Determining Solutions For 2x - 3y = 15 A Step By Step Guide

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In mathematics, determining whether an ordered pair is a solution to a given equation is a fundamental concept in algebra. This involves substituting the values of the variables in the ordered pair into the equation and verifying if the equation holds true. In this article, we will delve into how to determine if the given ordered pairs are solutions to the equation $2x - 3y = 15$. We will meticulously evaluate each pair, providing a step-by-step explanation to ensure a comprehensive understanding of the solution process. This exercise is not just about finding the right answers; it's about reinforcing your understanding of algebraic principles and problem-solving techniques. So, let's embark on this mathematical journey and unravel the solutions together.

Understanding Ordered Pairs and Linear Equations

Before we dive into the specifics, let's establish a solid understanding of the core concepts. An ordered pair, represented as $(x, y)$, consists of two values: the x-coordinate and the y-coordinate. These coordinates correspond to points on a coordinate plane. A linear equation in two variables, such as $2x - 3y = 15$, describes a relationship between x and y. The solutions to a linear equation are the ordered pairs that, when substituted into the equation, make the equation true. Geometrically, a linear equation represents a straight line on the coordinate plane, and the solutions are the points that lie on this line. To determine if an ordered pair is a solution, we substitute the x and y values into the equation and check if both sides of the equation are equal. This process is crucial in various mathematical applications, including graphing linear equations, solving systems of equations, and modeling real-world scenarios.

Method for Verifying Solutions

The method for verifying whether an ordered pair is a solution to a linear equation is straightforward but requires careful attention to detail. The process involves substituting the x and y values from the ordered pair into the equation and then simplifying both sides of the equation. If, after simplification, the left-hand side (LHS) of the equation equals the right-hand side (RHS), then the ordered pair is a solution. If the LHS does not equal the RHS, the ordered pair is not a solution. It is crucial to perform the substitution and simplification accurately, following the order of operations (PEMDAS/BODMAS) to avoid errors. This method is a fundamental tool in algebra and is used extensively in solving and analyzing linear equations and systems of equations. The ability to correctly verify solutions is essential for building a strong foundation in algebra and for tackling more advanced mathematical concepts.

Detailed Analysis of Each Ordered Pair

Let's proceed to the detailed analysis of each ordered pair to determine if it satisfies the equation $2x - 3y = 15$. For each pair, we will substitute the x and y values into the equation, simplify, and then check if the equation holds true.

a) Is $(-7, -9)$ a solution?

To determine if $(-7, -9)$ is a solution, we substitute $x = -7$ and $y = -9$ into the equation $2x - 3y = 15$. This gives us:

2(−7)−3(−9)=152(-7) - 3(-9) = 15

Simplifying the left side:

−14+27=15-14 + 27 = 15

13=1513 = 15

Since $13$ does not equal $15$, the ordered pair $(-7, -9)$ is not a solution to the equation. This detailed step-by-step approach ensures clarity and helps in understanding the process of verifying solutions.

b) Is $(18, 7)$ a solution?

Now, let's check if $(18, 7)$ is a solution. We substitute $x = 18$ and $y = 7$ into the equation $2x - 3y = 15$:

2(18)−3(7)=152(18) - 3(7) = 15

Simplifying the left side:

36−21=1536 - 21 = 15

15=1515 = 15

Since $15$ equals $15$, the ordered pair $(18, 7)$ is a solution to the equation. This confirms that the point lies on the line represented by the equation.

c) Is $(24, 9)$ a solution?

Next, we evaluate the ordered pair $(24, 9)$. Substituting $x = 24$ and $y = 9$ into the equation $2x - 3y = 15$:

2(24)−3(9)=152(24) - 3(9) = 15

Simplifying the left side:

48−27=1548 - 27 = 15

21=1521 = 15

Since $21$ does not equal $15$, the ordered pair $(24, 9)$ is not a solution to the equation. This indicates that this point does not lie on the line represented by the equation.

d) Is $(3, -3)$ a solution?

Finally, let's determine if $(3, -3)$ is a solution. We substitute $x = 3$ and $y = -3$ into the equation $2x - 3y = 15$:

2(3)−3(−3)=152(3) - 3(-3) = 15

Simplifying the left side:

6+9=156 + 9 = 15

15=1515 = 15

Since $15$ equals $15$, the ordered pair $(3, -3)$ is a solution to the equation. This confirms that the point lies on the line represented by the equation.

Summary of Solutions

In summary, after evaluating each ordered pair, we have determined the following:

  • (-7, -9)$ is **not** a solution.

  • (18, 7)$ **is** a solution.

  • (24, 9)$ is **not** a solution.

  • (3, -3)$ **is** a solution.

This exercise demonstrates the practical application of substituting values into an equation to verify solutions. Understanding this process is crucial for solving more complex algebraic problems and for grasping the relationship between equations and their graphical representations.

Importance of Verifying Solutions

Verifying solutions is a critical step in the problem-solving process in mathematics. It ensures the accuracy of the results and helps in identifying any errors made during the solution process. In the context of linear equations, verifying solutions confirms whether the ordered pair lies on the line represented by the equation. This process is not only important for academic purposes but also has practical applications in various fields, such as engineering, economics, and computer science. For example, in engineering, verifying solutions can help ensure the stability and reliability of structures and systems. In economics, it can help in forecasting economic trends and making informed decisions. In computer science, it can help in debugging programs and ensuring the correctness of algorithms. Therefore, mastering the skill of verifying solutions is essential for success in mathematics and its applications.

Conclusion

In conclusion, determining whether an ordered pair is a solution to a linear equation is a fundamental skill in algebra. By substituting the x and y values of the ordered pair into the equation and verifying if the equation holds true, we can accurately identify the solutions. This article has provided a detailed step-by-step explanation of how to verify solutions for the equation $2x - 3y = 15$, using the ordered pairs $(-7, -9)$, $(18, 7)$, $(24, 9)$, and $(3, -3)$. The process involves careful substitution, simplification, and comparison of the left-hand side and right-hand side of the equation. Mastering this skill is crucial for building a strong foundation in algebra and for tackling more advanced mathematical concepts. Moreover, the ability to verify solutions is essential for ensuring accuracy and confidence in problem-solving, not only in mathematics but also in various real-world applications.