Question

a line of best fit was drawn to the plotted points in a data set below. based on the line of best fit, for what y - value does ( x = 24 )?

Explanation:

Step1: Find the slope of the line

We use two points on the line of best fit, say \((0, 2)\) and \((4, 5)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5 - 2}{4 - 0}=\frac{3}{4}\)? Wait, no, let's check another pair. Wait, the points given are \((0,2)\), \((4,5)\), \((8,8)\), \((12,11)\). Let's take \((0,2)\) and \((4,5)\): \(m=\frac{5 - 2}{4 - 0}=\frac{3}{4}\)? Wait, no, \(5 - 2 = 3\), \(4 - 0 = 4\), but wait, \((4,5)\) and \((8,8)\): \(m=\frac{8 - 5}{8 - 4}=\frac{3}{4}\). And \((8,8)\) and \((12,11)\): \(m=\frac{11 - 8}{12 - 8}=\frac{3}{4}\). So the slope \(m=\frac{3}{4}\)? Wait, no, wait \((0,2)\) to \((4,5)\): \(y\) increases by 3 when \(x\) increases by 4. Wait, but let's check the y-intercept. The line passes through \((0,2)\), so the equation is \(y = mx + b\), where \(b = 2\). Wait, but when \(x = 4\), \(y = 5\): \(5 = m*4 + 2\) → \(4m = 3\) → \(m=\frac{3}{4}\). Wait, but when \(x = 8\), \(y = 8\): \(8=\frac{3}{4}*8 + 2=6 + 2 = 8\), correct. When \(x = 12\), \(y=\frac{3}{4}*12 + 2=9 + 2 = 11\), correct. So the equation of the line of best fit is \(y=\frac{3}{4}x + 2\).

Step2: Find y when x = 24

Substitute \(x = 24\) into the equation: \(y=\frac{3}{4}*24 + 2\). Calculate \(\frac{3}{4}*24 = 18\), then \(18 + 2 = 20\). Wait, but wait, maybe I made a mistake in the y-intercept. Wait, the point \((0,2)\) is given? Wait, the graph shows \((0,2)\) as a point on the line. Wait, but let's check the slope again. Alternatively, notice that from \(x = 0\) to \(x = 4\), \(y\) goes from 2 to 5 (increase by 3). From \(x = 4\) to \(x = 8\), \(y\) goes from 5 to 8 (increase by 3). From \(x = 8\) to \(x = 12\), \(y\) goes from 8 to 11 (increase by 3). So the slope is \(\frac{3}{4}\) (since for every 4 units increase in x, y increases by 3). So the rate of change is 3 per 4 x-units, or \(\frac{3}{4}\) per x-unit. Alternatively, the line has a constant slope, so the equation is linear. Let's use two points, say \((4,5)\) and \((8,8)\). The slope \(m=\frac{8 - 5}{8 - 4}=\frac{3}{4}\). Then using point-slope form: \(y - 5=\frac{3}{4}(x - 4)\). Simplify: \(y=\frac{3}{4}x - 3 + 5=\frac{3}{4}x + 2\), which matches the earlier equation. So when \(x = 24\), \(y=\frac{3}{4}*24 + 2 = 18 + 2 = 20\).

Wait, but let's check another way. The line passes through (0,2), (4,5), (8,8), (12,11). So the pattern is that when x increases by 4, y increases by 3. So from x=0 to x=4 (4 units), y=2 to 5 (3 increase). x=4 to 8 (4 units), y=5 to 8 (3 increase). x=8 to 12 (4 units), y=8 to 11 (3 increase). So the next interval: x=12 to 16 (4 units), y=11 to 14 (3 increase). x=16 to 20 (4 units), y=14 to 17 (3 increase). x=20 to 24 (4 units), y=17 to 20 (3 increase). So at x=24, y=20. That matches the earlier calculation.

Answer:

20