To determine if an equation represents a linear function, we look for two main criteria:
- It can be written in the form \(y = mx + b\), where \(m\) and \(b\) are constants, and \(x\) and \(y\) are variables.
- For each value of \(x\), there is exactly one value of \(y\).
Now, let's analyze the statements you've provided:
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"No, it is NOT a linear function because for each x-value there is exactly one y-value."
- This statement is contradictory. If there is exactly one \(y\)-value for each \(x\)-value, it typically supports that the equation might be linear.
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"Yes, it is a linear function because it will graph as a straight line."
- This statement supports that the equation is linear. If it graphs as a straight line, then it is indeed a linear function.
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"No, it is NOT a linear function because the variable is in the denominator. will be multiplied by."
- This can indicate a non-linear function. For instance, if you have a function of the form \(y = \frac{1}{x}\) or \(y = x^2\), they are not linear due to the variable being in the denominator or raised to a power.
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"Yes, it is a linear function because for each x-value there is exactly one y-value."
- This is another accurate explanation in favor of a linear function; it implies a consistent relationship.
Based on these analyses, the key indicators of whether or not an equation represents a linear function primarily focus on its form and the one-to-one mapping between x-values and y-values. If the equation contains a variable in the denominator or involves multiplication that causes it to no longer maintain a straight-line relationship, it would not represent a linear function.
Thus, the correct responses would depend on the nature of the equation in question. If the equation takes the form \(y = mx + b\) or a similar form ensuring one output for each input, it is a linear function. If it includes variables in non-linear ways (like in the denominator), it would not be a linear function.
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