To determine which of the given statements are identities, we need to verify whether each equation is true for all values of the involved variables.
Let’s analyze each statement:
Statement I: [tex]\( y = x \)[/tex]
- Analysis: This equation states that [tex]\( y \)[/tex] is equal to [tex]\( x \)[/tex]. For this to be an identity, it must hold true for any value of [tex]\( x \)[/tex].
- Conclusion: This is not an identity because it only holds true when [tex]\( y \)[/tex] and [tex]\( x \)[/tex] are specifically equal. Hence, it is not true for all possible values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Statement II: [tex]\( 4 = 3x^2 + 2 \)[/tex]
- Analysis: This equation can be rewritten as [tex]\( 3x^2 + 2 = 4 \)[/tex]. Subtracting 2 from both sides, we get [tex]\( 3x^2 = 2 \)[/tex]. Dividing both sides by 3, we find [tex]\( x^2 = \frac{2}{3} \)[/tex]. Solving for [tex]\( x \)[/tex], we get [tex]\( x = \pm \sqrt{\frac{2}{3}} \)[/tex].
- Conclusion: This equation holds true only for two specific values of [tex]\( x \)[/tex], namely [tex]\( x = \sqrt{\frac{2}{3}} \)[/tex] and [tex]\( x = -\sqrt{\frac{2}{3}} \)[/tex]. Thus, it is not an identity because it is not valid for all values of [tex]\( x \)[/tex].
Statement III: [tex]\( x = \sqrt{x} \)[/tex]
- Analysis: To find the solutions of this equation, we can square both sides to obtain [tex]\( x^2 = x \)[/tex]. Rearranging gives us [tex]\( x^2 - x = 0 \)[/tex]. Factoring out [tex]\( x \)[/tex], we get [tex]\( x(x - 1) = 0 \)[/tex]. So, [tex]\( x = 0 \)[/tex] or [tex]\( x = 1 \)[/tex].
- Conclusion: This equation holds true only for [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]. Therefore, it is not an identity because it is not valid for all values of [tex]\( x \)[/tex].
Based on the detailed analysis, none of the given statements are identities as they do not hold true for all values of the variables involved.
Correct answer: C. none
