Understanding Equivalent Equations: Simplifying y - 6 -12x 4

When dealing with linear equations, it's often necessary to simplify them to a more recognizable form. This process not only helps in solving the problem but also in understanding the relationship between variables. In this article, we will explore the process of simplifying the given equation, y - 6 -12x 4, step by step.

Step-by-Step Simplification

The given equation is:

y - 6 -12x 4

Distributing the Coefficient

The first step is to distribute the coefficient -12 on the right-hand side:

y - 6 -12x - 48

Isolating the Variable

To isolate the variable y, we need to add 6 to both sides of the equation:

y - 6 6 -12x - 48 6

y -12x - 42

Standard Form

The equation is now in a standard form, which is commonly used in algebra and graphing. The equation y -12x - 42 is the simplified and equivalent form of the original equation.

Alternative Forms

While the standard form is useful, there are other forms you can express the equation in:

Verifying Equivalence

To verify that the equations are equivalent, you can multiply both sides of the original equation by a constant. For instance, multiplying both sides by 2 gives:

2(y - 6) 2(-12x 4)

2y - 12 -24x - 96

2y -24x - 84

y -12x - 42

This confirms that the equation remains y -12x - 42.

Conclusion

Understanding the process of simplifying and verifying linear equations is crucial for both algebraic manipulation and real-world applications. Whether you're working with standard form or slope-intercept form, the key is to maintain equivalence throughout the transformations. The simplified form y -12x - 42 is the result of distributing, isolating the variable, and verifying the equivalence of the equation.