Question

Step-by-step explanation:

Here, the given points,

(-3,-40) (1,12) (5,72) and (7, 137),

Since, the equation of a line is,

y = ax+ b

Where,

a=(n(\sum xy)-(\sum x)(\sum y))/(n(\sum x^2)-(\sum x)^2)

b= ((\sum y)(\sum x^2)-(\sum x)(\sum xy))/(n(\sum x^2)-(\sum x)^2)

By the graphing regression tool,

We get,

a = 16.9237 ≈ 17,

b = 2.94068 ≈ 3

Hence, the line of best fit would be,

y=17x+3. Please break this down

Answer 1

Certainly! Let's go through the process step by step to derive the equation of the line of best fit using the given points \((-3,-40)\), \((1,12)\), \((5,72)\), and \((7, 137)\).

### Step 1: Organize the Data

We'll list down the x and y values from the given points:

- \(x_1 = -3\), \(y_1 = -40\)
- \(x_2 = 1\), \(y_2 = 12\)
- \(x_3 = 5\), \(y_3 = 72\)
- \(x_4 = 7\), \(y_4 = 137\)

### Step 2: Calculate Necessary Sums

We need to calculate several summations:

1. \( \sum x \)
2. \( \sum y \)
3. \( \sum xy \)
4. \( \sum x^2 \)
5. The number of points \( n \)

#### Calculations:

1. **Calculate \( n \)**:
\[
n = 4 \quad (\text{since we have 4 points})
\]

2. **Calculate \( \sum x \)**:
\[
\sum x = -3 + 1 + 5 + 7 = 10
\]

3. **Calculate \( \sum y \)**:
\[
\sum y = -40 + 12 + 72 + 137 = 181
\]

4. **Calculate \( \sum xy \)**:
- For each point, calculate \( xy \):
- \((-3)(-40) = 120\)
- \((1)(12) = 12\)
- \((5)(72) = 360\)
- \((7)(137) = 959\)

Now sum these products:
\[
\sum xy = 120 + 12 + 360 + 959 = 1451
\]

5. **Calculate \( \sum x^2 \)**:
- For each x value, calculate \( x^2 \):
- \((-3)^2 = 9\)
- \(1^2 = 1\)
- \(5^2 = 25\)
- \(7^2 = 49\)

Now sum these squares:
\[
\sum x^2 = 9 + 1 + 25 + 49 = 84
\]

### Step 3: Plug Values into Formulas for \( a \) and \( b \)

Using the formulas provided for \( a \) and \( b \):

1. **Calculate \( a \)**:
\[
a = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

Substitute the values:
\[
a = \frac{4(1451) - (10)(181)}{4(84) - (10)^2} = \frac{5804 - 1810}{336 - 100}
\]
\[
a = \frac{3994}{236} \approx 16.9237 \quad (\text{approximately } 17)
\]

2. **Calculate \( b \)**:
\[
b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2}
\]

Substitute the values:
\[
b = \frac{(181)(84) - (10)(1451)}{4(84) - (10)^2} = \frac{15164 - 14510}{336 - 100}
\]
\[
b = \frac{654}{236} \approx 2.94068 \quad (\text{approximately } 3)
\]

### Step 4: Write the Line of Best Fit

Now that we have \( a \) and \( b \), the equation of the line of best fit is:
\[
y = ax + b
\]
Substituting \( a \) and \( b \):
\[
y = 17x + 3
\]

### Conclusion

Thus, the final result for the line of best fit based on the given points is:
\[
\boxed{y = 17x + 3}
\]