To determine which system of equations corresponds to the values in the table, we can use the given pairs of \( x \) and \( y \) values to derive the equations.
From the given table:
For \( x = 0 \),
- \( y_a = 3 \)
- \( y_b = 18 \)
For \( x = 1 \),
- \( y_a = 6 \)
- \( y_b = 13.5 \)
For \( x = 2 \),
- \( y_a = 9 \)
- \( y_b = 9 \)
Finding the equations for \( y_a \)
- Using points \( (0, 3) \) and \( (1, 6) \) to find a pattern:
- Slope \( m = \frac{6 - 3}{1 - 0} = 3 \)
- Using point-slope form \( y - y_1 = m(x - x_1) \): \[ y - 3 = 3(x - 0) \implies y = 3x + 3 \]
Finding the equations for \( y_b \)
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Using points \( (0, 18) \) and \( (1, 13.5) \):
- Slope \( m = \frac{13.5 - 18}{1 - 0} = -4.5 \)
- Using point-slope form: \[ y - 18 = -4.5(x - 0) \implies y = -4.5x + 18 \]
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Using the points \( (1, 13.5) \) and \( (2, 9) \):
- Slope \( m = \frac{9 - 13.5}{2 - 1} = -4.5 \) (This confirms the slope).
- Again confirming from point \( (0, 18) \): \[ y - 18 = -4.5(x - 0) \implies y = -4.5x + 18 \]
None of the other options seem to fit when analyzing the \( y_b \). Given the multiple choices, we would need to see if the first found equation for \( y_a \) matches any of the given choices.
Filtering down the choices
The only linear equation derived for \( y_a \) creates the expression:
- \( y_a = 3x + 3 \)
Checking the available options, we see:
- {y=3x+3; y=4; 12x+18}
- This doesn't match our other, so let's check for the collective use.
In terms of the correct or closest equation:
- Given how the formula pairs appear and sample linear equations assess upward trend vs decline within typical data ranges.
From the analysis, the reasonable approach resembles that the pair of values must lead up, retaining consistent average specs of
I'm prompted to review which accurately represents the output structure:
The closest match appears to be:
- {y=3x+3; y=−4.5; 12x+18}
And from the derived characteristics, you can choose as follows based on calculations given:
Answer
Thus, the best fitting description based on observations should guide us as:
{y = 3x + 3; y = 4; 12x + 18}