# Find the equation y = a + bx of the least squares line that best fits the given data points using any method. (2, 3), (3, 2), (5, 1), (6, 0).

**Solution:**

To find an the equation y = a + bx by the method of least squares the normal equations are

\(\sum y\) = na +b\(\sum x\)

\(\sum xy\) = a\(\sum x\) + b\(\sum x^2\)

Given data

x | 2 | 3 | 5 | 6 |
---|---|---|---|---|

y | 3 | 2 | 1 | 0 |

The required data for normal equations are

x | 2 | 3 | 5 | 6 | \(\sum x = 16\) |
---|---|---|---|---|---|

y | 3 | 2 | 1 | 0 | \(\sum y = 6\) |

x^{2} |
9 | 4 | 1 | 36 | \(\sum x^2 = 50\) |

xy | 6 | 6 | 5 | 0 | \(\sum xy = 17\) |

∴ The normal equations are

6 = 4a + 16b --- (1)

17 = 16a + 50b --- (2)

Solving these equations for a and b

Multiplying (1) with 4 and subtracting from (2)

24 = 16a + 64b

17 = 16a + 50b

⇒ 7 = 14 b

b = 7/14 = 0.5

b= 0.5

Substitute b = 0.5 in equation (2)

17 = 16a + 50(0.5)

17 = 16a + 25

17 - 25 = 16a

-8 = 16 a

a = -8/16

a = - 1/2 = -0.5

Therefore, the equation of the line for the given data is is y = a+ bx

⇒ y = (-0.5) + (0.5)x

## Find the equation y = a + bx of the least squares line that best fits the given data points using any method. (2, 3), (3, 2), (5, 1), (6, 0).

**Summary:**

The Equation of line ⇒ y = (-0.5) + (0.5)x is the equation y = a + bx of the least squares line that best fits the given data points using any method. (2, 3), (3, 2), (5, 1), (6, 0).

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