Question

Line
j is shown on the coordinate grid. The table of ordered pairs represents some points on line
k.

x: -1,0,1,2
y: 8,5,2,-1
Which system of equations represents the line
j and line k?

Answer 1

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.