how do i determine the eqaution of a line of best fit using the formulas of y=mx+c and slope formula

To determine the equation of the line of best fit using the formula \( y = mx + c \), you'll need to follow these steps:

### 1. **Calculate the Slope (m):**
The slope of the line of best fit can be determined using the formula:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

Where:
- \( n \) is the number of data points.
- \( \sum xy \) is the sum of the products of corresponding \( x \) and \( y \) values.
- \( \sum x \) is the sum of all \( x \) values.
- \( \sum y \) is the sum of all \( y \) values.
- \( \sum x^2 \) is the sum of the squares of all \( x \) values.

### 2. **Calculate the Y-Intercept (c):**
Once you have the slope \( m \), use it to find the y-intercept \( c \) with the following formula:

\[
c = \frac{\sum y - m(\sum x)}{n}
\]

Where:
- \( \sum y \) is the sum of all \( y \) values.
- \( \sum x \) is the sum of all \( x \) values.
- \( n \) is the number of data points.

### 3. **Write the Equation:**
Now that you have \( m \) and \( c \), the equation of the line of best fit is:

\[
y = mx + c
\]

### Example:
Suppose you have the following data points: \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\).

Let's say:

- \( n = 5 \)
- \( \sum x = 15 \)
- \( \sum y = 20 \)
- \( \sum xy = 70 \)
- \( \sum x^2 = 55 \)

#### Calculate \( m \):

\[
m = \frac{5(70) - (15)(20)}{5(55) - (15)^2} = \frac{350 - 300}{275 - 225} = \frac{50}{50} = 1
\]

#### Calculate \( c \):

\[
c = \frac{20 - 1(15)}{5} = \frac{20 - 15}{5} = \frac{5}{5} = 1
\]

#### Equation of the Line:
\[
y = 1x + 1 \quad \text{or simply} \quad y = x + 1
\]

### Do you want more details or have any questions?

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### Related Questions:
1. How do you interpret the slope and y-intercept in the context of a real-world problem?
2. What is the significance of minimizing the sum of squared errors in finding the line of best fit?
3. Can the method of least squares be applied to non-linear models?
4. How do you calculate the correlation coefficient, and what does it tell you about the line of best fit?
5. How do you determine if a linear model is appropriate for a given set of data?

### Tip:
Always plot the data and the line of best fit to visually inspect how well the line represents the data before drawing conclusions.

Learn how to determine the equation of a line of best fit using the formulas y = mx + c and slope formula. This detailed guide includes step-by-step calculations for slope (m) and y-intercept (c) using real data points. Suitable for Grades 9-12.

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