Question

the tables of ordered pairs represent some points on the graphs of linear functions f and g.
first graph: x: 0, 1, 2, 3. f(x): -2, -5, -8, -11. Second graph: x: -3, -1, 1, 3. g(x): -11, -3, 5, 13.
what is the system of equations?

Answer 1

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:
\[
\begin{align*}
f(x) &= -3x - 2 \\
g(x) &= 4x + 1
\end{align*}
\]