Proving that a First-Degree Equation in x and y Represents a Straight Line

In the realm of coordinate geometry, a first-degree equation in the variables x and y plays a fundamental role in defining a straight line. This article will explore the steps and mathematical proofs to show why such an equation always represents a straight line. We will discuss the general form, rearrangement, slope-intercept form, and special cases.

The General Form and Rearrangement

The general form of a first-degree equation is given by:

Ax By C 0

Here, A, B, and C are constants, and it is crucial that A and B are not both zero. This constraint is necessary because a line cannot be defined by such an equation if at least one of these coefficients is zero.

Rearranging the Equation

To prove that this equation represents a straight line, we will rearrange it to the slope-intercept form:

By -Ax - C

Dividing everything by B (assuming B ≠ 0):

y -frac{A}{B}x - frac{C}{B}

Identifying the Slope-Intercept Form

The rearranged equation now closely resembles the slope-intercept form of a line, y mx b, where:

- The slope m -frac{A}{B} - The y-intercept b -frac{C}{B}

Understanding the Graphical Representation

In a Cartesian coordinate system, the slope-intercept form represents a straight line because:

For every value of x, there is a corresponding value of y. This indicates a linear relationship. The slope m represents the angle of the line, while b indicates where the line crosses the y-axis.

Special Cases

There are special cases to consider when discussing the first-degree equation in x and y: If A 0 and B ≠ 0, the equation simplifies to:

By C 0

Upon rearranging, we get:

y -frac{C}{B}

This is the equation of a horizontal line.

If B 0 and A ≠ 0, the equation simplifies to:

Ax C 0

Upon rearranging, we get:

x -frac{C}{A}

This is the equation of a vertical line.

Conclusion

Since any equation of the first degree in x and y can be rearranged into a form that clearly shows a linear relationship, we can conclude that it represents a straight line.

This holds true for all values of A and B as long as they are not both zero. Thus, the proof is solid and relies on the fundamental properties of linear equations and their graphical representation.

Keywords: first-degree equation, straight line, slope-intercept form