To determine if the two expressions \(-0.58x + 2.27\) and \(\frac{7.037 - 1.798x}{3.1}\) are equivalent, we can set them equal to each other and analyze the results.
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The first expression is: \[ E_1 = -0.58x + 2.27 \]
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The second expression is: \[ E_2 = \frac{7.037 - 1.798x}{3.1} \]
To simplify this, we can perform the division: \[ E_2 = \frac{7.037}{3.1} - \frac{1.798x}{3.1} \] which gives us: \[ E_2 \approx 2.27 - 0.58x \]
Now, substituting the constants: \[ \frac{7.037}{3.1} \approx 2.27 \quad \text{and} \quad \frac{1.798}{3.1} \approx 0.58 \]
Thus: \[ E_2 \approx 2.27 - 0.58x \]
Now, notice that both expressions simplify to the same form: \[ E_1 = -0.58x + 2.27 \quad \text{and} \quad E_2 \approx -0.58x + 2.27 \]
If we plug in \(x = 0\) into both:
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For \(E_1\): \[ E_1(0) = -0.58(0) + 2.27 = 2.27 \]
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For \(E_2\): \[ E_2(0) = \frac{7.037 - 1.798(0)}{3.1} = \frac{7.037}{3.1} \approx 2.27 \]
This validates that the expressions are equivalent when \(x = 0\).
Next, evaluating for \(x = 1\):
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For \(E_1\): \[ E_1(1) = -0.58(1) + 2.27 = -0.58 + 2.27 = 1.69 \]
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For \(E_2\): \[ E_2(1) = \frac{7.037 - 1.798(1)}{3.1} = \frac{7.037 - 1.798}{3.1} = \frac{5.239}{3.1} \approx 1.69 \]
Final Verification: Both expressions yield the same outcome of approximately \(1.69\) at \(x = 1\).
Conclusion:
The expressions \(-0.58x + 2.27\) and \(\frac{7.037 - 1.798x}{3.1}\) are equivalent for all values of \(x\). However, the other expression you compared, \(-0.58x + 2.27\) and \(\frac{7.037 - 2.798x}{3.1}\), likely is not equivalent as it alters the coefficient of \(x\), thus changing the relationship captured in the expressions.
So the correct statement would be:
- Equivalent Expressions: \(-0.58x + 2.27\) and \(\frac{7.037 - 1.798x}{3.1}\)
- Not Equivalent Expressions: \(-0.58x + 2.27\) and \(\frac{7.037 - 2.798x}{3.1}\)