Question

what is the slope of this line? (with a coordinate grid and a line plotted)

Explanation:

Step1: Identify two points on the line

We can see that the line passes through the origin \((0,0)\) and another point, for example, when \(x = 3\), let's find the corresponding \(y\)-value. From the graph, when \(x = 3\), \(y=-18\)? Wait, no, let's check the grid. Wait, the line crosses the \(y\)-axis at \((0, - 6)\)? Wait, no, looking at the graph, the line goes through \((0, - 6)\) and let's take another point. Let's take \(x=-3\), what's \(y\)? When \(x = - 3\), \(y = 12\)? Wait, maybe better to take two clear points. Let's see, the line passes through \((0,-6)\) and \((3, - 24)\)? Wait, no, let's count the slope formula. The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points: let's take \((0, - 6)\) and \((3, - 24)\)? Wait, no, maybe I made a mistake. Wait, looking at the graph, when \(x = 0\), \(y=-6\), and when \(x = 3\), what's \(y\)? Wait, the line is going down from left to right, so it's a negative slope. Let's take two points: \((- 3,12)\) and \((0, - 6)\). Let's use these two points. So \(x_1=-3\), \(y_1 = 12\); \(x_2 = 0\), \(y_2=-6\).

Step2: Apply the slope formula

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), substitute \(x_1=-3\), \(y_1 = 12\), \(x_2 = 0\), \(y_2=-6\).

So \(m=\frac{-6 - 12}{0-(-3)}=\frac{-18}{3}=-6\).

Wait, let's check another pair. Let's take \((0, - 6)\) and \((1, - 12)\)? Wait, no, maybe the grid is such that each square is 3 units? Wait, the \(x\)-axis has marks at -24, -21, ..., 0, 3, 6,... so each grid line is 3 units? Wait, the distance between \(x = 0\) and \(x = 3\) is 3 units. Similarly for \(y\)-axis. Let's take two points: \((- 3,12)\) and \((0, - 6)\). Then \(y_2 - y_1=-6 - 12=-18\), \(x_2 - x_1=0-(-3)=3\), so slope \(m=\frac{-18}{3}=-6\). Alternatively, take \((0, - 6)\) and \((3, - 24)\): \(y_2 - y_1=-24-(-6)=-18\), \(x_2 - x_1=3 - 0 = 3\), so \(m=\frac{-18}{3}=-6\).

Answer:

\(-6\)