Solving For Y In The Equation Y = 2x - 3 For Different Values Of X

In the realm of mathematics, equations serve as fundamental tools for describing relationships between variables. Among these, linear equations hold a prominent position due to their simplicity and wide range of applications. This article delves into the intricacies of a specific linear equation, y = 2x - 3, exploring how to determine the value of 'y' for different values of 'x'. By substituting various values for 'x' into the equation, we can unravel the corresponding values of 'y', gaining a deeper understanding of the relationship between these two variables.

Decoding the Equation: y = 2x - 3

At its core, the equation y = 2x - 3 represents a linear relationship between two variables, 'x' and 'y'. The equation's structure reveals that 'y' is dependent on 'x', meaning the value of 'y' changes as the value of 'x' varies. The coefficient '2' in front of 'x' signifies the slope of the line represented by the equation, indicating the rate at which 'y' changes for every unit change in 'x'. The constant term '-3' represents the y-intercept, the point where the line crosses the y-axis. This equation forms the basis for our exploration, and by manipulating it, we can uncover the hidden values of 'y' for different 'x' values.

Understanding the equation is paramount to solving it. We know that the equation y = 2x - 3 is a linear equation, meaning it represents a straight line when graphed. The '2' is the slope, indicating the line's steepness, and the '-3' is the y-intercept, the point where the line crosses the vertical y-axis. Think of it this way: for every increase of 1 in the value of 'x', the value of 'y' increases by 2. This consistent relationship allows us to predict the value of 'y' for any given 'x'. By understanding these core components, we can confidently tackle the task of finding 'y' for specific 'x' values.

Furthermore, let's consider the real-world implications of such an equation. Imagine 'x' represents the number of hours worked, and 'y' represents the total earnings after deducting expenses. The equation y = 2x - 3 could then model a scenario where an individual earns $2 per hour but has an initial expense of $3. By understanding the equation, we can determine the earnings for any number of hours worked. This illustrates the practical relevance of solving linear equations, highlighting their ability to model and predict real-world phenomena. The power of equations lies in their ability to translate abstract mathematical concepts into tangible and relatable scenarios.

Unveiling Y for X = 0

Let's embark on our journey to determine the value of 'y' when 'x' is 0. This scenario represents the starting point, the value of 'y' when there is no 'x' contribution. To find 'y', we substitute 'x = 0' into the equation y = 2x - 3. This substitution yields y = 2(0) - 3. Simplifying this expression, we find that y = -3. Therefore, when x = 0, y = -3. This point (0, -3) represents the y-intercept of the line, the point where the line intersects the y-axis. This is a critical piece of information for understanding the behavior of the equation.

Substituting x = 0 into the equation y = 2x - 3 is a straightforward process, but it yields a crucial result. By replacing 'x' with '0', we essentially isolate the constant term '-3'. This is because 2 multiplied by 0 equals 0, effectively eliminating the '2x' term. What remains is y = -3, a clear indication that when 'x' has no value, 'y' holds a negative value of 3. This concept is vital in understanding the starting point of the linear relationship described by the equation.

Consider again the real-world example of earnings. If 'x' represents hours worked and 'y' represents earnings after expenses, then when x = 0, y = -3 implies that before any work is done, there is a debt or initial expense of $3. This highlights how setting 'x' to zero can reveal the initial conditions or fixed costs in a given situation. This understanding adds depth to our interpretation of the equation and its practical applications.

Discovering Y for X = 1

Now, let's shift our focus to the case where 'x' equals 1. This represents a single unit of 'x', allowing us to observe how 'y' changes in response. Substituting 'x = 1' into the equation y = 2x - 3, we obtain y = 2(1) - 3. Simplifying this expression, we find that y = 2 - 3, which results in y = -1. Consequently, when x = 1, y = -1. This point (1, -1) lies on the line represented by the equation, providing another piece of the puzzle in understanding the relationship between 'x' and 'y'.

