Graphing The Equation Y-6=-(x-4) A Step-by-Step Guide

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In the realm of mathematics, understanding the relationship between equations and their graphical representations is a fundamental skill. Linear equations, with their straight-line graphs, serve as a cornerstone for more advanced mathematical concepts. In this article, we will delve into the equation y-6=-(x-4), dissecting its components and determining which graph accurately depicts its characteristics. Our exploration will encompass rewriting the equation in slope-intercept form, identifying key features like slope and intercepts, and ultimately selecting the correct graphical representation.

Unveiling the Equation: Transforming to Slope-Intercept Form

To effectively analyze the equation y-6=-(x-4), our initial step involves transforming it into the slope-intercept form, which is y = mx + b. This form provides immediate insights into the line's slope (m) and y-intercept (b). Let's embark on this transformation:

  1. Distribute the negative sign: y - 6 = -x + 4
  2. Isolate y: Add 6 to both sides of the equation: y = -x + 10

Now, the equation is elegantly expressed in slope-intercept form: y = -x + 10. This form reveals that the slope (m) is -1 and the y-intercept (b) is 10. The slope of -1 indicates that the line descends as it moves from left to right, while the y-intercept of 10 signifies that the line intersects the y-axis at the point (0, 10).

Key Features: Slope and Intercepts Demystified

The slope of a line quantifies its steepness and direction. A negative slope, as in our case (-1), indicates a downward trajectory, meaning the line falls as we move from left to right. The magnitude of the slope reflects the rate of change in y for every unit change in x. In simpler terms, for every 1 unit we move to the right along the x-axis, the line descends 1 unit along the y-axis.

The intercepts are the points where the line intersects the coordinate axes. The y-intercept, which we've already identified as (0, 10), is the point where the line crosses the y-axis. To find the x-intercept, we set y to 0 in the equation and solve for x:

  • 0 = -x + 10
  • x = 10

Thus, the x-intercept is (10, 0). These intercepts provide crucial anchor points for accurately plotting the line on a graph. To summarize, for the equation y = -x + 10:

  • Slope: -1
  • Y-intercept: (0, 10)
  • X-intercept: (10, 0)

Graph Analysis: Visualizing the Equation

Now that we've deciphered the equation's key features, we can analyze the provided graph options. Option A describes a linear graph function with a diagonal line passing through the second, third, and fourth quadrants and intercepts the x-axis at (-2,0) and y-axis at (0,-2). This description does not align with our calculated intercepts of (10, 0) and (0, 10), nor does it reflect the negative slope of -1. The intercepts provided in option A are negative, indicating the line would be in different quadrants than our equation suggests. Therefore, option A is an incorrect representation of the equation y-6=-(x-4).

To visually confirm, we can sketch the graph using the information we've gathered. Start by plotting the intercepts (10, 0) and (0, 10). Then, connect these points with a straight line. The line should have a negative slope, descending from left to right. This mental picture or quick sketch serves as a valuable tool for verifying the correctness of a given graph.

Conclusion: Selecting the Accurate Representation

By transforming the equation into slope-intercept form, meticulously identifying the slope and intercepts, and carefully analyzing the provided graph descriptions, we've arrived at a clear understanding of the equation y-6=-(x-4). Option A, with its negative intercepts, does not align with the characteristics of our equation. The correct graph should depict a line with a negative slope, intersecting the y-axis at (0, 10) and the x-axis at (10, 0). This process highlights the power of combining algebraic manipulation with graphical interpretation to accurately represent mathematical relationships.

Linear equations form the bedrock of algebra and are essential for understanding various mathematical and real-world phenomena. Mastering the art of solving linear equations opens doors to a deeper comprehension of more complex mathematical concepts. In this comprehensive guide, we will explore the fundamentals of linear equations, delve into different methods for solving them, and illustrate the concepts with examples.

What is a Linear Equation?

A linear equation is an algebraic equation in which the highest power of the variable is 1. A linear equation can be written in the general form: ax + b = 0, where x is the variable, and a and b are constants, with a not equal to 0. The graph of a linear equation is always a straight line. These equations can involve one or more variables, but we will focus primarily on linear equations with one variable for simplicity.

Key Characteristics of Linear Equations:

  • Variable Power: The variable (x) has an exponent of 1.
  • Straight Line Graph: When plotted on a coordinate plane, a linear equation forms a straight line.
  • Slope and Intercept: Linear equations can be expressed in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.

