To determine which system of linear equations corresponds to the table of values, we can analyze the relationship between \(x\) and \(y_a\) and between \(x\) and \(y_b\).
Analyzing \(y_a\):
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Given the points:
- \( (0, 3) \)
- \( (1, 6) \)
- \( (2, 9) \)
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We can compute the slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
- Using points \( (0, 3) \) and \( (1, 6) \): \[ m = \frac{6 - 3}{1 - 0} = \frac{3}{1} = 3 \]
- Using points \( (1, 6) \) and \( (2, 9) \): \[ m = \frac{9 - 6}{2 - 1} = \frac{3}{1} = 3 \]
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The linear equation can be formulated as: \[ y_a = 3x + b \] To find \(b\), we can use one of the points, say \( (0, 3) \): \[ 3 = 3(0) + b \implies b = 3 \] The equation is: \[ y_a = 3x + 3 \]
Analyzing \(y_b\):
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Given the points:
- \( (0, 18) \)
- \( (1, 1312) \)
- \( (2, 9) \)
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Calculate the slope (only using points \( (1, 1312) \) and \( (2, 9) \)): \[ m = \frac{9 - 1312}{2 - 1} = 9 - 1312 = -1303 \]
The difference between consecutive \(x\) values indicates that this relationship isn't linear between all points. Instead, it would not fit within the typical case of a linear equation format. The left logic fits a point but cannot finalize due to missing equations.
Now we can look into the possible systems provided:
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The only system that matches with our \(y_a\) equation is: \[ {y = 3x + 3} \]
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The terms of \(y_b\) remain ambiguous as it should correlate in the equations proposed.
Conclusion:
The options provided in the responses indicate the foundational structure for \(y_a\) would yield:
- \(y = 3x + 3\) in following all those options.
So we can conclude: Final Answers
- \( { y = 3x + 3, y = 4 } \) follows as consistent points set with calculation of slope.
Thus the relevant system of linear equations represented by the table of values is: {y=3x+3, y=4}