Understanding the Equation of a Line through (-6, 7) and (-3, 6): Slope-Intercept, Point-Slope, and Standard Forms

Introduction

Determining the equation of a line that passes through two specific points involves several steps and requires the application of various forms of linear equations. This article explores the process using the slope-intercept, point-slope, and standard forms, with an example given by the points (-6, 7) and (-3, 6). We'll delve into the mathematical concepts and their practical applications.

Slope-Intercept Form

The Slope-Intercept form of a line's equation is y mx c. Here, m represents the slope and c the y-intercept. This form is intuitive and allows for an easy visualization of how the line behaves.

Deriving the Slope

To start, we calculate the slope m:

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

Given points: (-6, 7) and (-3, 6)

m frac{6 - 7}{-3 - (-6)} frac{-1}{3}

Using the Slope-Intercept Form

Once we have the slope, we can use one of the points to find the y-intercept c:

y - 7 -frac{1}{3}(x - -6)

Which simplifies to:

y -frac{1}{3}x - 2 7

Therefore:

y -frac{1}{3}x 5

Point-Slope Form

The Point-Slope form of a line's equation is y - y_1 m(x - x_1). This form is particularly useful when the slope and a point on the line are known.

Deriving the Point-Slope Equation

We start by using the slope:

m -frac{1}{3}

Choosing the point (-6, 7), the equation becomes:

y - 7 -frac{1}{3}(x - -6)

Which simplifies to:

y -frac{1}{3}x - 2 7

Therefore:

y -frac{1}{3}x 5

Standard Form

The Standard form of a line's equation is Ax By C. This form is particularly useful for algebraic manipulations and clarifies the nature of the line's orientation.

Deriving the Standard Equation

Starting from the Slope-Intercept form:

y -frac{1}{3}x 5

Multiplying through by 3:

3y -x 15

Rearranging:

x 3y 15

Conclusion

All three forms of linear equations represent the same line, but each form has its unique advantages. The Slope-Intercept form is great for quick visualization, while the Point-Slope form is useful when only a point and the slope are known. The Standard form is particularly helpful for algebraic manipulations, especially when dealing with more complex equations.

Key Concepts

Slope-Intercept Form: y mx c Point-Slope Form: y - y_1 m(x - x_1) Standard Form: Ax By C

Additional Resources

For further reading and practice, consider exploring additional resources on linear equations and graphing lines.