Solving For Y In Linear Equations An Exploration Of Y = -3x + 2

In the realm of mathematics, understanding linear equations is a fundamental skill. Linear equations, often expressed in the form y = mx + b, represent straight lines on a graph and play a crucial role in various fields, including science, engineering, and economics. Today, we will delve into the linear equation y = -3x + 2 and explore how to determine the value of y for specific values of x. This exercise will not only reinforce our understanding of linear equations but also highlight their practical applications in solving real-world problems. We will methodically substitute different x values into the equation and calculate the corresponding y values, providing a step-by-step guide for anyone looking to master this essential mathematical concept. By the end of this exploration, you will have a solid grasp of how to manipulate linear equations and confidently find solutions for any given x value.

Before we dive into the calculations, let's break down the linear equation y = -3x + 2. This equation is in slope-intercept form, a standard way to represent a linear equation. In this form, y represents the dependent variable, x represents the independent variable, -3 is the slope (denoted as m), and 2 is the y-intercept (denoted as b). The slope indicates the rate of change of y with respect to x, meaning for every unit increase in x, y decreases by 3 units. The y-intercept is the point where the line crosses the y-axis, which in this case is at y = 2. Understanding these components is crucial for visualizing and interpreting the equation. The slope-intercept form allows us to quickly identify key characteristics of the line, such as its steepness and where it intersects the y-axis. This information is invaluable in various applications, from predicting trends to designing structures. By grasping the significance of each part of the equation, we can effectively use it to solve for unknown values and make informed decisions.

Let's tackle our first problem: finding the value of y when x = 3. To do this, we substitute x = 3 into the equation y = -3x + 2. This means replacing the x in the equation with the number 3. So, our equation becomes y = -3(3) + 2. Now, we perform the multiplication first, following the order of operations. -3 multiplied by 3 equals -9, so the equation simplifies to y = -9 + 2. Finally, we add -9 and 2, which gives us y = -7. Therefore, when x = 3, the value of y is -7. This simple substitution and calculation demonstrate the core process of solving linear equations for specific points. It's a fundamental skill that allows us to map the relationship between x and y and understand how they interact within the equation. By mastering this process, we can confidently determine the y value for any given x value in a linear equation.

Next, we'll find the value of y when x = -5/3. This problem involves working with a fraction, but the process remains the same. We substitute x = -5/3 into the equation y = -3x + 2. This gives us y = -3(-5/3) + 2. When we multiply -3 by -5/3, the negative signs cancel out, and the 3 in the numerator and denominator also cancel out, leaving us with 5. So, the equation simplifies to y = 5 + 2. Adding 5 and 2 gives us y = 7. Therefore, when x = -5/3, the value of y is 7. This example demonstrates that even when dealing with fractions, the fundamental principle of substitution remains the same. It also highlights the importance of understanding how to multiply and simplify fractions to arrive at the correct answer. By practicing these types of problems, we become more comfortable and proficient in working with linear equations involving various types of numbers.

Our final problem is to find the value of y when x = 0. Substituting x = 0 into the equation y = -3x + 2, we get y = -3(0) + 2. Any number multiplied by 0 is 0, so -3(0) equals 0. This simplifies the equation to y = 0 + 2. Adding 0 and 2 gives us y = 2. Therefore, when x = 0, the value of y is 2. This result is particularly significant because it represents the y-intercept of the line. The y-intercept is the point where the line crosses the y-axis, and it occurs when x is equal to 0. Finding the y-intercept is a crucial step in graphing a linear equation and understanding its behavior. It provides a fixed point from which we can plot the rest of the line using the slope. By recognizing the significance of the y-intercept, we gain a deeper understanding of linear equations and their graphical representation.

In conclusion, we have successfully found the values of y for the given x values in the equation y = -3x + 2. We found that when x = 3, y = -7; when x = -5/3, y = 7; and when x = 0, y = 2. This exercise has demonstrated the fundamental process of substituting values into a linear equation and solving for the unknown variable. Mastering this skill is essential for understanding and applying linear equations in various contexts. From graphing lines to solving real-world problems, the ability to manipulate linear equations is a valuable asset. By practicing these types of problems, we build confidence and proficiency in working with mathematical equations. The understanding gained from this exploration will serve as a strong foundation for more advanced mathematical concepts and applications. Remember, practice makes perfect, so continue to explore and solve linear equations to further enhance your skills.