To determine which system of linear equations corresponds to the given table of values, we will use the points provided to find models \( y_a \) and \( y_b \).
Given the values:
- When \( x = 0 \), \( y_a = 3 \) and \( y_b = 18 \)
- When \( x = 1 \), \( y_a = 6 \) and \( y_b = 1312 \)
- When \( x = 2 \), \( y_a = 9 \) and \( y_b = 9 \)
Analyzing the first function \( y_a \):
Using the points provided for \( y_a \):
- (0, 3)
- (1, 6)
- (2, 9)
Let's check the change in \( y_a \) relative to \( x \):
- From \( x = 0 \) to \( x = 1 \): \( 6 - 3 = 3 \) (change of 3)
- From \( x = 1 \) to \( x = 2 \): \( 9 - 6 = 3 \) (change of 3)
This shows a consistent difference, indicating a linear function. The slope (m) can be determined. The slope is calculated as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 3}{1 - 0} = 3 \]
We can use the slope-intercept form \( y = mx + b \):
Starting with the equation: \[ y_a = 3x + b \]
Using the point (0,3) to find \( b \): \[ 3 = 3(0) + b \implies b = 3 \]
Thus, the equation for \( y_a \) is: \[ y_a = 3x + 3 \]
Analyzing the second function \( y_b \):
Using the points provided for \( y_b \):
- (0, 18)
- (1, 1312)
- (2, 9)
Now we can see that the values for \( y_b \) appear to change drastically, particularly with the large jump to \( 1312 \). Hence we need to find if any linear pattern holds.
Since the values are too erratic, they don't suggest a clear linear equation from the ones given. However we analyze it next.
Looking for a possible equation, \( y_b \) decreases significantly moving from \( x = 1 \) to \( x = 2 \), which makes us believe it is likely linear but complicated.
Making an equation
We can try finding the equation through slope: Let us use two points: (0, 18) and (2, 9):
\[ m_{yb} = \frac{9 - 18}{2 - 0} = \frac{-9}{2} = -4.5 \]
Taking the same form: \[ y = -4.5x + b \]
Using (0,18): \[ 18 = -4.5(0) + b \implies b = 18 \] So this gives us: \[ y_b = -4.5x + 18 \]
Now, checking linearity from observed behavior might help, but given the complex nature from earlier, it doesn't fall simply under \( y = mx + b \).
Conclusion
We've established:
- \( y_a = 3x + 3 \)
- The highly variable behavior in \( y_b \) cannot easily match any standard simple equation without more context.
Choosing an Option
Looking for matches: None of the equations seem to precisely represent the lighter visible observation - but summarizing the closest:
- A linear progression in itself suggests \( y = 3x + 3, y = -4x \))
So evaluating the possible equations from the question leads to:
From the closest selections available as a conjecture looks like:
- No direct fit - but if claimed against the closest options may suggest \( y = 3x + 3, y = -4x + 18 \) under the given tabulated and from differences seen.
If forced to pick from simplistic linear indicating possible:
\({y = 3x + 3, y = -4x + 18}\) but based on both functions direct observations don’t fit. Evaluate against each response could check graphs or performance.
Evaluate outputs against response recalls closely matching to one of the delimited options. Thus could render:
Choosing: \({y=3x+3, y=-412x+18}\) from recalls tracking with the closest match from possible observations via slope policies against observed outcomes.
Final conclusion compacts to checking \({y=3x+3, y=−412x+18} \).
3 answers