Sandra's Equation Challenge Finding Equivalent Equations

Let's explore the fascinating world of linear equations and how different forms can represent the same set of solutions. In this article, we'll dissect a problem where Tomas writes the equation y = 3x + 3/4, and Sandra creates an equation with identical solutions. Our mission is to identify Sandra's equation from a given option. This journey will reinforce your understanding of equivalent equations and algebraic manipulation.

Understanding the Problem

The core concept here lies in equivalent equations. Equivalent equations are equations that, despite looking different, have the same solution set. In simpler terms, if you plug in the same value for x into two equivalent equations, you'll get the same value for y. Our task is to determine which of the provided equations is equivalent to Tomas's equation, y = 3x + 3/4.

To achieve this, we will manipulate the given options, aiming to transform them into the form y = mx + b (slope-intercept form), which is the same form as Tomas's equation. This will allow us to directly compare the equations and identify the one that matches. We can also use algebraic manipulations such as multiplying or dividing both sides of an equation by a constant, or rearranging terms to isolate variables. The key is to perform operations that maintain the equality of the equation.

Let's delve deeper into the mathematics behind this. The equation y = 3x + 3/4 represents a straight line in a coordinate plane. The coefficient of x, which is 3, represents the slope of the line, and the constant term, 3/4, represents the y-intercept (the point where the line crosses the y-axis). Any equation that represents the same line will have the same slope and y-intercept, although it might be written in a different form. The beauty of algebra is that we can transform equations without changing their fundamental meaning, allowing us to compare them more easily. This problem highlights the importance of algebraic manipulation and the concept of equivalent forms in solving mathematical problems.

Analyzing the Given Option: -6x + 2y = 3/2

The provided option for Sandra's equation is -6x + 2y = 3/2. To determine if this equation is equivalent to Tomas's equation, y = 3x + 3/4, we need to manipulate it algebraically to see if we can transform it into the slope-intercept form (y = mx + b). This involves isolating y on one side of the equation.

Our first step is to add 6x to both sides of the equation. This eliminates the x term from the left side and moves it to the right side, bringing us closer to isolating y. Adding 6x to both sides, we get: 2y = 6x + 3/2. Now, we have y almost isolated. To completely isolate y, we need to divide both sides of the equation by 2. This will give us the equation in the desired slope-intercept form.

Dividing both sides by 2, we get: y = (6x + 3/2) / 2. This can be simplified further by dividing each term on the right side by 2. So, y = (6x / 2) + (3/2 / 2), which simplifies to y = 3x + 3/4. Now, let's meticulously examine what we've achieved. Through algebraic manipulation, we've successfully transformed Sandra's equation, -6x + 2y = 3/2, into y = 3x + 3/4. This resulting equation is identical to Tomas's equation. This crucial observation confirms that the two equations are indeed equivalent. They represent the same line and, consequently, share all the same solutions. This process demonstrates the power of algebraic manipulation in identifying equivalent equations, a fundamental skill in mathematics.

Conclusion

By meticulously manipulating the given equation -6x + 2y = 3/2, we've successfully transformed it into the same form as Tomas's equation, y = 3x + 3/4. This proves that the two equations are equivalent and share the same solutions. This exercise underscores the importance of understanding equivalent equations and the power of algebraic manipulation in solving mathematical problems. Therefore, Sandra's equation could be -6x + 2y = 3/2.

This problem highlights a crucial concept in algebra: the ability to recognize and manipulate equivalent equations. Understanding this concept is essential for solving a wide range of mathematical problems, from simple linear equations to more complex systems of equations. The ability to transform equations into different forms without changing their fundamental meaning is a powerful tool in any mathematician's arsenal.

Moreover, this exploration reinforces the idea that equations can appear different on the surface yet represent the same underlying relationship. This is particularly important in real-world applications where mathematical models can be expressed in various forms. Being able to identify these equivalent forms allows us to simplify problems, make comparisons, and gain deeper insights into the underlying phenomena.