Determine Inequality Solution Does (-1, 3) Solve 5x - 3y > 1

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In mathematics, determining whether a specific point is a solution to an inequality is a fundamental concept, particularly in the realm of linear inequalities. This involves substituting the coordinates of the point into the inequality and verifying if the resulting statement holds true. This article provides a comprehensive guide on how to determine if the point (-1, 3) is a solution to the inequality 5x - 3y > 1. We will delve into the step-by-step process, offering clear explanations and insights to enhance your understanding of this mathematical concept. Whether you're a student learning algebra or simply seeking to refresh your knowledge, this guide will equip you with the tools necessary to tackle similar problems with confidence.

Understanding Linear Inequalities

Before diving into the specifics of our problem, let's first understand linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as >, <, ≥, or ≤. Unlike linear equations, which have a single solution or a set of discrete solutions, linear inequalities have a range of solutions. This range is often represented graphically as a shaded region on a coordinate plane, indicating all the points that satisfy the inequality. Grasping this concept is crucial because it lays the foundation for understanding how a single point relates to the solution set of an inequality.

Linear inequalities are prevalent in various real-world applications, from optimizing resource allocation to determining feasible regions in linear programming problems. They provide a powerful tool for modeling situations where constraints and limitations are involved. The ability to solve and interpret linear inequalities is thus a valuable skill in both academic and practical contexts. Understanding the graphical representation of inequalities, where the solution set is depicted as a shaded region, further enriches this skill by providing a visual understanding of the solution space. This visual interpretation aids in identifying points that satisfy the inequality and understanding the overall behavior of the system being modeled.

The solution to a linear inequality is not just a single number, but rather a set of numbers that make the inequality true. This set can be infinite, and it is often represented graphically on a number line or a coordinate plane. For example, the inequality x > 2 represents all numbers greater than 2, and this can be visualized as a shaded region on a number line extending to the right of 2. Similarly, in two dimensions, the solution to an inequality like y < x represents all points below the line y = x, which is depicted as a shaded area on the coordinate plane. The boundary line itself (y = x in this case) may or may not be included in the solution set, depending on whether the inequality is strict (< or >) or inclusive (≤ or ≥).

The Inequality: 5x - 3y > 1

Now, let's focus on the specific inequality we're working with: 5x - 3y > 1. This is a linear inequality in two variables, x and y. To determine whether a point is a solution to this inequality, we need to substitute the x and y coordinates of the point into the inequality and check if the resulting statement is true. This process is akin to verifying whether a particular key fits a lock – the point must satisfy the condition imposed by the inequality. The inequality 5x - 3y > 1 represents a region on the coordinate plane, and our task is to ascertain whether the point (-1, 3) falls within this region.

The inequality 5x - 3y > 1 can be interpreted as a constraint on the relationship between x and y. It defines a region on the Cartesian plane where the combination of x and y values satisfies the condition that five times x minus three times y is strictly greater than one. Graphically, this inequality represents the area above the line 5x - 3y = 1, but not including the line itself, since the inequality is strict (i.e., it does not include equality). This graphical representation offers a visual way to understand the set of all points that satisfy the inequality. Any point within this region, when its coordinates are substituted into the inequality, will yield a true statement.

The Point: (-1, 3)

The point we are interested in is (-1, 3). This is a coordinate pair representing a specific location on the Cartesian plane. The first number, -1, is the x-coordinate, which indicates the point's horizontal position relative to the origin. The second number, 3, is the y-coordinate, representing the point's vertical position. To determine if this point is a solution to the inequality 5x - 3y > 1, we need to substitute these values into the inequality. The x-coordinate will replace the x variable, and the y-coordinate will replace the y variable. This substitution transforms the inequality into a numerical statement that we can then evaluate for truth or falsehood. This process is fundamental to determining whether the point lies within the solution region of the inequality.

The coordinates of a point are its address in the two-dimensional plane, much like a street address locates a specific building in a city. The x-coordinate tells us how far to move horizontally from the origin (0, 0), with negative values indicating movement to the left and positive values to the right. Similarly, the y-coordinate indicates the vertical movement from the origin, with positive values signifying upward movement and negative values downward. The ordered pair (-1, 3), therefore, pinpoints a location one unit to the left of the origin and three units above it. This precise positioning is crucial because it allows us to assess whether this specific point satisfies the conditions set forth by the inequality 5x - 3y > 1.

Step-by-Step Solution

Now, let's go through the step-by-step solution to determine if the point (-1, 3) is a solution to the inequality 5x - 3y > 1:

  1. Substitute the x and y values: Replace x with -1 and y with 3 in the inequality: 5(-1) - 3(3) > 1

  2. Perform the multiplication: Multiply the numbers: -5 - 9 > 1

  3. Simplify the expression: Combine the terms on the left side: -14 > 1

  4. Evaluate the statement: Determine if the resulting inequality is true or false. Is -14 greater than 1?

Each of these steps is crucial to the solution process. Substitution is the foundation, replacing variables with their corresponding values. Multiplication simplifies the terms, making it easier to compare the expressions. Simplification further combines like terms, reducing the expression to its simplest form. Finally, evaluating the statement determines whether the original inequality holds true for the given point. In this case, the result will reveal whether (-1, 3) is indeed a solution to 5x - 3y > 1.

