Solutions For 4x - 5y = 24 Ordered Pairs And Equation Verification

Introduction

In mathematics, finding solutions to equations is a fundamental skill. This article will delve into the process of identifying ordered pairs $(x, y)$ that satisfy the linear equation $4x - 5y = 24$. We will examine several given ordered pairs and determine whether they are solutions to the equation. This involves substituting the $x$ and $y$ values of each ordered pair into the equation and checking if the equality holds true. Understanding how to solve such problems is crucial for various mathematical applications, including graphing linear equations, solving systems of equations, and understanding linear relationships in real-world contexts. This article aims to provide a clear and detailed explanation of the solution process, ensuring that readers can confidently tackle similar problems in the future. By mastering these techniques, you will gain a deeper appreciation for the elegance and utility of linear equations in mathematics and beyond.

Understanding Linear Equations and Ordered Pairs

Before we dive into the specific problem, let's establish a solid understanding of linear equations and ordered pairs. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations in two variables, such as $x$ and $y$, can be written in the general form $Ax + By = C$, where $A$, $B$, and $C$ are constants. The equation $4x - 5y = 24$ fits this form, with $A = 4$, $B = -5$, and $C = 24$. Ordered pairs, denoted as $(x, y)$, represent a specific point on a coordinate plane. The first value, $x$, is the abscissa (horizontal coordinate), and the second value, $y$, is the ordinate (vertical coordinate). To determine if an ordered pair is a solution to a linear equation, we substitute the $x$ and $y$ values into the equation and check if the equation holds true. If the substitution results in a true statement, the ordered pair is a solution; otherwise, it is not. This process is a cornerstone of solving linear equations and is essential for understanding their graphical representation. The graph of a linear equation is a straight line, and every point on the line corresponds to an ordered pair that is a solution to the equation. Conversely, every solution to the equation corresponds to a point on the line. This connection between algebraic equations and geometric representations is a fundamental concept in mathematics. Understanding it allows us to visualize solutions and gain a deeper insight into the nature of linear relationships. In the following sections, we will apply this understanding to the specific equation $4x - 5y = 24$ and determine which of the given ordered pairs are solutions.

Analyzing the Ordered Pair (4, 8)

Our first ordered pair to analyze is $(4, 8)$. To determine if this ordered pair is a solution to the equation $4x - 5y = 24$, we need to substitute $x = 4$ and $y = 8$ into the equation and verify if the equality holds. The substitution yields: $4(4) - 5(8) = 16 - 40$. Simplifying the expression, we get $16 - 40 = -24$. Now, we compare this result with the right-hand side of the equation, which is $24$. Since $-24 eq 24$, the ordered pair $(4, 8)$ does not satisfy the equation $4x - 5y = 24$. This means that the point $(4, 8)$ does not lie on the line represented by the equation. When we substitute the values of $x$ and $y$ and find that the equation does not hold, it is crucial to double-check our calculations to ensure accuracy. A small arithmetic error can lead to an incorrect conclusion. In this case, the arithmetic is straightforward, and the result clearly shows that $(4, 8)$ is not a solution. This process of substituting and verifying is a fundamental technique in algebra and is widely used to solve equations and inequalities. It allows us to systematically check potential solutions and determine their validity. In the context of linear equations, this method is particularly useful for identifying points that lie on the line represented by the equation. By understanding this process, we can confidently analyze other ordered pairs and determine whether they are solutions to the equation.

Evaluating the Ordered Pair (-9, -12)

Next, let's examine the ordered pair $(-9, -12)$ to see if it satisfies the equation $4x - 5y = 24$. We substitute $x = -9$ and $y = -12$ into the equation: $4(-9) - 5(-12) = -36 + 60$. Simplifying the expression, we get $-36 + 60 = 24$. Now, we compare this result with the right-hand side of the equation, which is $24$. Since $24 = 24$, the ordered pair $(-9, -12)$ is indeed a solution to the equation $4x - 5y = 24$. This means that the point $(-9, -12)$ lies on the line represented by the equation. This confirms that the ordered pair is a valid solution, and it provides us with a specific point on the line defined by the equation. It's essential to note that there are infinitely many solutions to a linear equation in two variables, and each solution corresponds to a point on the line. Finding one solution, like $(-9, -12)$, helps us understand the nature of the equation and its graphical representation. The process of verifying solutions by substitution is a cornerstone of algebra and is crucial for solving a wide range of mathematical problems. By carefully substituting the values and simplifying the expression, we can confidently determine whether an ordered pair satisfies the given equation. This skill is fundamental for further studies in mathematics, including solving systems of equations, graphing linear equations, and understanding linear relationships in various contexts. In the next sections, we will continue to analyze the remaining ordered pairs using the same approach.

