Understanding the Equation of a Line Passing Through Two Points

Finding the equation of a line that passes through two points can be a straightforward task using basic algebra and the concept of slope. In this article, we will walk through the steps of determining the equation of a line passing through the points (13, 45) and (4, -5) and express this equation in terms of (x). This process involves determining the slope (m) and using a point to write the equation in different forms.

Step 1: Calculate the Slope (m)

To find the slope of the line passing through the points (13, 45) and (4, -5), we use the slope formula: [ m frac{y_2 - y_1}{x_2 - x_1} ] Substituting the coordinates into the formula, we get: [ m frac{-5 - 45}{4 - 13} ] Calculating the numerator and the denominator separately, we have: [ m frac{-50}{-9} frac{50}{9} ] Therefore, the slope (m) of the line is ( frac{50}{9} ).

Step 2: Find the Equation of the Line in Point-Slope Form

Now that we have the slope, we can use the point-slope form of the equation of a line, which is: [ y - y_1 m(x - x_1) ] Using the point (13, 45), the equation becomes: [ y - 45 frac{50}{9}(x - 13) ] Expanding and simplifying this equation, we get: [ y - 45 frac{50}{9}x - frac{650}{9} ] Rearranging terms, we obtain the equation in point-slope form: [ y frac{50}{9}x frac{405 - 650}{9} ] [ y frac{50}{9}x - frac{245}{9} ]

Step 3: Convert to General Form

The general form of the equation of a line is (Ax By C). To convert the point-slope form to the general form, we first eliminate the fraction by multiplying every term by 9: [ 9y 5 - 245 ] Rearranging the terms to match the general form, we have: [ 5 - 9y 245 ]

Step 4: Express in Terms of x

To express the line in terms of (x), we solve the general form equation for (x): [ 5 - 9y 245 ] Adding (9y) to both sides, we get: [ 5 9y 245 ] Dividing every term by 50, we finally obtain the equation of the line in terms of (x): [ x frac{9y 245}{50} ] This equation can also be written as: [ x frac{9}{50}y frac{245}{50} ] Or simply in functional notation as (x f(y)) where (f(y) frac{9}{50}y frac{49}{10}).

Conclusion

By following these steps, we have found that the equation of the line passing through the points (13, 45) and (4, -5) in terms of (x) is (x frac{9}{50}y frac{49}{10}). This process demonstrates how to use algebraic manipulation and the concept of slope to find and express the equation of a line in different forms.

Additional Resources for SEO

- Understanding Slope in Geometry - Algebraic Manipulation Techniques - Common Forms of Line Equations

For more information on these topics, readers can refer to the additional resources provided.