X And Y Intercepts Of 4x - 2y = -12 Step-by-Step Guide

In the realm of mathematics, particularly in coordinate geometry, intercepts play a crucial role in understanding the behavior and characteristics of lines and curves. The x-intercept and y-intercept are specific points where a line or curve intersects the x-axis and y-axis, respectively. These intercepts provide valuable information about the graph of the equation and its relationship to the coordinate axes. In this comprehensive guide, we will delve into the process of identifying the x-intercept and y-intercept of the line represented by the equation 4x - 2y = -12. Understanding these intercepts is fundamental to graphing linear equations and interpreting their significance in various mathematical contexts.

To master the concept of intercepts, we must first define what they represent graphically. The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. Conversely, the y-intercept is the point where the line crosses the y-axis, and at this point, the x-coordinate is always zero. This simple yet crucial understanding forms the basis for our method of finding these intercepts. To find the x-intercept, we set y = 0 in the equation and solve for x. Similarly, to find the y-intercept, we set x = 0 and solve for y. This method is universally applicable to any linear equation and provides a straightforward way to determine the points where the line interacts with the coordinate axes. The ability to quickly and accurately find these intercepts is a cornerstone of linear algebra and graphical analysis. It not only helps in visualizing the line's position but also aids in solving systems of equations and understanding the behavior of linear functions. Furthermore, mastering this skill lays the groundwork for more advanced concepts in calculus and analytical geometry, where intercepts play a crucial role in sketching curves and understanding the behavior of functions.

The x-intercept is the point where the line intersects the x-axis. To find this point, we set y = 0 in the equation 4x - 2y = -12. This substitution simplifies the equation, allowing us to solve for x, which will give us the x-coordinate of the x-intercept. By setting y to zero, we are essentially focusing on the scenario where the line's height above or below the x-axis is zero, pinpointing the exact location where it crosses this axis. The process involves basic algebraic manipulation, a cornerstone of mathematical problem-solving. We replace the variable y with the numerical value 0, transforming the original equation into a simpler one that involves only one variable, x. This step is crucial because it allows us to isolate x and determine its value, which directly corresponds to the x-coordinate of the intercept. Once we have this value, we pair it with the y-coordinate (which we know is 0) to define the x-intercept as a specific point on the coordinate plane. This point is of significant importance as it provides one of the two essential landmarks needed to accurately graph the line. In addition to graphical representation, the x-intercept also holds practical significance in real-world applications, such as determining break-even points in economics or finding the roots of a function in calculus. Understanding how to calculate the x-intercept efficiently and accurately is therefore a fundamental skill in mathematics, with applications extending far beyond the classroom. Let's delve into the step-by-step calculation to solidify this concept.

Substituting y = 0 into the equation 4x - 2y = -12, we get:

4x - 2(0) = -12

This simplifies to:

4x = -12

To solve for x, we divide both sides of the equation by 4:

x = -12 / 4

x = -3

Therefore, the x-intercept is the point (-3, 0). This result indicates that the line crosses the x-axis at the point where x is -3 and y is 0. Graphically, this means we can locate this point on the coordinate plane by moving 3 units to the left of the origin along the x-axis. This point serves as a crucial reference for sketching the line, as it provides one of the two essential points needed to define its position and orientation. Understanding the significance of this intercept extends beyond mere plotting; it also plays a vital role in interpreting the behavior of the linear function. For instance, in real-world scenarios, the x-intercept might represent a point of equilibrium or a zero-crossing, offering valuable insights into the system being modeled. The ability to accurately calculate and interpret the x-intercept is therefore a cornerstone of mathematical literacy, with applications spanning various fields and disciplines. The negative value of x further tells us that the intersection occurs on the left side of the y-axis, which is a key piece of information for visualizing the line's position in the coordinate plane.

