Question

what is the equation of the trend line in the scatter plot?
use the two yellow points to write the equation in slope - intercept form. write any coefficients as integers, proper fractions, or improper fractions in simplest form.

Explanation:

Step1: Identify the two yellow points

From the scatter plot, the two yellow points are \((4, 9)\) and \((8, 3)\).

Step2: Calculate the slope (\(m\))

The formula for slope is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(4,9)\) and \((x_2,y_2)=(8,3)\). Then \(m=\frac{3 - 9}{8 - 4}=\frac{-6}{4}=-\frac{3}{2}\).

Step3: Find the y - intercept (\(b\))

The slope - intercept form of a line is \(y = mx + b\). We know \(m = -\frac{3}{2}\) and we can use the point \((4,9)\) to find \(b\). Substitute \(x = 4\), \(y = 9\) and \(m=-\frac{3}{2}\) into \(y=mx + b\):
\(9=-\frac{3}{2}(4)+b\)
\(9=-6 + b\)
Add 6 to both sides: \(b=9 + 6=15\)? Wait, no, wait. Wait, let's recalculate. Wait, \(-\frac{3}{2}\times4=-6\), then \(9=-6 + b\), so \(b=9 + 6 = 15\)? Wait, that can't be right. Wait, no, maybe I mixed up the points. Wait, let's check the points again. Wait, the first yellow point: when \(x = 4\), \(y = 9\); the second yellow point: when \(x = 8\), \(y = 3\). Wait, let's recalculate the slope: \(\frac{3 - 9}{8 - 4}=\frac{-6}{4}=-\frac{3}{2}\). Then using the point \((4,9)\): \(y=-\frac{3}{2}x + b\). Substitute \(x = 4\), \(y = 9\): \(9=-\frac{3}{2}(4)+b\) \(9=-6 + b\), so \(b = 15\)? Wait, but when \(x = 10\), \(y = 0\), let's check with \(b = 15\) and \(m=-\frac{3}{2}\), \(y=-\frac{3}{2}(10)+15=-15 + 15 = 0\), which matches the end - point. Wait, but let's check the other point \((8,3)\): \(y=-\frac{3}{2}(8)+15=-12 + 15 = 3\), which also matches. So the slope - intercept form is \(y=-\frac{3}{2}x + 15\)? Wait, no, wait, maybe I made a mistake in the points. Wait, looking at the graph, the first yellow point is at \(x = 4\), \(y = 9\); the second is at \(x = 8\), \(y = 3\). Wait, but let's check the line. The line goes from \((4,9)\) to \((8,3)\) to \((10,0)\). So the slope is \(\frac{0 - 9}{10 - 4}=\frac{-9}{6}=-\frac{3}{2}\), and the y - intercept is 15? Wait, when \(x = 0\), \(y = 15\)? But the graph's y - axis only goes up to 10. Wait, maybe the points are misidentified. Wait, no, the yellow points are at \((4,9)\) and \((8,3)\). Let's recalculate the slope again: \(m=\frac{3 - 9}{8 - 4}=\frac{-6}{4}=-\frac{3}{2}\). Then using the point \((4,9)\): \(y=-\frac{3}{2}x + b\). \(9=-\frac{3}{2}\times4 + b\) \(9=-6 + b\), so \(b = 15\). But when \(x = 4\), \(y = 9\); \(x = 8\), \(y = 3\); \(x = 10\), \(y = 0\). Let's check \(x = 10\): \(y=-\frac{3}{2}\times10+15=-15 + 15 = 0\), which is correct. So the equation of the line is \(y=-\frac{3}{2}x + 15\)? Wait, no, that seems off because the y - intercept is 15, but the graph's y - axis is up to 10. Wait, maybe the first yellow point is at \(x = 3\), \(y = 10\)? No, the yellow point is at \(x = 4\), \(y = 9\). Wait, maybe I made a mistake in the slope calculation. Wait, let's take two points on the line: \((4,9)\) and \((8,3)\). The change in \(y\) is \(3 - 9=-6\), change in \(x\) is \(8 - 4 = 4\), so slope is \(\frac{-6}{4}=-\frac{3}{2}\). Then the equation is \(y=-\frac{3}{2}x + b\). Plugging in \((4,9)\): \(9=-\frac{3}{2}(4)+b\) \(9=-6 + b\), so \(b = 15\). So the equation is \(y=-\frac{3}{2}x + 15\).

Answer:

\(y = -\frac{3}{2}x + 15\)