Methods for Solving Linear Equations

Solving a linear equation involves isolating the variable on one side of the equation. This is achieved by applying algebraic operations to both sides of the equation while maintaining equality. Several methods can be used to solve linear equations, including:

  1. Using Inverse Operations: This is the most common method for solving linear equations. It involves performing the opposite operation to undo the operations affecting the variable. For example, if the variable is being added to a constant, we subtract that constant from both sides of the equation. The inverse operations method is a cornerstone of algebraic manipulation, allowing us to systematically isolate the variable and determine its value. The power of inverse operations lies in their ability to unravel the equation, step by step, until the variable stands alone, revealing its solution. This process requires a meticulous approach, ensuring that each operation is applied equally to both sides of the equation, maintaining the delicate balance that defines equality.

    For instance, consider the equation 2x + 5 = 11. To isolate x, we first undo the addition of 5 by subtracting 5 from both sides, resulting in 2x = 6. Next, we undo the multiplication by 2 by dividing both sides by 2, yielding x = 3. This systematic application of inverse operations demonstrates the elegance and effectiveness of this method. It's not just about finding the answer; it's about understanding the underlying principles of algebraic manipulation and the preservation of equality. Mastering this method lays a solid foundation for tackling more complex equations and mathematical problems in the future.

  2. Combining Like Terms: If the equation contains like terms (terms with the same variable or constants), we combine them to simplify the equation before applying inverse operations. Combining like terms is a crucial step in simplifying complex equations, making them more manageable and easier to solve. It involves identifying terms with the same variable and exponent (e.g., 3x and 5x) or constant terms (e.g., 7 and -2) and then adding or subtracting them accordingly. This process not only reduces the number of terms in the equation but also reveals the underlying structure, paving the way for the application of inverse operations. The beauty of combining like terms lies in its ability to transform a seemingly daunting equation into a more palatable form, allowing us to focus on the essential steps required to isolate the variable.

    For example, in the equation 4x + 2 + x - 5 = 12, we can combine 4x and x to get 5x and combine 2 and -5 to get -3, resulting in the simplified equation 5x - 3 = 12. This simplification makes it easier to apply inverse operations and solve for x. Furthermore, understanding how to effectively combine like terms is a valuable skill that extends beyond solving linear equations; it is a fundamental technique used throughout algebra and other branches of mathematics.

  3. Using the Distributive Property: If the equation contains parentheses, we use the distributive property to eliminate them before solving. The distributive property is a powerful tool in algebra that allows us to simplify expressions involving parentheses. It states that for any numbers a, b, and c, a(b + c) = ab + ac. In the context of solving equations, the distributive property enables us to eliminate parentheses by multiplying the term outside the parentheses by each term inside. This process transforms the equation into a more manageable form, often revealing like terms that can be combined. The importance of the distributive property cannot be overstated; it is a fundamental skill that unlocks the door to solving a wide range of algebraic problems.

    For example, consider the equation 3(x + 2) - 5 = 10. Applying the distributive property, we multiply 3 by both x and 2, resulting in 3x + 6 - 5 = 10. This eliminates the parentheses and allows us to combine like terms and proceed with solving the equation. Moreover, the distributive property is not limited to simple linear equations; it is a cornerstone of algebraic manipulation used in more complex contexts, such as simplifying polynomial expressions and solving systems of equations. Mastering this property is essential for building a strong foundation in algebra.

Solving Linear Equations: Step-by-Step Examples

Let's illustrate the methods discussed above with some examples:

Example 1: Solve the equation 3x - 7 = 8

  1. Add 7 to both sides: 3x = 15
  2. Divide both sides by 3: x = 5

Example 2: Solve the equation 2(x + 3) - 5 = 9

  1. Apply the distributive property: 2x + 6 - 5 = 9
  2. Combine like terms: 2x + 1 = 9
  3. Subtract 1 from both sides: 2x = 8
  4. Divide both sides by 2: x = 4

Real-World Applications of Linear Equations

Linear equations are not confined to the realm of abstract mathematics; they have numerous applications in real-world scenarios. From calculating distances and speeds to determining costs and profits, linear equations provide a powerful framework for modeling and solving problems. The ability to translate real-world situations into linear equations and then solve them is a valuable skill that enhances problem-solving capabilities in various domains. Real-world applications of linear equations span a vast spectrum, demonstrating their versatility and practical utility.

