To determine if the relationship \( y = \sqrt{x - 5} \) is linear, we look at the properties of linear equations. A linear relationship can be described by the equation:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
For the given equation \( y = \sqrt{x - 5} \):
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Identifying the Form: The equation involves a square root, which is not a linear term. Linear equations only involve first-degree terms (the highest power of \( x \) is 1).
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Graphing: If you graph \( y = \sqrt{x - 5} \), you will find that it is a curve that starts at the point \( (5, 0) \) and increases but does so in a non-linear manner.
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Behavior: Linear equations produce straight lines, whereas the square root function produces a curve. The slope of a linear function remains constant, while the slope of a square root function changes depending on the value of \( x \).
Therefore, the relationship \( y = \sqrt{x - 5} \) is not a linear relationship.