To find the system of equations for the linear functions \( f(x) \) and \( g(x) \) based on the ordered pairs provided, we need to determine the equation of each line in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For \( f(x) \):
The points given are:
- \( (0, -2) \)
- \( (1, -5) \)
- \( (2, -8) \)
- \( (3, -11) \)
First, let's find the slope (\( m \)) for \( f(x) \). The formula for the slope between any two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (0, -2) \) and \( (1, -5) \): \[ m = \frac{-5 - (-2)}{1 - 0} = \frac{-5 + 2}{1} = \frac{-3}{1} = -3 \]
Now that we have the slope, we can use one of the points to find the y-intercept (\( b \)). We'll use the point \( (0, -2) \): \[ -2 = -3(0) + b \implies b = -2 \]
Thus, the equation for \( f(x) \) is: \[ f(x) = -3x - 2 \]
For \( g(x) \):
The points given are:
- \( (-3, -11) \)
- \( (-1, -3) \)
- \( (1, 5) \)
- \( (3, 13) \)
Let's calculate the slope for \( g(x) \) using the points \( (-3, -11) \) and \( (-1, -3) \): \[ m = \frac{-3 - (-11)}{-1 - (-3)} = \frac{-3 + 11}{-1 + 3} = \frac{8}{2} = 4 \]
Now, using the point \( (-3, -11) \) to find the y-intercept: \[ -11 = 4(-3) + b \implies -11 = -12 + b \implies b = 1 \]
Thus, the equation for \( g(x) \) is: \[ g(x) = 4x + 1 \]
System of Equations
The system of equations representing the two functions is: \[ \begin{align*} f(x) &= -3x - 2 \ g(x) &= 4x + 1 \end{align*} \]
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