To verify if the ordered pair \((\frac{5}{3}, -\frac{2}{3})\) is a solution to the system of equations, we will substitute \(x = \frac{5}{3}\) and \(y = -\frac{2}{3}\) into both equations.
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Check the first equation: \[ y = \frac{1}{2}x - \frac{3}{2} \] Substituting the values: \[ -\frac{2}{3} = \frac{1}{2} \cdot \frac{5}{3} - \frac{3}{2} \]
Calculate the right side: \[ \frac{1}{2} \cdot \frac{5}{3} = \frac{5}{6} \]
Now, substituting: \[ -\frac{2}{3} = \frac{5}{6} - \frac{3}{2} \] To subtract, we need a common denominator. The common denominator between 6 and 2 is 6. Rewrite \(-\frac{3}{2}\) as \(-\frac{9}{6}\): \[ -\frac{2}{3} = \frac{5}{6} - \frac{9}{6} \] Simplifying the right side: \[ -\frac{2}{3} = \frac{5 - 9}{6} = -\frac{4}{6} = -\frac{2}{3} \]
The first equation holds true.
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Check the second equation: \[ y = -x + 1 \] Substituting the values: \[ -\frac{2}{3} = -\left(\frac{5}{3}\right) + 1 \]
Calculate the right side: \[ -\left(\frac{5}{3}\right) + 1 = -\frac{5}{3} + 1 \] Rewrite 1 as \(\frac{3}{3}\): \[ -\frac{5}{3} + \frac{3}{3} = -\frac{5 - 3}{3} = -\frac{2}{3} \]
The second equation holds true.
Since substituting the values into both equations yields true statements, the correct answer is:
Yes, because substituting these values into both equations forms two true statements.
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