To determine whether the graphs of the given pairs of equations are parallel, perpendicular, or neither, we first need to rewrite each equation in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope.
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The first equation is \(y = 2x + 12\) (this is already in slope-intercept form).
- Slope (\(m_1\)) = 2
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The second equation is \(-y = 3\). We can solve for \(y\): \[ y = -3 \]
- This is a horizontal line, and the slope (\(m_2\)) of a horizontal line is 0.
Now we can compare the slopes:
- The slopes are \(m_1 = 2\) and \(m_2 = 0\).
Parallel Lines: The slopes of the two lines would need to be equal. Since \(2 \neq 0\), they are not parallel.
Perpendicular Lines: The slopes of the two lines must satisfy the condition \(m_1 \cdot m_2 = -1\). In this case: \[ 2 \cdot 0 = 0 \neq -1 \] Thus, the lines are not perpendicular either.
Since neither condition for parallelism nor perpendicularity is satisfied, we conclude that the lines are:
Neither.