The y-intercept is the point where the line intersects the y-axis. To find this point, we set x = 0 in the equation 4x - 2y = -12. This is the inverse of the process we used to find the x-intercept, and it's a crucial step in fully understanding the line's behavior. Setting x to zero effectively isolates the term involving y, allowing us to solve for the y-coordinate of the point where the line crosses the vertical axis. This point provides another key reference for graphing the line, and it often holds significant meaning in practical applications. For instance, in a linear cost function, the y-intercept might represent the fixed costs, the costs that are incurred even when the production level is zero. In a supply and demand model, it could represent the price at which the quantity supplied or demanded is zero. The process of finding the y-intercept mirrors that of the x-intercept, but the different setup allows us to extract different information about the line. It's a demonstration of the power of algebraic manipulation to reveal different aspects of the same underlying relationship. The y-intercept, combined with the x-intercept, provides a complete picture of how the line interacts with the coordinate axes, and it's a fundamental concept in linear algebra and graphical analysis. Let's proceed with the calculation to solidify this understanding.

Substituting x = 0 into the equation 4x - 2y = -12, we get:

4(0) - 2y = -12

This simplifies to:

-2y = -12

To solve for y, we divide both sides of the equation by -2:

y = -12 / -2

y = 6

Thus, the y-intercept is the point (0, 6). This means the line intersects the y-axis at the point where y is 6 and x is 0. Graphically, this point is located 6 units above the origin on the y-axis. Just like the x-intercept, the y-intercept is a vital reference point for sketching the line. It provides a second anchor, along with the x-intercept, that allows us to accurately draw the line on the coordinate plane. The value of the y-intercept also carries significance beyond graphical representation. In many real-world contexts, it represents the initial value or starting point of a linear relationship. For example, in a savings account model, the y-intercept might represent the initial deposit, and the slope of the line would represent the rate of interest. Understanding the y-intercept in these contexts can provide valuable insights into the situation being modeled, making its calculation and interpretation a key skill in mathematical problem-solving. In our specific case, the positive value of 6 indicates that the line intersects the y-axis above the origin, which further refines our understanding of the line's position in the coordinate plane.

In summary, by setting y = 0 and solving for x, we found the x-intercept to be (-3, 0). Conversely, by setting x = 0 and solving for y, we determined the y-intercept to be (0, 6). These two points uniquely define the line 4x - 2y = -12 on the coordinate plane. Understanding the significance of intercepts goes beyond the ability to plot a graph. Intercepts are fundamental in various mathematical applications, including solving systems of equations, analyzing linear functions, and modeling real-world phenomena. The x-intercept, as the point where the line crosses the x-axis, often represents a crucial threshold or equilibrium point in the system being modeled. For instance, in a profit-loss scenario, the x-intercept might represent the break-even point, where the company neither makes a profit nor incurs a loss. Similarly, the y-intercept, as the point where the line crosses the y-axis, frequently represents the initial state or starting value of the system. In a distance-time graph, the y-intercept might represent the initial distance of an object from a reference point. The ability to quickly and accurately identify and interpret intercepts is therefore a valuable skill in both academic and professional settings. It allows for a deeper understanding of mathematical relationships and their applications in real-world contexts. The process of finding intercepts also reinforces the fundamental algebraic techniques of substitution and equation-solving, which are essential building blocks for more advanced mathematical concepts. By mastering the concept of intercepts, students develop a stronger foundation in mathematics and gain a powerful tool for problem-solving and critical thinking.

Therefore, the correct answer is:

A. The x-intercept is (-3, 0) and the y-intercept is (0, 6).

Understanding intercepts is a cornerstone of linear algebra and graphical analysis. By mastering the techniques to find x and y-intercepts, you gain valuable insights into the behavior of linear equations and their graphical representations. This knowledge not only helps in solving mathematical problems but also provides a solid foundation for more advanced topics in mathematics and related fields. The ability to interpret intercepts in real-world contexts enhances problem-solving skills and critical thinking abilities. This fundamental concept serves as a building block for a deeper understanding of mathematical relationships and their applications in various disciplines.