To find the system of linear equations that the table of values represents, we need to analyze the values in the table. Here are the pairs of \( (x, ya) \) and \( (x, yb) \):
- For \( ya \):
- At \( x = 0 \), \( ya = 0 \)
- At \( x = 1 \), \( ya = 2 \)
- At \( x = 2 \), \( ya = 4 \)
From this, we can see that \( ya \) appears to be a linear function of \( x \). To find its equation:
- The slope \( m \) can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{1 - 0} = 2 \] This means \( ya = 2x \) (intercept is 0).
- For \( yb \):
- At \( x = 0 \), \( yb = 3 \)
- At \( x = 1 \), \( yb = 3.5 \)
- At \( x = 2 \), \( yb = 4 \)
From these values, we can see that \( yb \) is also a linear function:
- The slope \( m \) can be calculated similarly: \[ m = \frac{3.5 - 3}{1 - 0} = 0.5 \] Thus, the equation for \( yb \) can be calculated as:
- Using the point (0, 3), we find \( yb = 0.5x + 3 \).
Now we have:
- \( ya = 2x \)
- \( yb = 0.5x + 3 \)
To express this in the form of a system of equations for the provided options, we match equations accordingly.
Checking the options given, since:
- \( ya = 2x \)
- \( yb \) must be \( y = 0.5x + 3 \).
None of the provided options exactly match \( {y=2x, y=0.5x+3} \).
However, assuming these equations are relabeled as:
- \( y = 2x \) is consistent with \( ya \)
- \( y = 0.5x + 3 \) is consistent with the form \( yb \)
The correct system based on the given options is not explicitly listed among the choices. However, we can interpret that the correct equations derived from the table do not match any of the provided selections.
If I had to select based upon closest alignment with derived equations, I would highlight the relationships provided in the left clauses of the options, especially considering \( y=2x \).
In conclusion, none of the choices directly correlate with both derived equations accurately. Please recheck the options against values derived or if there were potentially miscommunication in equation establishment.