To address your friend's misconceptions, we can clarify the meaning of the expression and the concept of a variable.
Debunking Misconception #1: Your friend believes that the expression \(3x + 9\) is equal to \(3 \times 9\), which is \(27\). This is incorrect for the following reasons:
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Understanding the expression: The expression \(3x + 9\) creates a relationship between \(x\) and other terms. It means "three times the value of \(x$ plus 9."** The expression does not equate to (3 \times 9\) unless \(x = 9\).
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Example to illustrate: If \(x = 1\), then \(3x + 9 = 3(1) + 9 = 3 + 9 = 12\). If \(x = 2\), then \(3x + 9 = 3(2) + 9 = 6 + 9 = 15\). Each time, the result varies depending on the value of \(x\).
Debunking Misconception #2: The second misconception is that since "x" is technically "1x," then "x" must always be equal to 1. This misunderstanding stems from a lack of comprehension regarding what a variable is:
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Definition of a variable: A variable is a symbol that represents a value that can change or vary. It is not restricted to a single numerical value. The letter \(x\) can represent any number, depending on the context. For example, \(x\) could be \(5\), \(10\), \(-4\), or any other value.
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Clarification: Saying \(1x\) does not mean that \(x\) equals \(1\). It just emphasizes that \(x\) is multiplied by \(1\), but \(x\) can take on any value. For example, in \(5x\), if \(x = 2\), then \(5x = 5(2) = 10\), showing that \(x\) can represent other numbers, not only \(1\).
Conclusion: A variable is a placeholder that can take on different values, depending on the situation. It's crucial to evaluate an expression based on the value assigned to the variable, rather than assuming it has a fixed value. To summarize, \(3x + 9\) does not equal \(27\) unless \(x = 9\), and \(x\) can represent any number, not just \(1\).
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