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Raquel throws darts at a coordinate grid centered at the origin. Her goal is to create a line of darts. Her darts actually hit the coordinate grid at [tex](-5,0)[/tex], [tex](1,-3)[/tex], [tex](4,5)[/tex], [tex](-8,-6)[/tex], [tex](0,2)[/tex], and [tex](9,6)[/tex].

Which equation best approximates the line of best fit of the darts?

A. [tex]y = 0.6x + 0.6[/tex]
B. [tex]y = 0.1x + 0.8[/tex]
C. [tex]y = 0.8x + 0.1[/tex]
D. [tex]y = 0.5x + 0.6[/tex]



Answer :

GinnyAnswer
To determine which equation best approximates the line of best fit for the darts Raquel threw, we need to follow these steps:

### Step 1: Collect the Data Points
Raquel's darts landed at the following coordinates:
- [tex]\((-5, 0)\)[/tex]
- [tex]\((1, -3)\)[/tex]
- [tex]\((4, 5)\)[/tex]
- [tex]\((−8, −6)\)[/tex]
- [tex]\((0, 2)\)[/tex]
- [tex]\((9, 6)\)[/tex]

### Step 2: Calculate the Line of Best Fit
The line of best fit [tex]\( y = mx + b \)[/tex] is determined by calculating the slope ([tex]\(m\)[/tex]) and the intercept ([tex]\(b\)[/tex]).

Based on the data points, the actual calculated slope and intercept are:
- Slope [tex]\( m \approx 0.633 \)[/tex]
- Intercept [tex]\( b \approx 0.561 \)[/tex]

### Step 3: Evaluate Candidate Equations
We have four candidate equations:
1. [tex]\( y = 0.6x + 0.6 \)[/tex]
2. [tex]\( y = 0.1x + 0.8 \)[/tex]
3. [tex]\( y = 0.8x + 0.1 \)[/tex]
4. [tex]\( y = 0.5x + 0.6 \)[/tex]

We need to determine which of these is closest to the actual line of best fit.

### Step 4: Compare Differences
We'll find the total difference for each candidate equation by considering the differences in slope and intercept:

1. For [tex]\( y = 0.6x + 0.6 \)[/tex]:
- Difference in slope: [tex]\( |0.6 - 0.633| = 0.033 \)[/tex]
- Difference in intercept: [tex]\( |0.6 - 0.561| = 0.039 \)[/tex]
- Total difference: [tex]\( 0.033 + 0.039 = 0.072 \)[/tex]

2. For [tex]\( y = 0.1x + 0.8 \)[/tex]:
- Difference in slope: [tex]\( |0.1 - 0.633| = 0.533 \)[/tex]
- Difference in intercept: [tex]\( |0.8 - 0.561| = 0.239 \)[/tex]
- Total difference: [tex]\( 0.533 + 0.239 = 0.772 \)[/tex]

3. For [tex]\( y = 0.8x + 0.1 \)[/tex]:
- Difference in slope: [tex]\( |0.8 - 0.633| = 0.167 \)[/tex]
- Difference in intercept: [tex]\( |0.1 - 0.561| = 0.461 \)[/tex]
- Total difference: [tex]\( 0.167 + 0.461 = 0.628 \)[/tex]

4. For [tex]\( y = 0.5x + 0.6 \)[/tex]:
- Difference in slope: [tex]\( |0.5 - 0.633| = 0.133 \)[/tex]
- Difference in intercept: [tex]\( |0.6 - 0.561| = 0.039 \)[/tex]
- Total difference: [tex]\( 0.133 + 0.039 = 0.172 \)[/tex]

### Step 5: Select the Best Fit Equation
The candidate equation with the smallest total difference is the best fit. Here are the total differences:
- Equation 1: [tex]\( 0.072 \)[/tex]
- Equation 2: [tex]\( 0.772 \)[/tex]
- Equation 3: [tex]\( 0.628 \)[/tex]
- Equation 4: [tex]\( 0.172 \)[/tex]

The smallest total difference is [tex]\( 0.072 \)[/tex], corresponding to the equation:
[tex]\[ y = 0.6x + 0.6 \][/tex]

### Conclusion
The equation that best approximates the line of best fit for Raquel's darts is:
[tex]\[ \boxed{y = 0.6x + 0.6} \][/tex]

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