Rewrite The Equation Y - 1 = (1/2)(x + 4) In Standard Form Or Slope-intercept Form.
In the realm of mathematics, equations serve as fundamental tools for describing relationships between variables. Among the various forms in which linear equations can be expressed, the standard form and the slope-intercept form hold significant importance due to their unique properties and applications. Transforming an equation from one form to another is a crucial skill in algebra, enabling us to analyze and interpret the equation's characteristics more effectively. In this article, we will delve into the process of rewriting the given equation, y - 1 = (1/2)(x + 4), into both standard form and slope-intercept form, elucidating the steps involved and highlighting the significance of each form. This equation represents a linear relationship, and by converting it into standard and slope-intercept forms, we can easily identify its key features, such as its slope and intercepts. The ability to manipulate equations and express them in different forms is a cornerstone of mathematical problem-solving, allowing for a deeper understanding of the underlying relationships between variables.
The standard form of a linear equation is typically represented as Ax + By = C, where A, B, and C are constants, and A and B are not both zero. This form is particularly useful for identifying the x- and y-intercepts of the line, which are the points where the line crosses the x- and y-axes, respectively. The slope-intercept form, on the other hand, is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. This form is advantageous for quickly determining the slope and y-intercept of the line, which are crucial for graphing the line and understanding its direction and position on the coordinate plane. The given equation, y - 1 = (1/2)(x + 4), is currently in point-slope form, which is another common form for linear equations. Point-slope form is particularly useful when we know a point on the line and the slope of the line. However, to fully analyze and utilize the equation, it is often necessary to convert it into standard form or slope-intercept form. This conversion process involves algebraic manipulation, including distributing, combining like terms, and rearranging the equation to match the desired form. By mastering these techniques, we can gain a comprehensive understanding of linear equations and their applications in various mathematical and real-world scenarios. The process of converting between different forms of linear equations not only enhances our algebraic skills but also deepens our understanding of the relationships between variables and their graphical representations. This understanding is essential for solving a wide range of problems in mathematics, science, and engineering.
The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. To rewrite the given equation, y - 1 = (1/2)(x + 4), in slope-intercept form, we need to isolate y on one side of the equation. This involves a series of algebraic manipulations to simplify and rearrange the terms.
First, distribute the (1/2) on the right side of the equation:
y - 1 = (1/2)x + 2
This step expands the equation, making it easier to isolate y. Distributing the (1/2) involves multiplying it by both terms inside the parentheses, which are x and 4. This is a fundamental algebraic operation that helps to simplify equations and make them easier to work with. The distributive property is a key concept in algebra and is used extensively in solving equations and simplifying expressions. By applying the distributive property correctly, we can transform equations into more manageable forms and ultimately solve for the unknown variables.
Next, add 1 to both sides of the equation to isolate y:
y - 1 + 1 = (1/2)x + 2 + 1
y = (1/2)x + 3
This step completes the conversion to slope-intercept form. Adding 1 to both sides of the equation maintains the equality and moves the constant term to the right side, leaving y isolated on the left side. This is a crucial step in solving for y and expressing the equation in slope-intercept form. The slope-intercept form, y = (1/2)x + 3, immediately reveals the slope and y-intercept of the line. The slope, m, is (1/2), which means that for every 2 units increase in x, y increases by 1 unit. The y-intercept, b, is 3, which means that the line crosses the y-axis at the point (0, 3). This information is invaluable for graphing the line and understanding its behavior. The slope-intercept form provides a clear and concise representation of the line's characteristics, making it a powerful tool for analyzing and interpreting linear equations.
Therefore, the equation y - 1 = (1/2)(x + 4) in slope-intercept form is y = (1/2)x + 3. This form clearly shows that the slope of the line is 1/2 and the y-intercept is 3. The slope-intercept form is particularly useful for graphing linear equations and for quickly identifying the slope and y-intercept, which are essential characteristics of the line. By understanding the slope and y-intercept, we can easily visualize the line and its position on the coordinate plane. This form also allows us to compare different linear equations and understand their relative steepness and positions. The slope-intercept form is a fundamental concept in algebra and is used extensively in various mathematical applications.
The standard form of a linear equation is expressed as Ax + By = C, where A, B, and C are integers, and A is non-negative. To rewrite the given equation, y - 1 = (1/2)(x + 4), in standard form, we need to rearrange the terms so that the equation matches this format. This involves several steps of algebraic manipulation.