The substitution of x = 1 into the equation y = 2x - 3 allows us to see the direct impact of the slope. The term '2x' becomes '2(1)', which equals 2. This value is then reduced by the constant term '-3', resulting in y = -1. The difference between y = -3 when x = 0 and y = -1 when x = 1 demonstrates the effect of the slope: for every increase of 1 in 'x', 'y' increases by 2. This fundamental characteristic of linear equations is clearly illustrated in this example.

Returning to our earnings scenario, if 'x' is hours worked and 'y' is earnings after expenses, then when x = 1, y = -1 suggests that after working one hour, the individual still has a debt of $1. This highlights the initial expense of $3 and the hourly earnings of $2. This concrete example reinforces the connection between the mathematical equation and real-world situations.

Determining Y for X = 2

Next, we explore the scenario where 'x' equals 2. This allows us to further examine the relationship between 'x' and 'y' as 'x' increases. Substituting 'x = 2' into the equation y = 2x - 3, we get y = 2(2) - 3. Simplifying this expression, we find that y = 4 - 3, which equals 1. Therefore, when x = 2, y = 1. The point (2, 1) further solidifies our understanding of the line's trajectory.

Substituting x = 2 into the equation y = 2x - 3 reinforces the concept of the slope. The term '2x' now becomes '2(2)', which equals 4. This value, reduced by the constant term '-3', results in y = 1. We see a further increase of 2 in the value of 'y' compared to when x = 1, again demonstrating the consistent effect of the slope. This consistent pattern allows us to predict the value of 'y' for any given 'x'.

In the earnings context, when x = 2, y = 1 signifies that after working two hours, the individual has earned a net profit of $1. This means they have overcome the initial expense and are now earning a positive amount. This point is significant as it represents the break-even point, where earnings surpass expenses. This illustrates how solving for 'y' at different 'x' values can provide valuable insights into the financial dynamics of the situation.

Calculating Y for X = 3

Finally, let's investigate the case where 'x' equals 3. This final substitution will provide us with a comprehensive understanding of the equation's behavior within the specified range of 'x' values. Substituting 'x = 3' into the equation y = 2x - 3, we obtain y = 2(3) - 3. Simplifying this expression, we find that y = 6 - 3, which equals 3. Consequently, when x = 3, y = 3. The point (3, 3) further extends our understanding of the line's path and its relationship between 'x' and 'y'.

The substitution of x = 3 into the equation y = 2x - 3 provides a conclusive illustration of the slope's impact. The term '2x' becomes '2(3)', which equals 6. Subtracting the constant term '-3', we arrive at y = 3. This further increase of 2 in the value of 'y' compared to when x = 2 reaffirms the consistent nature of the linear relationship. The slope of 2 dictates the constant rate of change between 'x' and 'y'.

Relating this to the earnings example, when x = 3, y = 3 indicates that after working three hours, the individual has earned a net profit of $3. This continued increase in earnings further highlights the profitability of the situation after the initial expenses have been covered. This final calculation reinforces the practical applicability of solving linear equations in real-world scenarios.

Conclusion: Unveiling the Interplay of X and Y

Through our exploration of the equation y = 2x - 3, we have successfully determined the value of 'y' for various values of 'x'. By substituting x = 0, 1, 2, and 3 into the equation, we found the corresponding values of y to be -3, -1, 1, and 3, respectively. These findings demonstrate the linear relationship between 'x' and 'y', where 'y' changes consistently in response to changes in 'x'. This understanding of linear equations is fundamental to various mathematical and real-world applications.

In summary, by systematically substituting different values for 'x' into the equation y = 2x - 3, we have unveiled the corresponding 'y' values and gained a deeper appreciation for the relationship between these variables. The process involved a straightforward substitution and simplification, highlighting the power of basic algebraic manipulation. Furthermore, by connecting the equation to a real-world earnings scenario, we have demonstrated the practical relevance of solving linear equations. This exercise underscores the importance of understanding mathematical concepts and their ability to model and predict real-world phenomena. The ability to solve such equations is a cornerstone of mathematical literacy and a valuable skill in various fields of study and professional endeavors. The consistent increase in the 'y' value for each unit increase in 'x' emphasizes the importance of understanding the slope in linear equations, making it easier to predict outcomes and make informed decisions based on the relationship between variables.