Questions: How many solutions does this system - have? - -x+3 y=3 - 2 x-6 y=30 None One Infinite

How many solutions does this system - have? - -x+3 y=3 - 2 x-6 y=30 None One Infinite
Transcript text: How many solutions does this system \[ \left\{\begin{array}{c} \text { have? } \\ -x+3 y=3 \\ 2 x-6 y=30 \end{array}\right. \] None One Infinite
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Solution

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Solution Steps

Step 1: Rewrite the system of equations

The given system of equations is: \[ \left\{\begin{array}{c} -x + 3y = 3 \\ 2x - 6y = 30 \end{array}\right. \]

Step 2: Simplify the second equation

Divide the second equation by 2 to simplify: \[ x - 3y = 15 \]

Step 3: Compare the two equations

The first equation is: \[ -x + 3y = 3 \] The simplified second equation is: \[ x - 3y = 15 \] Notice that the left-hand side of the second equation is the negative of the left-hand side of the first equation. This implies: \[ -x + 3y = 3 \quad \text{and} \quad -(-x + 3y) = 15 \] \[ -x + 3y = 3 \quad \text{and} \quad x - 3y = 15 \] These two equations are inconsistent because they cannot be true simultaneously. Therefore, the system has no solution.

Final Answer

\(\boxed{\text{No solutions}}\)

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