raquel throws darts at a coordinate grid centered at the origin. her goal is to create a line of darts. her…

Explanation:

Step1: Recall line - of - best - fit concept

The line of best fit for a set of data points $(x_i,y_i)$ has the form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. One way to estimate is by using the mean of $x$ and $y$ values and the relationship between them. But an alternative is to test the given points in each equation. Let's take two points, say $(0,2)$ and $(4,5)$.

Step2: Test point $(0,2)$ in each equation

For $y = 0.6x+0.6$:

When $x = 0$, $y=0.6\times0 + 0.6=0.6\neq2$.

For $y = 0.1x + 0.8$:

When $x = 0$, $y=0.1\times0+0.8 = 0.8\neq2$.

For $y = 0.8x+0.1$:

When $x = 0$, $y=0.8\times0 + 0.1=0.1\neq2$.

For $y = 0.5x+0.6$:

When $x = 0$, $y=0.5\times0+0.6 = 0.6\neq2$. Let's use another approach. We can use a statistical software or a calculator with linear - regression capabilities. But if we don't have that, we can calculate the slope $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^{2}-(\sum_{i = 1}^{n}x_i)^{2}}$ and $b=\bar{y}-m\bar{x}$, where $n = 6$, $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$ and $\bar{y}=\frac{\sum_{i = 1}^{n}y_i}{n}$. $\sum_{i = 1}^{6}x_i=-5 + 1+4-8 + 0+9=1$ $\sum_{i = 1}^{6}y_i=0-3 + 5-6 + 2+6=4$ $\sum_{i = 1}^{6}x_iy_i=(-5)\times0+1\times(-3)+4\times5+(-8)\times(-6)+0\times2+9\times6=0 - 3+20 + 48+0+54=119$ $\sum_{i = 1}^{6}x_i^{2}=(-5)^{2}+1^{2}+4^{2}+(-8)^{2}+0^{2}+9^{2}=25 + 1+16+64+0+81=187$

$m=\frac{6\times119-1\times4}{6\times187 - 1^{2}}=\frac{714 - 4}{1122-1}=\frac{710}{1121}\approx0.63$ $\bar{x}=\frac{1}{6}\approx0.17$, $\bar{y}=\frac{4}{6}\approx0.67$ $b=\bar{y}-m\bar{x}\approx0.67-0.63\times0.17\approx0.67 - 0.11=0.56\approx0.6$

Answer:

$y = 0.6x + 0.6$