Detailed Explanation of the Steps

Step 1: Substitute the x and y values

The initial step in determining whether a point is a solution to an inequality is to substitute the values of x and y from the given point into the inequality. This process involves replacing the variable 'x' in the inequality with the x-coordinate of the point and the variable 'y' with the y-coordinate. For the point (-1, 3) and the inequality 5x - 3y > 1, this means replacing 'x' with -1 and 'y' with 3. The resulting expression will be a numerical statement that can be evaluated to determine its truthfulness. This substitution is a crucial bridge between the abstract inequality and the concrete point, allowing us to assess their relationship. It transforms the inequality into a simple numerical comparison, which is the basis for the subsequent steps.

The act of substitution is not merely a mechanical replacement; it is the core of the verification process. It translates the symbolic language of algebra into a numerical expression that can be easily understood and assessed. Without substitution, the inequality remains a general statement about the relationship between x and y, but with it, we gain the ability to test the validity of this relationship for a specific pair of values. The accuracy of the substitution is paramount, as any error at this stage will propagate through the rest of the solution, leading to an incorrect conclusion. Therefore, careful attention must be paid to ensure that the values are correctly placed in the expression.

Step 2: Perform the multiplication

Once the variables have been substituted with their corresponding numerical values, the next step involves performing the necessary multiplications. In the context of our inequality 5x - 3y > 1 with the point (-1, 3), after substitution, we have 5(-1) - 3(3) > 1. The multiplication step requires us to evaluate the products 5 times -1 and 3 times 3. These operations are fundamental arithmetic calculations that simplify the expression and bring us closer to a state where we can determine the validity of the inequality. Accurate multiplication is essential, as these products directly influence the subsequent simplification and evaluation. The result of this step sets the stage for combining like terms and ultimately assessing whether the point satisfies the inequality.

Multiplication, in this context, serves to resolve the coefficients that multiply the variables in the original inequality. It is a critical step because it transforms the symbolic coefficients into actual numerical values that can be combined and compared. The multiplication of a positive number with a negative number (such as 5 times -1) yields a negative result, which is a crucial rule to remember in this process. Similarly, the multiplication of two positive numbers (such as 3 times 3) results in a positive number. These calculations must be performed with precision, as even a small error can alter the outcome and lead to a false conclusion about whether the point is a solution to the inequality.

Step 3: Simplify the expression

After performing the multiplications, the next crucial step is to simplify the expression. This involves combining any like terms on each side of the inequality to reduce it to its simplest form. In our case, after multiplying, we have -5 - 9 > 1. Simplifying this expression involves combining the two constant terms on the left side: -5 and -9. Adding these two negative numbers together is a basic arithmetic operation that yields a single numerical value. The goal of simplification is to make the inequality as straightforward as possible, so that the comparison in the next step is clear and unambiguous. The simplified expression allows us to see the relationship between the two sides of the inequality more directly.

Simplification is a process of condensing information while preserving its essential meaning. In the context of algebraic expressions, this means performing arithmetic operations such as addition, subtraction, multiplication, and division to combine like terms and reduce the expression to its most basic form. For inequalities, this process is vital because it eliminates unnecessary complexity and allows for a clear comparison between the two sides. The simplified form of the inequality makes it easier to determine whether the inequality holds true or false. Accuracy in simplification is critical; any mistake in combining terms can lead to an incorrect assessment of the inequality and, consequently, a wrong conclusion about whether the point is a solution.

Step 4: Evaluate the statement

The final step in determining whether a point is a solution to an inequality is to evaluate the statement resulting from the previous steps. This involves examining the simplified inequality and determining if the comparison it makes is true or false. In our example, after substituting, multiplying, and simplifying, we arrive at the inequality -14 > 1. Evaluating this statement requires us to consider whether the number -14 is indeed greater than the number 1. This comparison is straightforward and relies on our understanding of the number line and the relative positions of numbers. The outcome of this evaluation will definitively answer whether the point (-1, 3) is a solution to the original inequality 5x - 3y > 1.

Evaluation is the culmination of all the preceding steps; it is the moment where we interpret the mathematical result and draw a conclusion. The process of evaluation requires a clear understanding of the meaning of inequality symbols, such as '>', which signifies 'greater than.' In the context of numbers, 'greater than' means further to the right on the number line. Therefore, to evaluate the statement -14 > 1, we must determine whether -14 is positioned to the right of 1 on the number line. Since negative numbers are to the left of positive numbers, -14 is clearly not greater than 1. This evaluation provides the definitive answer to our original question: Is the point (-1, 3) a solution to the inequality 5x - 3y > 1? The accuracy of this evaluation hinges on a firm grasp of numerical relationships and the meaning of mathematical symbols.

Conclusion: Is (-1, 3) a Solution?

After performing the steps, we arrived at the statement -14 > 1. This statement is false because -14 is not greater than 1. Therefore, the point (-1, 3) is not a solution to the inequality 5x - 3y > 1. This conclusion is based on the logical progression of substituting the point's coordinates into the inequality, simplifying the resulting expression, and evaluating the truthfulness of the final statement. The point (-1, 3) does not lie in the region defined by the inequality 5x - 3y > 1 on the coordinate plane.

This process highlights the fundamental principle that a point is a solution to an inequality if and only if its coordinates satisfy the inequality when substituted. Our step-by-step approach provides a clear methodology for verifying this condition. The ability to determine whether a point satisfies an inequality is a foundational skill in algebra and is essential for understanding more advanced concepts such as linear programming and systems of inequalities. The conclusion that (-1, 3) is not a solution is not just an end result, but also a validation of the method used and an affirmation of the understanding of inequality concepts.

Final Answer

No, the point (-1, 3) is not a solution to the inequality 5x - 3y > 1.

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Determine Inequality Solution Does (-1, 3) Solve 5x - 3y > 1?