Introduction to Line Equations

In mathematics, the equation of a line is a fundamental concept, especially when dealing with coordinate geometry. This article will delve into the process of determining the equation of a straight line passing through two given points and finding the equations of lines parallel to this line, passing through a specified point. Let’s start by defining the equation of a line in two different contexts.

Equation of a Line through Two Points

The equation of a straight line passing through two points ((x_1, y_1)) and ((x_2, y_2)) can be expressed as:

y - y1 m(x - x1)

where m is the slope of the line, calculated as:

m frac{y_2 - y_1}{x_2 - x_1}

This slope is a measure of how steep the line is and how it changes as x and y values change.

Calculating the Equation of the Line Through Points A and B

Given the points A ((2, 4)) and B ((-2, -5)), let’s calculate the equation of the line passing through these points.

First, calculate the slope m.

m frac{-5 - 4}{-2 - 2} frac{-9}{-4} frac{9}{4}

Now, use the point-slope form of the equation and the point A ((2, 4)) to find the equation.

y - 4 frac{9}{4}(x - 2)

Convert this equation to the standard form.

4y - 16 9x - 18

9x - 4y - 2 0

Equation of Parallel Lines

Parallel lines have the same slope. Hence, the second line parallel to the first, passing through the point (1, 6), will have the same slope m frac{9}{4}.

Write the equation for the parallel line using point (1, 6).

y - 6 frac{9}{4}(x - 1)

Convert this equation into the standard form.

4y - 24 9x - 9

9x - 4y - 15 0

Applying the Slope-Intercept Form

The equation of a line can also be written in the slope-intercept form:

y mx b

where m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis).

Given Y2 -5, Y1 4, X2 -2, and X1 2, calculate the slope m:

m frac{-5 - 4}{-2 - 2} frac{-9}{-4} 2.25

To find b, solve with one of the points, such as (2, 4):

4 2.25(2) b

4 4.5 b

b 4 - 4.5 -0.5

Therefore, the equation of the line is:

y 2.25x - 0.5

Verify this by checking the y-value for x 2:

4 2.25(2) - 0.5

4 4.5 - 0.5

4 4

This confirms the equation is correct.

Conclusion

Understanding how to determine the equation of a line and the equations of lines parallel to it is crucial for solving geometric and algebraic problems. Whether using the point-slope form or the slope-intercept form, these methods provide a systematic approach to finding the right equation. By applying these principles, you can efficiently describe and analyze linear relationships in various contexts.