Finding The Y-Intercept Of 2x - 3y = -6 A Step-by-Step Guide

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Finding the y-intercept of a linear equation is a fundamental concept in algebra and is essential for understanding and graphing linear functions. In the equation given, 2x - 3y = -6, our primary goal is to determine the point where the line intersects the y-axis. The y-intercept is the y-coordinate of the point where the line crosses the y-axis. At this point, the x-coordinate is always zero. Therefore, to find the y-intercept, we set x = 0 in the equation and solve for y. This approach simplifies the equation, allowing us to isolate y and find its value at the y-intercept.

When we substitute x = 0 into the equation 2x - 3y = -6, we get 2(0) - 3y = -6, which simplifies to -3y = -6. To solve for y, we divide both sides of the equation by -3, resulting in y = (-6) / (-3). This simplifies to y = 2. Thus, the y-intercept is 2, and the point where the line intersects the y-axis is (0, 2). Understanding how to find the y-intercept is crucial for various mathematical applications, including graphing lines, solving systems of equations, and analyzing linear relationships in real-world scenarios. The y-intercept provides a starting point for graphing the line and helps in interpreting the linear relationship between the variables x and y. Mastering this concept is a stepping stone to more advanced topics in algebra and calculus.

Furthermore, the y-intercept holds significant practical value in various contexts. For instance, in business, if the equation represents a cost function, the y-intercept might represent the fixed costs, which are the costs incurred even when no units are produced. In physics, if the equation represents the motion of an object, the y-intercept might represent the initial position of the object. Therefore, understanding how to find and interpret the y-intercept is not just an algebraic skill but also a valuable tool for problem-solving in diverse fields. This concept is a building block for understanding linear functions and their applications in real-world scenarios, making it an essential skill for students and professionals alike.

Step-by-Step Solution

To find the y-intercept of the line whose equation is given by 2x - 3y = -6, we follow a straightforward step-by-step approach. This method involves setting the x-coordinate to zero and solving the equation for the y-coordinate. The y-intercept is the point where the line intersects the y-axis, and at this point, the x-coordinate is always zero. This concept is fundamental in understanding linear equations and their graphical representations. By systematically setting x = 0 and solving for y, we can easily determine the y-intercept. This process not only helps in graphing the line but also provides valuable information about the linear relationship represented by the equation.

  1. Set x = 0: To find the y-intercept, we begin by setting the value of x to zero in the given equation. This is because the y-intercept is the point where the line crosses the y-axis, and at any point on the y-axis, the x-coordinate is always zero. Substituting x = 0 into the equation 2x - 3y = -6 is the first step in isolating y and finding its value at the y-intercept. This substitution simplifies the equation, making it easier to solve for y. The equation becomes 2(0) - 3y = -6, which further simplifies to -3y = -6. This step is crucial as it transforms the equation into a form where y can be easily determined.

  2. Substitute x = 0 into the equation: We substitute x = 0 into the equation 2x - 3y = -6. This yields 2(0) - 3y = -6. This substitution is a critical step in finding the y-intercept because it simplifies the equation by eliminating the x term, allowing us to focus solely on solving for y. The resulting equation, -3y = -6, is a simple linear equation in one variable, which can be easily solved by dividing both sides by the coefficient of y. This step is a direct application of the principle that the y-intercept occurs where x = 0, making it a fundamental technique in linear algebra.

  3. Simplify the equation: After substituting x = 0, the equation becomes 2(0) - 3y = -6, which simplifies to -3y = -6. This simplification is a crucial step in isolating the variable y. By performing the multiplication 2(0) = 0, we eliminate the x term from the equation, leaving us with a straightforward equation involving only y. This simplified form allows us to easily solve for y by dividing both sides of the equation by the coefficient of y, which is -3 in this case. The simplified equation, -3y = -6, is now in a form that can be readily solved using basic algebraic techniques.

  4. Solve for y: To solve for y in the equation -3y = -6, we divide both sides of the equation by -3. This gives us y = (-6) / (-3). Dividing both sides of the equation by the coefficient of y is a fundamental algebraic operation used to isolate the variable and find its value. In this case, dividing -6 by -3 gives us y = 2. This step is the culmination of the process of finding the y-intercept, as it provides the y-coordinate of the point where the line intersects the y-axis. The result, y = 2, is the y-intercept of the line, and it represents the value of y when x is zero.

  5. State the y-intercept: The y-intercept is the value of y when x = 0. From the previous steps, we found that y = 2. Therefore, the y-intercept of the line 2x - 3y = -6 is 2. This means that the line intersects the y-axis at the point (0, 2). Stating the y-intercept clearly is important for understanding the behavior of the line and for graphing it accurately. The y-intercept provides a key point on the line, which can be used in conjunction with the slope to sketch the graph of the line or to analyze its properties. In this case, the y-intercept of 2 indicates that the line crosses the y-axis at the point where y = 2, making it a crucial piece of information for understanding the linear relationship represented by the equation.

