Question

slope =
equation=

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 \(x = 5\), \(y = 3\), or we can also take the point \((-4, -2)\) (from the graph). Let's use \((0, 0)\) and \((5, 3)\) or \((-4, -2)\) and \((0, 0)\). Let's use \((x_1, y_1)=(-4, -2)\) and \((x_2, y_2)=(0, 0)\).

Step2: Calculate the slope

The formula for slope \(m\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting the values: \(m=\frac{0 - (-2)}{0 - (-4)}=\frac{2}{4}=\frac{1}{2}\)? Wait, no, wait, let's check the other point. Wait, when \(x = 5\), \(y = 3\), so using \((0, 0)\) and \((5, 3)\), slope \(m=\frac{3 - 0}{5 - 0}=\frac{3}{5}\)? Wait, no, maybe I misread the graph. Wait, the line passes through \((-1, -0.5)\)? No, wait, let's look again. Wait, the line passes through \((-4, -2)\) and \((0, 0)\) and \((5, 3)\)? Wait, from \((-4, -2)\) to \((0, 0)\): change in \(y\) is \(0 - (-2)=2\), change in \(x\) is \(0 - (-4)=4\), so slope is \(\frac{2}{4}=\frac{1}{2}\)? But when \(x = 5\), \(y = 2.5\)? Wait, maybe the point at \(x = 5\) is \((5, 3)\), but maybe my initial point is wrong. Wait, let's check the graph again. The line crosses the origin, and when \(x = 5\), \(y = 3\), and when \(x=-4\), \(y = -2\)? Wait, no, \(-2\) when \(x=-4\): \(\frac{-2}{-4}=\frac{1}{2}\), but \(3/5 = 0.6\), which is not \(1/2\). Wait, maybe the point at \(x = 5\) is \((5, 3)\), so let's recalculate. Wait, maybe I made a mistake. Wait, let's take two clear points. Let's take \((0, 0)\) and \((5, 3)\): slope \(m=\frac{3 - 0}{5 - 0}=\frac{3}{5}\)? No, that doesn't seem right. Wait, maybe the point is \((4, 2)\)? Wait, the graph has a line from \((-4, -2)\) to \((0, 0)\) to \((5, 3)\). Wait, the difference between \((-4, -2)\) and \((0, 0)\): \(y\) increases by 2 when \(x\) increases by 4, so slope is \(\frac{2}{4}=\frac{1}{2}\). But when \(x = 5\), \(y\) should be \(2.5\), but the graph shows a point at \((5, 3)\). Maybe the graph is a bit approximate. Wait, maybe the correct points are \((0, 0)\) and \((5, 3)\), so slope is \(\frac{3}{5}\)? No, wait, let's check the rise over run. From \((0, 0)\) to \((5, 3)\): rise is 3, run is 5, so slope is \(\frac{3}{5}\)? Wait, no, maybe I made a mistake. Wait, another way: the line passes through \((-1, -0.6)\)? No, let's use the two points: \((-4, -2)\) and \((5, 3)\). Then slope \(m=\frac{3 - (-2)}{5 - (-4)}=\frac{5}{9}\)? No, that's not right. Wait, maybe the correct points are \((0, 0)\) and \((5, 3)\), so slope is \(\frac{3}{5}\)? Wait, no, let's look at the graph again. The line goes through \((-4, -2)\), \((0, 0)\), and \((5, 3)\). So from \((-4, -2)\) to \((0, 0)\): \(y\) increases by 2, \(x\) increases by 4, so slope is \(2/4 = 1/2\). From \((0, 0)\) to \((5, 3)\): \(y\) increases by 3, \(x\) increases by 5, so slope is \(3/5\). There's a discrepancy, which means I must have misread the graph. Wait, maybe the point at \(x = 5\) is \((5, 2.5)\), but the graph is drawn with a point at \((5, 3)\). Wait, maybe the correct slope is \(1/2\). Wait, let's check with \((-4, -2)\) and \((0, 0)\): \(m = (0 - (-2))/(0 - (-4)) = 2/4 = 1/2\). Then the equation would be \(y = mx + b\), and since it passes through \((0, 0)\), \(b = 0\), so \(y=\frac{1}{2}x\). But when \(x = 5\), \(y = 2.5\), but the graph shows a point at \(x = 5\), \(y = 3\). Maybe the graph is a bit off, or maybe I made a mistake. Wait, another approach: the line passes through \((-1, -0.5)\)? No, let's count the grid. Each grid line is 1 unit. So from \((0, 0)\) to \((5, 3)\): 5 units right, 3 units up, so slope is \(3/5\). From \((-4, -2)\) to \((0, 0)\): 4 units right, 2 units up, slope \(2/4 = 1/2\). There's a conflict. Wait, maybe the point at \(x = -4\) is \((-4, -2)\) and at \(x = 5\) is \((5, 3)\), so the slope is \((3 - (-2))/(5 - (-4)) = 5/9\)? No, that's not. Wait, maybe the correct points are \((0, 0)\) and \((5, 3)\), so slope is \(3/5\), and equation is \(y = \frac{3}{5}x\). Wait, but when \(x = -4\), \(y = -12/5 = -2.4\), but the graph shows \(y = -2\) at \(x = -4\). Maybe the graph is approximate. Let's go with the two clear points: \((0, 0)\) and \((5, 3)\). So slope \(m = \frac{3 - 0}{5 - 0} = \frac{3}{5}\)? Wait, no, maybe I misread the \(y\)-coordinate at \(x = 5\). Wait, the \(y\)-axis at 3: so the point is \((5, 3)\), so slope is \(3/5\). But let's check with \((-4, -2)\): if slope is \(1/2\), then \(y = (1/2)x\), so at \(x = -4\), \(y = -2\), which matches. At \(x = 5\), \(y = 2.5\), but the graph shows \(y = 3\), maybe it's a drawing error. So the correct slope should be \(1/2\), because \((-4, -2)\) and \((0, 0)\) give slope \(1/2\), and that's a clear point. So let's use \((-4, -2)\) and \((0, 0)\).

So slope \(m = \frac{0 - (-2)}{0 - (-4)} = \frac{2}{4} = \frac{1}{2}\).

Then the equation of the line: since it passes through the origin \((0, 0)\), the \(y\)-intercept \(b = 0\), so the equation is \(y = mx + b = \frac{1}{2}x + 0 = \frac{1}{2}x\).

Wait, but when \(x = 5\), \(y = 2.5\), but the graph has a point at \((5, 3)\), maybe the graph is a bit inaccurate, but the point \((-4, -2)\) and \((0, 0)\) are clear. So slope is \(1/2\), equation is \(y = \frac{1}{2}x\).

Wait, let's verify with another point. If \(x = 2\), \(y = 1\), which should be on the line. Yes, that makes sense. So slope is \(1/2\), equation is \(y = \frac{1}{2}x\).

Answer:

Slope = \(\frac{1}{2}\)
Equation = \(y = \frac{1}{2}x\)