Finding The Y-intercept Of The Equation -2x + 5y = -10

The $y$-intercept is a fundamental concept in algebra and coordinate geometry. It represents the point where a line or curve intersects the $y$-axis on a graph. Understanding how to find the $y$-intercept is crucial for graphing linear equations and analyzing their behavior. In this article, we will delve into the process of determining the $y$-intercept for the equation $-2x + 5y = -10$. This equation represents a linear relationship between $x$ and $y$, and our goal is to find the $y$-coordinate of the point where this line crosses the $y$-axis. To accomplish this, we will employ a simple yet powerful technique: setting $x$ to zero and solving for $y$. This method leverages the fact that any point on the $y$-axis has an $x$-coordinate of 0. By substituting $x = 0$ into the equation, we eliminate the $x$ term and are left with an equation involving only $y$. Solving this equation will directly give us the $y$-coordinate of the $y$-intercept. This article will provide a step-by-step guide to this process, ensuring that you not only understand the mechanics of finding the $y$-intercept but also grasp the underlying concept and its significance in linear equations.

Understanding the $y$-intercept

In the realm of coordinate geometry, the $y$-intercept holds a significant position as a key feature of a graph. Specifically, the $y$-intercept is the point where a line or curve intersects the $y$-axis. The $y$-axis, which is the vertical axis in the Cartesian coordinate system, represents all points where the $x$-coordinate is zero. Therefore, any point lying on the $y$-axis has the form $(0, y)$, where $y$ is the $y$-coordinate. The $y$-intercept is the specific $y$-coordinate of the point where the graph crosses this axis. Identifying the $y$-intercept is crucial for several reasons. First, it provides a visual anchor point on the graph, making it easier to sketch the line or curve. Second, it gives valuable information about the relationship between the variables represented by the equation. In many real-world applications, the $y$-intercept has a meaningful interpretation. For instance, in a linear equation representing the cost of a service, the $y$-intercept might represent the fixed cost or initial fee. Similarly, in a graph of population growth, the $y$-intercept could indicate the initial population size. Therefore, understanding the concept of the $y$-intercept and how to find it is essential for both mathematical proficiency and practical problem-solving. By grasping this concept, you can gain deeper insights into the behavior of functions and their graphical representations. The $y$-intercept, therefore, serves as a fundamental building block for more advanced topics in algebra and calculus.

The Equation -2x + 5y = -10

The equation $-2x + 5y = -10$ represents a linear relationship between the variables $x$ and $y$. This equation is in standard form, which is $Ax + By = C$, where $A$, $B$, and $C$ are constants. In this specific equation, $A = -2$, $B = 5$, and $C = -10$. Linear equations of this form create straight lines when graphed on the coordinate plane. The coefficients $A$ and $B$ determine the slope and orientation of the line, while the constant $C$ influences its position on the plane. To fully understand the behavior of this line, we need to determine key features such as its slope and intercepts. The slope, which represents the steepness and direction of the line, can be found by rearranging the equation into slope-intercept form ($y = mx + b$), where $m$ is the slope. The intercepts, which are the points where the line crosses the $x$ and $y$ axes, provide additional anchor points for graphing and analysis. The $y$-intercept, as discussed earlier, is the point where the line intersects the $y$-axis, and it occurs when $x = 0$. The $x$-intercept, conversely, is the point where the line intersects the $x$-axis, and it occurs when $y = 0$. By finding these intercepts, we can easily sketch the graph of the line and gain a visual understanding of the relationship between $x$ and $y$ as defined by the equation $-2x + 5y = -10$. This linear equation, therefore, is a foundational element in understanding linear relationships and their graphical representations.

Step-by-Step Guide to Finding the $y$-intercept

To find the $y$-intercept of the equation $-2x + 5y = -10$, we follow a straightforward, step-by-step process. This process leverages the fundamental property of the $y$-intercept: it is the point where the line crosses the $y$-axis, and at this point, the $x$-coordinate is always zero. Therefore, our first step is to substitute $x = 0$ into the equation. This substitution eliminates the $x$ term and simplifies the equation, leaving us with an equation involving only $y$. The simplified equation will be of the form $By = C$, where $B$ and $C$ are constants. Our next step is to solve this equation for $y$. This involves isolating $y$ on one side of the equation, which is typically done by dividing both sides of the equation by the coefficient of $y$ (in this case, $B$). Once we have solved for $y$, the resulting value is the $y$-coordinate of the $y$-intercept. The $y$-intercept is then expressed as the ordered pair $(0, y)$, where $y$ is the value we just calculated. This ordered pair represents the point on the graph where the line intersects the $y$-axis. By following these steps, we can systematically determine the $y$-intercept for any linear equation. This method is not only effective but also provides a clear understanding of the underlying concept. The ability to find the $y$-intercept is a crucial skill in algebra and is essential for graphing linear equations and analyzing their properties. This step-by-step guide ensures that you can confidently find the $y$-intercept for various linear equations.

Step 1: Substitute $x = 0$ into the Equation

The first critical step in finding the $y$-intercept of the equation $-2x + 5y = -10$ is to substitute $x = 0$ into the equation. This substitution is based on the fundamental understanding that the $y$-intercept is the point where the line intersects the $y$-axis, and at any point on the $y$-axis, the $x$-coordinate is always zero. By setting $x$ to zero, we are essentially looking for the $y$-value that corresponds to the point where the line crosses the $y$-axis. When we substitute $x = 0$ into the equation $-2x + 5y = -10$, we replace the $x$ term with 0, resulting in the equation $-2(0) + 5y = -10$. The term $-2(0)$ simplifies to 0, effectively eliminating the $x$ term from the equation. This simplification is crucial because it leaves us with an equation that contains only the variable $y$, making it much easier to solve for $y$. The equation now becomes $0 + 5y = -10$, which simplifies further to $5y = -10$. This simplified equation represents a direct relationship between $y$ and a constant, allowing us to isolate $y$ and find its value. The substitution step is a key technique in finding intercepts and is widely used in algebra and coordinate geometry. By understanding this step, you can confidently simplify equations and move closer to finding the $y$-intercept. This initial substitution is the foundation for the subsequent steps in the process.

