Finding Solutions To Y=2x A Step By Step Guide

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In the realm of mathematics, equations reign supreme as the language of relationships between variables. These relationships often manifest as ordered pairs, which serve as potential solutions to the equation. Understanding how to identify these solutions is a fundamental skill, a cornerstone upon which more advanced concepts are built. In essence, an ordered pair (x, y) is deemed a solution to an equation if, upon substituting the x and y values into the equation, the equation holds true. This means that the left-hand side of the equation must equal the right-hand side after the substitution. To master this concept, let's delve deeper into the process of verifying solutions and apply it to the equation y=2x.

Our focus will be on the equation y=2x, a simple yet powerful linear equation that illustrates a direct relationship between x and y. We'll explore how to determine whether given ordered pairs satisfy this equation, essentially unlocking the secrets to identifying solutions. This exploration will not only enhance your understanding of equation-solving but also solidify your grasp of fundamental algebraic principles. So, let's embark on this journey, where we dissect the equation, analyze ordered pairs, and ultimately, master the art of identifying solutions. This will involve substituting the x and y values from each ordered pair into the equation y=2x and checking if the equation holds true. If the equation remains balanced after the substitution, the ordered pair is indeed a solution. This meticulous process ensures that we accurately identify all the solutions from the given options.

The equation y=2x is a linear equation, and its solutions are the ordered pairs that, when substituted into the equation, make the equation true. This equation represents a straight line when graphed on a coordinate plane. Every point on this line corresponds to an ordered pair that satisfies the equation. To determine if an ordered pair is a solution, we substitute the x and y values into the equation and check if the left-hand side (LHS) equals the right-hand side (RHS). If they are equal, the ordered pair is a solution; otherwise, it is not. Understanding this fundamental principle is crucial for solving equations and understanding the relationship between variables. This is the backbone of algebra, forming the basis for more complex equations and mathematical models.

Let's meticulously evaluate each ordered pair against the equation y = 2x to determine which ones qualify as solutions. This process involves substituting the x and y values from each pair into the equation and verifying if the equality holds true. This hands-on approach will not only help us identify the solutions but also solidify our understanding of the concept.

Ordered Pair A: (2, 3)

For the ordered pair (2, 3), we substitute x = 2 and y = 3 into the equation y = 2x. This yields 3 = 2 * 2, which simplifies to 3 = 4. This statement is false. Therefore, the ordered pair (2, 3) is not a solution to the equation y = 2x. The key here is the principle of equality – if substituting the values does not result in a true statement, then the ordered pair does not lie on the line represented by the equation.

Ordered Pair B: (4, 6)

Next, let's consider the ordered pair (4, 6). Substituting x = 4 and y = 6 into the equation y = 2x, we get 6 = 2 * 4, which simplifies to 6 = 8. This statement is also false. Consequently, the ordered pair (4, 6) is not a solution to the equation y = 2x. This reinforces the importance of careful substitution and evaluation when determining solutions.

Ordered Pair C: (6, 9)

Now, let's analyze the ordered pair (6, 9). Substituting x = 6 and y = 9 into the equation y = 2x, we have 9 = 2 * 6, which simplifies to 9 = 12. This statement is false, making (6, 9) not a solution to the equation y = 2x. This pattern of substitution and verification is a fundamental technique in algebra.

Ordered Pair D: (8, 16)

Finally, let's examine the ordered pair (8, 16). Substituting x = 8 and y = 16 into the equation y = 2x, we obtain 16 = 2 * 8, which simplifies to 16 = 16. This statement is true. Therefore, the ordered pair (8, 16) is a solution to the equation y = 2x. This positive result highlights the significance of finding values that satisfy the equation, as these points lie on the line represented by the equation.

Based on our evaluations, only the ordered pair (8, 16) is a solution to the equation y = 2x. Now, let's look at the given options:

  • (A) A and B only
  • (B) A, B, and C only
  • (C) A, B, and D only
  • (D) D only

Since only the ordered pair D (8, 16) is a solution, the correct answer is (D) D only. This conclusion underscores the importance of meticulously analyzing each option and applying the principles of equation-solving to arrive at the accurate answer.

In conclusion, determining whether an ordered pair is a solution to an equation is a fundamental skill in mathematics. By substituting the x and y values into the equation and verifying if the equation holds true, we can accurately identify solutions. In the context of the equation y = 2x, only the ordered pair (8, 16) satisfied the equation, making option (D) the correct answer. This process not only reinforces the understanding of algebraic principles but also lays the foundation for tackling more complex equations and mathematical problems. Remember, practice and a thorough understanding of these core concepts are the keys to mathematical proficiency.

By systematically evaluating each ordered pair, we've not only solved the specific question but also gained a deeper appreciation for the process of identifying solutions to equations. This knowledge will undoubtedly serve as a valuable tool in your mathematical journey, empowering you to confidently tackle future challenges. The ability to substitute values, simplify equations, and verify results is a cornerstone of algebra and beyond. So, embrace this skill, practice it diligently, and watch your mathematical abilities flourish.