To find the equations representing line \( j \) and line \( k \), we first need to derive the equation for line \( k \) using the ordered pairs provided in the table:
- Points on line \( k \):
- \((-1, 8)\)
- \((0, 5)\)
- \((1, 2)\)
- \((2, -1)\)
Step 1: Determine the slope of line \( k \)
The slope (\( m \)) can be found using any two points on line \( k \). Let's take the first two points: \((-1, 8)\) and \((0, 5)\).
The formula for slope is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points:
\[ m = \frac{5 - 8}{0 - (-1)} = \frac{-3}{1} = -3 \]
Step 2: Write the point-slope form of the equation
Now, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using the point \((0, 5)\):
\[ y - 5 = -3(x - 0) \]
This simplifies to:
\[ y - 5 = -3x \ y = -3x + 5 \]
So, the equation of line \( k \) is:
\[ y = -3x + 5 \]
Step 3: Determine the equation of line \( j \)
Since we do not have specific information about line \( j \), we need either coordinates or defined characteristics of this line to formulate its equation. Therefore, to complete the answer we could represent line \( j \) with a general form, such as:
\[ y = mx + b \]
Where \( m \) represents the slope of line \( j \) and \( b \) the y-intercept.
Final System of Equations
Assuming line \( j \)'s equation is represented by \( y = mx + b \) for some slope \( m \) and y-intercept \( b \), the system of equations representing both lines would be:
\[ y = -3x + 5 \quad (\text{for line } k) \ y = mx + b \quad (\text{for line } j) \]
If you later obtain specific information about line \( j \), you can amend it accordingly. Without that, it remains a variable representation.