For instance, consider a scenario where you want to calculate the total cost of renting a car, which includes a fixed daily rate plus a per-mile charge. This situation can be modeled using a linear equation, where the total cost is a linear function of the number of miles driven. Similarly, in physics, the relationship between distance, speed, and time for an object moving at a constant velocity can be expressed using a linear equation. In finance, linear equations are used to model simple interest calculations and to analyze linear depreciation of assets. These are just a few examples of how linear equations serve as a powerful tool for understanding and solving real-world problems. The more adept one becomes at recognizing and formulating linear relationships, the better equipped they are to make informed decisions and solve practical challenges in various aspects of life.

Conclusion: Mastering Linear Equations

Linear equations are a fundamental concept in algebra and are essential for solving a wide range of mathematical and real-world problems. By understanding the key characteristics of linear equations and mastering the various methods for solving them, you can build a strong foundation in algebra and enhance your problem-solving skills. Linear equations are not just abstract mathematical constructs; they are powerful tools that can be applied to understand and model the world around us. The journey to mastering linear equations is a rewarding one, paving the way for deeper exploration of mathematical concepts and enhancing problem-solving abilities.

It's not just about memorizing formulas and procedures; it's about developing a conceptual understanding of what linear equations represent and how they can be manipulated to reveal solutions. This understanding involves recognizing the relationship between the equation and its graphical representation, grasping the significance of slope and intercepts, and appreciating the power of inverse operations. Furthermore, the ability to translate real-world scenarios into linear equations is a crucial skill that bridges the gap between abstract mathematics and practical applications. The more one engages with linear equations, the more they become a natural and intuitive tool for analysis and problem-solving. This mastery not only unlocks the door to more advanced mathematical topics but also equips individuals with a valuable asset for navigating the complexities of the world around them.

Graphing linear equations is a fundamental skill in algebra that allows us to visualize the relationship between variables and understand the behavior of linear functions. A graph provides a visual representation of all the solutions to a linear equation, making it easier to identify key features such as intercepts and slope. In this comprehensive guide, we will explore different methods for graphing linear equations and illustrate the concepts with examples.

Understanding the Coordinate Plane

Before we delve into graphing linear equations, it's crucial to understand the coordinate plane, also known as the Cartesian plane. The coordinate plane is formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, which represents the point (0, 0). Every point on the coordinate plane can be uniquely identified by an ordered pair (x, y), where x represents the horizontal distance from the origin (positive to the right, negative to the left) and y represents the vertical distance from the origin (positive upwards, negative downwards).

The coordinate plane is divided into four quadrants, numbered I through IV in a counterclockwise direction. Quadrant I has positive x and y values, Quadrant II has negative x and positive y values, Quadrant III has negative x and y values, and Quadrant IV has positive x and negative y values. A thorough understanding the coordinate plane is the cornerstone of graphical representation in mathematics. It provides the framework for plotting points, visualizing relationships between variables, and interpreting the behavior of functions. The coordinate plane, with its two perpendicular axes, is not merely a grid; it's a powerful tool for translating abstract mathematical concepts into tangible visual forms.

Each point on the plane is uniquely defined by an ordered pair *(x, y)*, where *x* represents the horizontal displacement from the origin and *y* represents the vertical displacement. This system allows us to map any equation or relationship onto the plane, creating a visual representation that can reveal patterns, trends, and solutions. Furthermore, the coordinate plane is divided into four quadrants, each with its own distinct characteristics based on the signs of the *x* and *y* coordinates. This division provides a valuable context for interpreting the position and behavior of points and graphs. A firm grasp of the coordinate plane is not only essential for graphing linear equations but also serves as a gateway to understanding more advanced topics in mathematics, such as trigonometry, calculus, and analytic geometry.

Methods for Graphing Linear Equations

Several methods can be used to graph linear equations, each with its own advantages and applications. Here, we will explore three common methods:

  1. Using the Slope-Intercept Form: This method is particularly useful when the equation is in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. To graph the equation, we first plot the y-intercept (0, b) on the y-axis. Then, we use the slope (m) to find another point on the line. The slope can be interpreted as