -4x - 10y = -2 \] This means the two equations represent the same line.

Step 3: Determine the Nature of the Solution

Since both equations are proportional, the system has infinitely many solutions. The solutions lie on the line represented by \(2x + 5y = 1\).

Final Answer

The system has infinitely many solutions. The solutions lie on the line represented by \(2x + 5y = 1\).

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