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 = 15?

Explanation:

Step1: Find the slope of the line

The line passes through points \((0, 2)\) and \((3, 4)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 2}{3 - 0}=\frac{2}{3}\)? Wait, no, looking at the points: \((0,2)\), \((3,4)\)? Wait, no, the points on the line of best fit: \((0,2)\), \((3,4)\)? Wait, actually, let's check the points given: \((0,2)\), \((3,4)\)? Wait, no, the line has points like \((0,2)\), \((3,4)\)? Wait, no, looking at the graph, the line passes through \((0, 2)\), \((3, 4)\)? Wait, no, let's take two clear points: \((0, 2)\) and \((6, 6)\). Then slope \(m=\frac{6 - 2}{6 - 0}=\frac{4}{6}=\frac{2}{3}\)? Wait, no, \((0,2)\) and \((3,4)\): \(m=\frac{4 - 2}{3 - 0}=\frac{2}{3}\). Wait, but another point: \((6,6)\), so from \((0,2)\) to \((6,6)\), change in \(y\) is \(4\), change in \(x\) is \(6\), so slope \(m = \frac{4}{6}=\frac{2}{3}\). Wait, but let's check \((3,4)\): when \(x=3\), \(y=4\). So the equation of the line: using point-slope form, \(y - y_1 = m(x - x_1)\). Using \((0,2)\), \(y - 2 = \frac{2}{3}(x - 0)\), so \(y=\frac{2}{3}x + 2\). Wait, but when \(x=6\), \(y=\frac{2}{3}(6)+2 = 4 + 2 = 6\), which matches \((6,6)\). When \(x=9\), \(y=\frac{2}{3}(9)+2 = 6 + 2 = 8\), which matches \((9,8)\). When \(x=12\), \(y=\frac{2}{3}(12)+2 = 8 + 2 = 10\), which matches \((12,10)\). So the equation is \(y=\frac{2}{3}x + 2\). Now, when \(x = 15\), \(y=\frac{2}{3}(15)+2 = 10 + 2 = 12\)? Wait, no, wait: \(\frac{2}{3}\times15 = 10\), plus 2 is 12? Wait, but let's check the slope again. Wait, maybe the slope is 1? Wait, no, \((0,2)\), \((3,4)\): difference in \(y\) is 2, difference in \(x\) is 3, so slope 2/3. But wait, \((0,2)\), \((6,6)\): difference in \(y\) is 4, difference in \(x\) is 6, slope 4/6=2/3. \((3,4)\), \((6,6)\): difference in \(y\) is 2, difference in \(x\) is 3, slope 2/3. So equation is \(y = \frac{2}{3}x + 2\). Wait, but when \(x=15\), \(y=\frac{2}{3}\times15 + 2 = 10 + 2 = 12\). Wait, but let's check the pattern: from \(x=0\) (y=2), \(x=3\) (y=4), \(x=6\) (y=6), \(x=9\) (y=8), \(x=12\) (y=10). So the pattern is that for every increase of 3 in \(x\), \(y\) increases by 2. So from \(x=12\) (y=10) to \(x=15\) (increase by 3), \(y\) increases by 2, so \(10 + 2 = 12\). So the y-value when \(x=15\) is 12.

Wait, let's re-examine the slope. The line passes through (0,2), (3,4), (6,6), (9,8), (12,10). So the slope is (4-2)/(3-0)=2/3, but also (6-4)/(6-3)=2/3, (8-6)/(9-6)=2/3, (10-8)/(12-9)=2/3. So the slope is 2/3. The equation is \(y = \frac{2}{3}x + 2\). When \(x=15\), \(y = \frac{2}{3}(15) + 2 = 10 + 2 = 12\).

Step1: Identify the pattern or equation of the line of best fit

The line of best fit passes through points \((0, 2)\), \((3, 4)\), \((6, 6)\), \((9, 8)\), \((12, 10)\). We can see that for every increase of 3 in \(x\), \(y\) increases by 2. This is a linear relationship with slope \(m=\frac{2}{3}\) and y-intercept \(b = 2\) (from the point \((0, 2)\)). The equation of the line is \(y=\frac{2}{3}x + 2\).

Step2: Calculate \(y\) when \(x = 15\)

Substitute \(x = 15\) into the equation:
\(y=\frac{2}{3}(15)+2\)
First, calculate \(\frac{2}{3}\times15\): \(\frac{2\times15}{3}=10\)
Then, add 2: \(10 + 2 = 12\)

Answer:

12