To determine which system of linear equations the table of values represents, we can analyze the data in the table. The table shows two sets of values for \( y_a \) and \( y_b \) corresponding to values of \( x \).
To find the equations for both \( y_a \) and \( y_b \), we can identify the pattern in both columns.
For \( y_a \):
- When \( x = 0 \), \( y_a = 3 \)
- When \( x = 1 \), \( y_a = 6 \)
- When \( x = 2 \), \( y_a = 9 \)
It looks like \( y_a \) is increasing by 3 for each increase in \( x \). This is a linear relationship, and we can describe it with the equation: \[ y_a = 3x + 3 \]
To confirm:
- For \( x = 0 \): \( y_a = 3(0) + 3 = 3 \) ✅
- For \( x = 1 \): \( y_a = 3(1) + 3 = 6 \) ✅
- For \( x = 2 \): \( y_a = 3(2) + 3 = 9 \) ✅
For \( y_b \):
- When \( x = 0 \), \( y_b = 18 \)
- When \( x = 1 \), \( y_b = 13 \)
- When \( x = 2 \), \( y_b = 9 \)
Here, \( y_b \) is decreasing as \( x \) increases. The changes in \( y_b \) are:
- From \( 18 \) to \( 13 \): decrease of 5
- From \( 13 \) to \( 9 \): decrease of 4
To find a linear equation, we can use the first and last points: Using points \( (0, 18) \) and \( (2, 9) \), the slope \( m \) can be calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 18}{2 - 0} = \frac{-9}{2} = -\frac{9}{2} \]
Thus, the equation of the line can be expressed in point-slope form: Using point \( (0, 18) \): \[ y_b - 18 = -\frac{9}{2}(x - 0) \Rightarrow y_b = -\frac{9}{2}x + 18 \]
But we need an integer slope. Therefore, for simplicity: \[ y_b = -4.5x + 18 \]
Depending on how the equations are simplified in the possible responses, we can express it, using integer coefficients as: \[ y_b = -\frac{9}{2}x + 18 \text{ or } 9x + y_b = 36 \]
Now let's match the original options:
- Option \( {y = 3x + 3, y = 4x + 12} \) (not correct)
- Option \( {y = 3x + 3, y = 4x - 18} \) (not correct)
- Option \( {y = -3x + 3, y = 4x + 12} \) (not correct)
None of the options provided match our equations directly, indicating there may have been a mistake in obtaining the answers from the choices provided.
If the correct equations don't perfectly match any, the answer might be: { y=3x+3 , y=4x-12 } if it were matched correctly based on original numerical transformations and minimizing errors.
If necessary, please clarify table values if they differ or re-verify data points.
2 answers