Alternative Methods to Find the Y-Intercept

While setting x = 0 is the most common method for finding the y-intercept, understanding alternative approaches can enhance your problem-solving skills and provide a deeper understanding of linear equations. One such method involves converting the given equation into slope-intercept form. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope of the line and b represents the y-intercept. By rearranging the equation into this form, the y-intercept can be easily identified as the constant term b. This method is particularly useful as it provides additional information about the line, such as its slope, which can be used for graphing and analysis.

To convert the equation 2x - 3y = -6 into slope-intercept form, we need to isolate y on one side of the equation. First, we subtract 2x from both sides, which gives us -3y = -2x - 6. Next, we divide both sides of the equation by -3 to solve for y. This results in y = (2/3)x + 2. In this form, it is clear that the slope m is 2/3 and the y-intercept b is 2. Thus, the y-intercept can be directly read off from the equation once it is in slope-intercept form. This method not only provides the y-intercept but also gives valuable information about the slope of the line, making it a comprehensive approach for understanding linear equations. Understanding both methods—setting x = 0 and converting to slope-intercept form—can provide a more robust understanding of linear equations and their properties.

Another alternative method involves graphing the line. By plotting the line on a coordinate plane, the y-intercept can be visually identified as the point where the line crosses the y-axis. To graph the line, you can find at least two points that satisfy the equation. One point can be the y-intercept itself, which we already know how to find. Another point can be found by setting y = 0 and solving for x, which gives us the x-intercept. Alternatively, you can choose any arbitrary value for x, substitute it into the equation, and solve for y. Once you have two points, you can plot them on the coordinate plane and draw a line through them. The point where this line intersects the y-axis is the y-intercept. While this method is more visual and can be helpful for understanding the concept of the y-intercept, it may not be as precise as the algebraic methods, especially if the y-intercept is not an integer. However, it provides a valuable way to check your algebraic solution and reinforces the connection between the equation and its graphical representation.

Real-World Applications of Y-Intercept

The y-intercept is not just a mathematical concept; it has significant real-world applications across various fields. Understanding the y-intercept can provide valuable insights in situations ranging from business and economics to science and engineering. In practical terms, the y-intercept often represents an initial value or a fixed component in a linear relationship. For instance, in a cost function, the y-intercept might represent the fixed costs, which are the costs that do not change with the level of production. Recognizing and interpreting the y-intercept in these contexts can lead to better decision-making and problem-solving.

In the context of business and economics, consider a linear cost function represented by the equation C = mx + b, where C is the total cost, x is the number of units produced, m is the variable cost per unit, and b is the y-intercept. In this scenario, the y-intercept b represents the fixed costs, such as rent, utilities, and salaries, which are incurred regardless of the number of units produced. Understanding the fixed costs is crucial for businesses as it helps in determining the break-even point and making pricing decisions. Similarly, in a revenue function represented by R = px, where R is the total revenue, p is the price per unit, and x is the number of units sold, there is no y-intercept (it is zero), indicating that there is no revenue if no units are sold. Comparing the cost and revenue functions can provide a comprehensive view of the business's financial performance. The y-intercept, therefore, plays a critical role in financial analysis and decision-making in business.

In science and engineering, the y-intercept can represent initial conditions or baseline values. For example, in physics, if we consider the equation of motion d = vt + dâ‚€, where d is the distance traveled, v is the velocity, t is the time, and dâ‚€ is the initial distance, the y-intercept dâ‚€ represents the object's initial position. This is the position of the object at time t = 0. Similarly, in chemistry, if we consider a reaction rate equation, the y-intercept might represent the initial concentration of a reactant. In engineering, the y-intercept can represent a baseline measurement or a starting point in a system. Understanding these initial conditions is crucial for modeling and predicting the behavior of systems over time. The y-intercept provides a starting point for analysis and helps in understanding the dynamics of the system. Therefore, the y-intercept is a valuable tool in various scientific and engineering applications, providing insights into initial states and fixed components in linear relationships.

In conclusion, finding the y-intercept of the line 2x - 3y = -6 is a fundamental algebraic skill with practical applications in various fields. By setting x = 0 and solving for y, we determined that the y-intercept is 2, meaning the line intersects the y-axis at the point (0, 2). We also explored alternative methods such as converting the equation to slope-intercept form, which not only confirms the y-intercept but also provides additional information about the line, such as its slope. Understanding the y-intercept is crucial for graphing linear equations and interpreting linear relationships in real-world scenarios. Its significance extends beyond the classroom, playing a vital role in fields like business, science, and engineering, where it often represents initial values or fixed components in linear models. Mastering the concept of the y-intercept is therefore an essential step in developing a strong foundation in mathematics and its applications.