Question

what is the slope of this line? (with a coordinate grid and a line plotted, y-axis labeled with numbers from -48 to 48, x-axis labeled with numbers from -48 to 48)

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through the points \((0, 12)\) and \((1, -12)\) (we can also use other points, but these are easy to identify). Let's confirm: when \(x = 0\), \(y = 12\) (the y - intercept), and when \(x = 1\), we can see the change in \(y\). Alternatively, we can use two clear points. Let's take \((0, 12)\) and \((- 1,24)\) (since when \(x=-1\), \(y = 24\) as per the grid).

Step2: Use the slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(0,12)\) and \((x_2,y_2)=(-1,24)\). Then \(m=\frac{24 - 12}{-1-0}=\frac{12}{-1}=- 12\)? Wait, no, let's check another pair. Wait, maybe better to take \((0,12)\) and \((1, - 12)\). Then \(m=\frac{-12 - 12}{1 - 0}=\frac{-24}{1}=-24\)? Wait, no, let's look at the grid again. Each grid square is 6 units? Wait, no, the y - axis has 6, 12, 18, etc. Wait, the distance between 0 and 6 on y is one grid, so each grid is 6 units? Wait, no, the coordinates: when \(x = 0\), \(y = 12\) (so the point is \((0,12)\)). When \(x = 1\), what's \(y\)? Let's see the line: from \((0,12)\), going to the right 1 unit (x increases by 1), y decreases by 24? Wait, no, maybe the grid is 6 units per square. Wait, the y - axis: 6, 12, 18, 24, 30, 36, 42, 48. So each grid line is 6 units. So let's take two points: \((0,12)\) and \((1, - 12)\) is wrong. Wait, let's take \((0,12)\) and \((-1,24)\): the difference in \(x\) is \(-1-0=-1\), difference in \(y\) is \(24 - 12 = 12\), so slope \(m=\frac{12}{-1}=-12\)? No, that can't be. Wait, maybe the grid is 6 units per square. Wait, when \(x = 0\), \(y = 12\) (so 2 grid squares up from 0, since each square is 6). When \(x=-1\), \(y = 24\) (4 grid squares up). When \(x = 1\), \(y=0\)? No, the line is steep. Wait, let's use the formula correctly. Let's take two points: \((0,12)\) and \((1, - 12)\) is incorrect. Wait, let's look at the line: from \((0,12)\), if we move 1 unit to the right (x + 1), how much does y change? Looking at the graph, the line goes from (0,12) down to (1, - 12)? No, that's a big drop. Wait, maybe the x - axis and y - axis have each grid as 6 units. So the point (0,12) is (0,12), and the point (1, - 12) is (1, - 12), but the slope would be \(\frac{-12 - 12}{1-0}=\frac{-24}{1}=-24\). Wait, let's check with another pair. Let's take (0,12) and (-1,36). Then \(x_1 = 0,y_1 = 12\); \(x_2=-1,y_2 = 36\). Then \(m=\frac{36 - 12}{-1-0}=\frac{24}{-1}=-24\). Yes, that makes sense. Because from \(x = 0\) to \(x=-1\) (x decreases by 1), y increases by 24. So the slope is \(\frac{y_2 - y_1}{x_2 - x_1}=\frac{36 - 12}{-1 - 0}=\frac{24}{-1}=-24\). Alternatively, from (0,12) to (1, - 12): \(m=\frac{-12 - 12}{1-0}=\frac{-24}{1}=-24\). So the slope is - 24.

Wait, let's verify: the slope formula is \(m=\frac{\Delta y}{\Delta x}\). If we take two points \((x_1,y_1)\) and \((x_2,y_2)\), the slope is the change in y over change in x. Let's take (0,12) and (1, - 12): \(\Delta y=-12 - 12=-24\), \(\Delta x = 1-0 = 1\), so \(m=\frac{-24}{1}=-24\). That seems correct.

Answer:

The slope of the line is \(-24\)