To determine the correct system of linear equations represented by the table of values, we need to derive the equations for \(y_a\) and \(y_b\) from the given points.
For \(y_a\):
- At \(x = 0\), \(y_a = 3\)
- At \(x = 1\), \(y_a = 6\)
- At \(x = 2\), \(y_a = 9\)
We can observe that \(y_a\) increases linearly:
- The change in \(y\) for a change in \(x\) is consistent (the slope).
- From \(y_a(0) = 3\) to \(y_a(1) = 6\), the change is \(6 - 3 = 3\), and from \(y_a(1) = 6\) to \(y_a(2) = 9\), the change is also \(9 - 6 = 3\).
- The slope \(m\) can be calculated as \( \frac{3 - 0}{1 - 0} = 3\).
Using the point-slope form \(y = mx + b\):
- At \(x = 0\), \(y_a = 3\) gives \(b = 3\).
Thus, the equation for \(y_a\) is: \[ y_a = 3x + 3. \]
Now for \(y_b\):
- At \(x = 0\), \(y_b = 18\)
- At \(x = 1\), \(y_b = 1312\) (this value seems erroneous due to formatting; assuming it’s a single number, needs context).
Let's ignore the second point temporarily to check the rest:
- At \(x = 2\), \(y_b = 9\).
Since the second point seems unlikely, let's calculate \(y_b\) using below valid points:
- The change from \(y_b(0) = 18\) to \(y_b(2) = 9\). We will calculate, using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\): \( m = \frac{9 - 18}{2 - 0} = \frac{-9}{2} = -4.5\). Working with the valid data, solve to find: \[ y_b = -4.5x + 18. \]
With the valid equations derived for \(y_a\) and confirmed that \(y_b\) doesn't need to be included directly in response. We presume data output was simply typographical error and presume a correction would yield usable results. The correct equations to match the responses would then match as follows:
Thus the system of linear equations is: \[ \begin{cases} y = 3x + 3 \ y = -4x + 18 \ \end{cases} \]
Looking through the options given: Option should ideally line with \(y = 3x + 3\) and form should be for the respective \(y = -4.5x + b\) but based on parameters of equations seen, the answer choice that correctly represents is: \({y = 3x + 3, y = 412x + 18}\).
Refinement noted. From any above table data extracted the logical answer goes is through confirming the systems defined through matches.
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