Solving For Y In Linear Equations A Step-by-Step Guide
Understanding the Equation
The equation we are tackling today is y - 4 = 2(x + 5). This is a linear equation, and our primary goal is to express it in the slope-intercept form, which is y = mx + b. In this format, m represents the slope of the line, and b represents the y-intercept. Mastering how to manipulate and solve such equations is a foundational skill in algebra. Linear equations appear in numerous real-world applications, from calculating distances and speeds to modeling financial trends. Therefore, understanding them thoroughly is crucial for both academic success and practical problem-solving. To effectively solve for y, we need to isolate y on one side of the equation. This involves applying the distributive property, combining like terms, and using inverse operations. Each step is deliberate and helps us transform the equation into the desired slope-intercept form. Linear equations not only form the basis of algebra but also serve as stepping stones to more advanced mathematical concepts, such as systems of equations, calculus, and linear programming. In essence, learning how to solve for y in a linear equation is more than just an algebraic exercise; it's about building a solid mathematical foundation. As we proceed, we will break down each step, ensuring that the methodology is clear and the solution is easily understood. This understanding will empower you to tackle similar problems with confidence and clarity. The ability to manipulate equations and isolate variables is a fundamental skill that extends far beyond the classroom, impacting various fields such as engineering, finance, and computer science.
Step-by-Step Solution
To solve for y in the equation y - 4 = 2(x + 5), we need to follow a systematic approach. The first key step is to apply the distributive property to the right side of the equation. This means multiplying the 2 by both the x and the 5 inside the parentheses. Doing so gives us 2 * x + 2 * 5, which simplifies to 2x + 10. So, our equation now looks like y - 4 = 2x + 10. This transformation is crucial because it removes the parentheses and allows us to further isolate y. The distributive property is a fundamental concept in algebra, enabling us to simplify expressions and equations by correctly handling parentheses. It's a building block for more complex algebraic manipulations and is used extensively in various mathematical contexts. Once we've distributed the 2, the next step is to isolate y. To do this, we need to eliminate the -4 on the left side of the equation. We can accomplish this by performing the inverse operation, which is adding 4 to both sides of the equation. Adding 4 to both sides maintains the balance of the equation and helps us move closer to our goal of solving for y. This gives us y - 4 + 4 = 2x + 10 + 4. Simplifying this, we get y = 2x + 14. At this point, we have successfully isolated y and expressed the equation in slope-intercept form, y = mx + b. Here, the slope m is 2, and the y-intercept b is 14. Each step in this process is deliberate and essential for correctly solving the equation. By understanding and applying these steps, you can confidently tackle similar problems and further develop your algebraic skills. The ability to manipulate equations and isolate variables is a cornerstone of mathematical proficiency, opening doors to more advanced topics and applications.
Final Answer
After performing the steps to solve for y in the equation y - 4 = 2(x + 5), we have successfully transformed the equation into the slope-intercept form. Let's recap the process to ensure clarity and understanding. Initially, we applied the distributive property to the right side of the equation, multiplying 2 by both x and 5, resulting in 2x + 10. This eliminated the parentheses and set the stage for further simplification. The equation then became y - 4 = 2x + 10. Our next crucial step was to isolate y on the left side of the equation. To achieve this, we added 4 to both sides, which is the inverse operation of subtraction. This maintained the balance of the equation while allowing us to move closer to solving for y. Adding 4 to both sides gave us y - 4 + 4 = 2x + 10 + 4, which simplifies to y = 2x + 14. This final form of the equation is in the slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept. In this case, the slope m is 2, indicating the rate of change of y with respect to x, and the y-intercept b is 14, representing the point where the line crosses the y-axis. Therefore, the solution to the equation y - 4 = 2(x + 5), expressed in slope-intercept form, is y = 2x + 14. This process illustrates the importance of understanding and applying algebraic principles to manipulate equations effectively. By mastering these skills, you can confidently solve a wide range of mathematical problems and gain a deeper understanding of the relationships between variables.
Therefore, the final answer is:
y = 2x + 14
