Question

a coordinate plane with x - axis from -12 to 12, y - axis from -12 to 12, a grid, and a blue line passing through the origin - related area (visually, intersecting y - axis at -6 and extending with arrows).

Explanation:

To determine the equation of the line, we can use the slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

Step 1: Find the y - intercept (\(b\))

The line crosses the y - axis at the point \((0,-6)\). In the slope - intercept form \(y=mx + b\), when \(x = 0\), \(y=b\). So, \(b=-6\).

Step 2: Calculate the slope (\(m\))

We can use two points on the line to find the slope. Let's take two points, for example, \((0,-6)\) and \((6,-5)\) (we can also use other pairs of points on the line).
The formula for the slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
Substitute \(x_1 = 0,y_1=-6,x_2 = 6,y_2=-5\) into the slope formula:
\(m=\frac{-5-(-6)}{6 - 0}=\frac{-5 + 6}{6}=\frac{1}{6}\)

Step 3: Write the equation of the line

Now that we know \(m=\frac{1}{6}\) and \(b=-6\), we substitute these values into the slope - intercept form \(y=mx + b\)
We get \(y=\frac{1}{6}x-6\)

If we want to write it in the standard form \(Ax+By = C\) (where \(A\), \(B\), and \(C\) are integers and \(A\geq0\)):
Starting from \(y=\frac{1}{6}x-6\)
Multiply through by 6 to get rid of the fraction: \(6y=x - 36\)
Rearrange to get \(x-6y=36\)

If the question was to find the equation of the line, the answer in slope - intercept form is \(y=\frac{1}{6}x - 6\) and in standard form is \(x-6y = 36\)

(If the original question was different, for example, to find the slope or the y - intercept, we can adjust the answer accordingly. Since the original problem was not fully stated, but based on the graph, a common question is to find the equation of the line)

Answer:

The equation of the line in slope - intercept form is \(\boldsymbol{y=\frac{1}{6}x-6}\) (or in standard form \(\boldsymbol{x - 6y=36}\))