First, we use the result from the previous section, which is the slope-intercept form:
y = (1/2)x + 3
This step simplifies the process by starting from a more manageable form of the equation. The slope-intercept form, y = (1/2)x + 3, provides a clear starting point for converting to standard form. We have already isolated y on one side of the equation, which is a crucial step in this process. Now, we need to move the x term to the left side of the equation and eliminate any fractions. These steps will bring the equation closer to the standard form, Ax + By = C. The ability to convert between different forms of linear equations is a valuable skill in algebra, allowing us to analyze and interpret equations in different ways.
Next, subtract (1/2)x from both sides of the equation:
-(1/2)x + y = 3
This step moves the x term to the left side of the equation, which is necessary for standard form. Subtracting (1/2)x from both sides maintains the equality and brings the equation closer to the desired format. However, the standard form requires that the coefficient of x be an integer, so we need to eliminate the fraction. This is a common step in converting to standard form, as it ensures that the equation meets the required criteria. The standard form provides a consistent format for representing linear equations, making it easier to compare and analyze them.
To eliminate the fraction, multiply both sides of the equation by 2:
2 * [-(1/2)x + y] = 2 * 3
-x + 2y = 6
This step eliminates the fraction and ensures that all coefficients are integers. Multiplying both sides of the equation by 2 maintains the equality and transforms the equation into a form where all coefficients are integers. This is a crucial step in converting to standard form, as it satisfies the requirement that A, B, and C be integers. The standard form, Ax + By = C, is particularly useful for identifying the x- and y-intercepts of the line. To find the x-intercept, we set y to 0 and solve for x. To find the y-intercept, we set x to 0 and solve for y. This makes the standard form a valuable tool for graphing linear equations and understanding their behavior.
Finally, multiply both sides by -1 to make the coefficient of x positive:
-1 * (-x + 2y) = -1 * 6
x - 2y = -6
This step ensures that the coefficient of x is positive, which is a standard convention for the standard form of a linear equation. Multiplying both sides of the equation by -1 maintains the equality and transforms the equation into the final standard form. The standard form, x - 2y = -6, provides a clear and concise representation of the linear equation. This form is particularly useful for identifying the intercepts of the line and for comparing different linear equations. The standard form is a fundamental concept in algebra and is used extensively in various mathematical applications.
Thus, the equation y - 1 = (1/2)(x + 4) in standard form is x - 2y = -6. This form is useful for quickly identifying the x- and y-intercepts of the line. The x-intercept is the point where the line crosses the x-axis, and it can be found by setting y to 0 and solving for x. In this case, the x-intercept is (-6, 0). The y-intercept is the point where the line crosses the y-axis, and it can be found by setting x to 0 and solving for y. In this case, the y-intercept is (0, 3). The standard form provides a convenient way to determine these intercepts, which are essential for graphing the line and understanding its position on the coordinate plane.
In conclusion, we have successfully rewritten the equation y - 1 = (1/2)(x + 4) into both slope-intercept form and standard form. The slope-intercept form, y = (1/2)x + 3, reveals that the line has a slope of 1/2 and a y-intercept of 3. This form is particularly useful for graphing the line and quickly identifying its slope and y-intercept. The standard form, x - 2y = -6, provides an alternative representation of the equation, which is useful for identifying the x- and y-intercepts of the line. These intercepts are the points where the line crosses the x- and y-axes, respectively. The ability to convert between these forms is a fundamental skill in algebra, allowing us to analyze and interpret linear equations in different ways. Each form provides unique insights into the properties of the line, and understanding these forms enhances our ability to solve a wide range of mathematical problems.
The process of converting between different forms of linear equations not only reinforces our algebraic skills but also deepens our understanding of the relationships between variables and their graphical representations. By mastering these techniques, we can effectively analyze and solve problems involving linear equations in various contexts. The slope-intercept form and standard form are essential tools in mathematics, and their applications extend beyond the classroom to various real-world scenarios. Understanding these forms allows us to model and solve problems involving linear relationships, such as determining the cost of a service based on a fixed fee and an hourly rate, or predicting the distance traveled by an object moving at a constant speed. The ability to manipulate and interpret linear equations is a valuable skill that can be applied in numerous fields, including science, engineering, economics, and finance. Therefore, mastering the conversion between different forms of linear equations is a crucial step in developing a strong foundation in mathematics and its applications.