Checking the Ordered Pair (1, -4)

Now, let's consider the ordered pair $(1, -4)$ and determine if it is a solution to the equation $4x - 5y = 24$. We substitute $x = 1$ and $y = -4$ into the equation: $4(1) - 5(-4) = 4 + 20$. Simplifying the expression, we get $4 + 20 = 24$. Comparing this result with the right-hand side of the equation, which is $24$, we find that $24 = 24$. Therefore, the ordered pair $(1, -4)$ is a solution to the equation $4x - 5y = 24$. This indicates that the point $(1, -4)$ lies on the line represented by the equation. The fact that this ordered pair satisfies the equation reinforces the concept that linear equations have infinitely many solutions, each corresponding to a point on the line. It is crucial to accurately perform the substitution and simplification steps to avoid errors in determining whether an ordered pair is a solution. A common mistake is to miscalculate the product of a negative number and a negative number, so careful attention to detail is essential. By verifying solutions like $(1, -4)$, we gain confidence in our understanding of linear equations and their properties. This skill is not only valuable for solving mathematical problems but also for applying mathematical concepts in real-world scenarios. In the next section, we will analyze the final ordered pair, $(6, 0)$, to complete our investigation.

Verifying the Ordered Pair (6, 0)

Finally, let's examine the ordered pair $(6, 0)$ to see if it is a solution to the equation $4x - 5y = 24$. We substitute $x = 6$ and $y = 0$ into the equation: $4(6) - 5(0) = 24 - 0$. Simplifying the expression, we get $24 - 0 = 24$. Comparing this result with the right-hand side of the equation, which is $24$, we find that $24 = 24$. Thus, the ordered pair $(6, 0)$ is a solution to the equation $4x - 5y = 24$. This means that the point $(6, 0)$ lies on the line represented by the equation. This particular solution is interesting because it represents the x-intercept of the line. The x-intercept is the point where the line crosses the x-axis, and it occurs when $y = 0$. Finding intercepts is a useful technique for graphing linear equations, as it provides two specific points on the line. By verifying this solution, we further solidify our understanding of how ordered pairs relate to linear equations. The ability to substitute values and check for equality is a fundamental skill in algebra and is crucial for solving various mathematical problems. In this case, we have successfully verified that $(6, 0)$ is a solution, adding to our collection of points that lie on the line defined by the equation $4x - 5y = 24$. With this final verification, we have completed our analysis of all the given ordered pairs.

Conclusion: Identifying Solutions to Linear Equations

In conclusion, we have thoroughly examined the process of identifying ordered pairs that are solutions to the linear equation $4x - 5y = 24$. By substituting the $x$ and $y$ values of each ordered pair into the equation and verifying if the equality holds, we were able to determine which pairs are solutions. We found that the ordered pairs $(-9, -12)$, $(1, -4)$, and $(6, 0)$ are solutions to the equation, while $(4, 8)$ is not. This exercise highlights the fundamental concept that a solution to a linear equation in two variables is an ordered pair that, when substituted into the equation, makes the equation true. The graphical interpretation of this concept is that each solution corresponds to a point on the line represented by the equation. Understanding how to find and verify solutions is crucial for various mathematical applications, including graphing linear equations, solving systems of equations, and modeling real-world scenarios with linear relationships. The ability to manipulate equations and solve for variables is a fundamental skill in mathematics, and this article has provided a comprehensive guide to one aspect of this skill. By mastering these techniques, you will be well-equipped to tackle more complex mathematical problems and gain a deeper appreciation for the power and versatility of linear equations. The principles discussed here are applicable to a wide range of algebraic problems, making this a valuable skill for any student of mathematics.