Given the set of data points \( \{ (2,5), (4,10), (6,15), (8,20) \} \), find the equation of the line of best fit. Then, factor out the greatest common factor (GCF) from each term in the equation.
Answer
7. Now factor out the greatest common factor (GCF) from each term in the equation. The GCF of 0.5 and 10 is 0.5, so the equation becomes \( y = 0.5(x + 20) \).
Step 1 ：1. Use the formula for the slope of the line of best fit, which is \( m = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{n(\Sigma x^2) - (\Sigma x)^2} \). Plug in the given data points to find the slope.
Step 2 ：2. The sum of the x-values (\( \Sigma x \)) is \(2+4+6+8=20\). The sum of the y-values (\( \Sigma y \)) is \(5+10+15+20=50\). The sum of the product of the x and y values (\( \Sigma xy \)) is \(2*5+4*10+6*15+8*20=260\). The sum of the square of the x-values (\( \Sigma x^2 \)) is \(2^2+4^2+6^2+8^2=120\). Plug these numbers into the formula to find the slope.
Step 3 ：3. The slope \( m \) equals \( \frac{4(260) - 20*50}{4*120 - 20^2} = \frac{1040 - 1000}{480 - 400} = \frac{40}{80} = 0.5 \).
Step 4 ：4. Use the formula for the y-intercept of the line of best fit, which is \( b = \frac{\Sigma y - m\Sigma x}{n} \). Plug in the numbers we have calculated to find the y-intercept.
Step 5 ：5. The y-intercept \( b \) equals \( \frac{50 - 0.5*20}{4} = \frac{40}{4} = 10 \).
Step 6 ：6. So, the equation of the line of best fit is \( y = 0.5x + 10 \).
Step 7 ：7. Now factor out the greatest common factor (GCF) from each term in the equation. The GCF of 0.5 and 10 is 0.5, so the equation becomes \( y = 0.5(x + 20) \).
