Question

the graph shows the distance a different microcar travels over time. what is the equation of the line? $y = \frac{1}{70}x$, $y = \frac{1}{35}x$, $y = 35x$, $y = 70x$ (graph: y-axis distance (mi) with 0,140,280,420,560; x-axis time (h) with 0,2,4,6,8; line through (0,0) and other points)

Explanation:

Step1: Recall the slope-intercept form

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. Since the line passes through the origin \((0,0)\), \(b = 0\), so the equation is \(y=mx\), and we need to find the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Choose two points on the line

From the graph, we can see that the line passes through \((0,0)\) and let's take another point, for example, when \(x = 4\), \(y = 140\) (we can also check other points like when \(x = 6\), \(y\) should be \(210\) or when \(x=8\), \(y = 280\)).

Step3: Calculate the slope

Using the formula for slope \(m=\frac{y - 0}{x - 0}=\frac{y}{x}\) (since \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(x,y)\)). Let's use the point \((4,140)\). Then \(m=\frac{140}{4}=35\)? Wait, no, wait. Wait, when \(x = 4\), \(y = 140\)? Wait, no, looking at the graph, the y - axis is distance (mi) and x - axis is time (h). Wait, when \(x = 4\), the y - value: let's check the grid. The vertical lines are at \(x = 0,2,4,6,8\) and horizontal lines at \(y = 0,140,280,420,560\)? Wait, no, the graph: when \(x = 4\), the line is at \(y = 140\)? Wait, no, maybe I misread. Wait, when \(x = 2\), what's \(y\)? Wait, the line passes through \((0,0)\) and let's take \(x = 4\), \(y = 140\)? No, wait, if we take \(x = 2\), \(y\) should be \(70\)? Wait, no, let's recalculate. Wait, the correct way: let's take two points. Let's take \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(4,140)\)? No, that can't be. Wait, maybe the point is \((4,140)\) is wrong. Wait, looking at the graph, when \(x = 4\), the y - coordinate: the horizontal lines are at \(y = 0,140,280,420,560\)? Wait, no, the labels are \(0,140,280,420,560\)? Wait, the first horizontal line above 0 is 140, then 280, etc. So when \(x = 4\), the line is at \(y = 140\)? No, that would make slope \(m=\frac{140}{4}=35\), but wait, when \(x = 2\), \(y = 70\), \(x = 6\), \(y = 210\), \(x = 8\), \(y = 280\)? Wait, no, 280 at \(x = 8\)? Wait, \(\frac{280}{8}=35\). Wait, but let's check the options. The options are \(y=\frac{1}{70}x\), \(y=\frac{1}{35}x\), \(y = 35x\), \(y = 70x\). Wait, if \(x = 2\), then for \(y = 35x\), \(y=35\times2 = 70\), for \(y = 70x\), \(y = 140\). Wait, looking at the graph, when \(x = 2\), the line is at \(y = 70\)? No, maybe I misread the graph. Wait, let's look again. The graph: the y - axis is distance (mi), x - axis is time (h). The line starts at (0,0) and goes up. Let's take \(x = 2\), what's the y - value? If we look at the grid, the vertical lines are at \(x = 0,2,4,6,8\) and horizontal lines at \(y = 0,140,280,420,560\). Wait, so when \(x = 2\), the line is at \(y = 70\)? No, that's not on the grid. Wait, maybe the horizontal lines are at \(y = 0,70,140,210,280\)? No, the label is 140, 280, etc. Wait, maybe the correct point is \((x = 4,y = 140)\) is wrong. Wait, let's take \(x = 4\), \(y = 140\): slope \(m=\frac{140}{4}=35\), so \(y = 35x\). But wait, let's check \(x = 2\): \(y=35\times2 = 70\), which is halfway between 0 and 140, that makes sense. \(x = 6\): \(y = 35\times6=210\), which is halfway between 140 and 280. \(x = 8\): \(y = 35\times8 = 280\), which matches the graph (since at \(x = 8\), the line is at \(y = 280\)). So the slope \(m = 35\), so the equation is \(y = 35x\)? Wait, no, wait, 35x when x = 4 is 140, which is on the graph. Wait, but let's check the options. The options are \(y=\frac{1}{70}x\), \(y=\frac{1}{35}x\), \(y = 35x\), \(y = 70x\). So if the slope is 35, then the equation is \(y = 35x\).

Answer:

\(y = 35x\) (the option with \(y = 35x\))