Step 2: Solve for $y$

After substituting $x = 0$ into the equation $-2x + 5y = -10$, we arrived at the simplified equation $5y = -10$. The next step is to solve this equation for $y$. Solving for $y$ means isolating $y$ on one side of the equation, which will give us the $y$-coordinate of the $y$-intercept. To isolate $y$, we need to undo the operation that is being applied to it. In this case, $y$ is being multiplied by 5. The inverse operation of multiplication is division, so we will divide both sides of the equation by 5. Dividing both sides of the equation $5y = -10$ by 5 gives us rac{5y}{5} = rac{-10}{5}. On the left side of the equation, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with just $y$. On the right side of the equation, we have rac{-10}{5}, which simplifies to $-2$. Therefore, the equation becomes $y = -2$. This result tells us that the $y$-coordinate of the $y$-intercept is $-2$. This means that the line intersects the $y$-axis at the point where $y = -2$. Solving for $y$ is a fundamental algebraic skill, and it is crucial for finding intercepts and understanding the behavior of equations. By mastering this step, you can confidently determine the $y$-coordinate of the $y$-intercept.

Step 3: Express the $y$-intercept as an Ordered Pair

Now that we have found the $y$-coordinate of the $y$-intercept, which is $y = -2$, the final step is to express this intercept as an ordered pair. An ordered pair is a pair of numbers written in the form $(x, y)$, where $x$ represents the $x$-coordinate and $y$ represents the $y$-coordinate. As we discussed earlier, the $y$-intercept is the point where the line intersects the $y$-axis, and at this point, the $x$-coordinate is always zero. Therefore, the $x$-coordinate of the $y$-intercept is 0. We have already found that the $y$-coordinate of the $y$-intercept is $-2$. Combining these two pieces of information, we can express the $y$-intercept as the ordered pair $(0, -2)$. This ordered pair represents a specific point on the coordinate plane, and it is the point where the line defined by the equation $-2x + 5y = -10$ crosses the $y$-axis. Expressing the $y$-intercept as an ordered pair is important because it clearly and concisely identifies the location of the intercept on the graph. It provides a visual anchor point that can be used to sketch the graph of the line or to analyze its behavior. The ordered pair notation is a standard convention in mathematics, and understanding how to use it is essential for communicating mathematical ideas effectively. By expressing the $y$-intercept as an ordered pair, we have completed the process of finding the $y$-intercept for the given equation.

The $y$-intercept for -2x + 5y = -10

In conclusion, by following the steps outlined above, we have successfully determined the $y$-intercept for the equation $-2x + 5y = -10$. We began by understanding the concept of the $y$-intercept as the point where a line intersects the $y$-axis. We then substituted $x = 0$ into the equation, which simplified it to $5y = -10$. Solving for $y$, we found that $y = -2$. Finally, we expressed the $y$-intercept as the ordered pair $(0, -2)$. This ordered pair represents the specific point on the coordinate plane where the line crosses the $y$-axis. Understanding how to find the $y$-intercept is a crucial skill in algebra and coordinate geometry. It allows us to identify a key feature of a linear equation and its graph, providing valuable information about the relationship between the variables $x$ and $y$. The $y$-intercept, along with the slope and other intercepts, helps us to fully understand the behavior of a linear equation and its graphical representation. This process not only provides a specific answer but also reinforces the fundamental concepts of linear equations and coordinate geometry. By mastering these concepts, you can confidently tackle more complex problems and gain a deeper understanding of mathematical relationships.

Importance of $y$-intercept

The $y$-intercept is a crucial concept in mathematics, particularly in algebra and coordinate geometry, due to its significant role in understanding and analyzing linear equations and their graphs. The $y$-intercept provides a fundamental anchor point on the graph of a line or curve, marking the location where the graph intersects the $y$-axis. This intersection point offers valuable insights into the behavior of the equation and the relationship between the variables it represents. One of the primary reasons the $y$-intercept is so important is its role in graphing linear equations. Knowing the $y$-intercept, along with the slope, allows us to easily sketch the graph of a line. The $y$-intercept serves as a starting point on the $y$-axis, and the slope dictates the direction and steepness of the line. This makes graphing linear equations much simpler and more intuitive. Furthermore, the $y$-intercept often has a meaningful interpretation in real-world applications. In many scenarios, the $y$-intercept represents the initial value or starting point of a situation. For example, in a linear equation representing the cost of a service, the $y$-intercept might represent the fixed cost or initial fee. Similarly, in a graph of population growth, the $y$-intercept could indicate the initial population size. This real-world relevance makes the $y$-intercept a valuable tool for problem-solving and decision-making. In addition to its practical applications, the $y$-intercept is also essential for understanding more advanced mathematical concepts. It plays a crucial role in the study of functions, calculus, and other areas of mathematics. The ability to find and interpret the $y$-intercept is a foundational skill that is necessary for success in these higher-level courses. Therefore, understanding the importance of the $y$-intercept is crucial for both mathematical proficiency and practical problem-solving. It is a fundamental concept that provides valuable insights into the behavior of equations and their graphical representations.