Solving The Sum Of -12y And 13y Equals -5 An Algebraic Solution

Mathematics often presents us with challenges, and one of the fundamental skills in this field is the ability to translate phrases into algebraic equations and solve them. In this comprehensive guide, we will meticulously dissect the phrase "The sum of -12y and 13y is -5" to construct an algebraic equation and subsequently determine the value of y. This step-by-step approach will not only provide the solution to this specific problem but also equip you with the tools to tackle similar mathematical conundrums. Understanding the underlying principles of algebraic manipulation is crucial for success in higher-level mathematics and its applications in various scientific and engineering disciplines. This article aims to make the process clear and accessible, even for those who may find algebra daunting. Let's embark on this journey together, demystifying the world of equations and empowering you with the confidence to solve them.

Translating the Phrase into an Algebraic Equation

In the realm of algebra, translating verbal phrases into mathematical expressions is a cornerstone skill. The phrase "The sum of -12y and 13y is -5" is a classic example of such a translation. To accurately represent this statement algebraically, we need to break it down into its constituent parts and convert them into corresponding mathematical symbols and operations. The phrase "the sum of" immediately suggests the operation of addition. Therefore, we will be adding two terms together. The terms are "-12y" and "13y," which are algebraic expressions involving the variable y. The word "is" in this context signifies equality, indicating that the sum of the two terms is equal to a specific value, which in this case is -5. Now, let's piece these elements together. We start with the sum of -12y and 13y, which can be written as -12y + 13y. The phrase states that this sum "is -5," so we equate the expression to -5. This leads us to the algebraic equation: -12y + 13y = -5. This equation is the mathematical representation of the given verbal phrase. It concisely captures the relationship between the terms involving y and the constant value -5. Mastering this translation process is fundamental to solving a wide range of algebraic problems. It allows us to convert real-world scenarios and word problems into a mathematical framework that can be analyzed and solved. The ability to translate phrases into equations is not just a mathematical skill; it is a critical thinking skill that enhances problem-solving abilities in various contexts.

Simplifying the Equation

Before we can isolate y and determine its value, we must first simplify the equation -12y + 13y = -5. Simplification in algebra involves combining like terms to reduce the equation to its most manageable form. In this equation, we have two terms that contain the variable y: -12y and 13y. These are considered like terms because they have the same variable raised to the same power (in this case, y to the power of 1). To combine like terms, we simply add or subtract their coefficients. The coefficient of a term is the numerical factor that multiplies the variable. In -12y, the coefficient is -12, and in 13y, the coefficient is 13. Therefore, we need to perform the operation -12 + 13. When we add -12 and 13, we get 1. So, -12 + 13 = 1. This means that -12y + 13y simplifies to 1y. It is a common practice in algebra to omit the coefficient 1 when it multiplies a variable. So, 1y is typically written simply as y. Now, let's rewrite the simplified equation. We started with -12y + 13y = -5. After combining like terms, we have y = -5. This simplified equation tells us directly the value of y. The process of simplification is crucial in solving algebraic equations because it reduces the complexity of the equation, making it easier to isolate the variable and find its value. By combining like terms, we eliminate unnecessary clutter and bring the equation to its most fundamental form. This skill is essential for tackling more complex equations and problems in algebra and beyond.

Solving for y

After simplifying the equation -12y + 13y = -5 to y = -5, we have effectively solved for y. The equation y = -5 directly states the value of y, which is -5. In this case, the solution is straightforward because the simplification process led us directly to the value of the variable. However, it's important to understand the underlying principle of solving for a variable in more complex equations. Solving for a variable means isolating the variable on one side of the equation. This is achieved by performing operations on both sides of the equation to undo any operations that are affecting the variable. The goal is to manipulate the equation until the variable stands alone on one side, with its value on the other side. For example, if we had an equation like 2y = 10, we would need to divide both sides by 2 to isolate y. This would give us y = 5. In the case of our equation, -12y + 13y = -5, the simplification process itself isolated y. By combining the like terms -12y and 13y, we arrived at y = -5. This means that no further operations were needed to isolate y. The solution y = -5 is the value that, when substituted back into the original equation, makes the equation true. To verify this, we can substitute -5 for y in the original equation: -12(-5) + 13(-5) = -5. This simplifies to 60 - 65 = -5, which is true. Therefore, we can confidently conclude that the solution to the equation is y = -5. Understanding the process of solving for a variable is a fundamental skill in algebra. It allows us to find the unknown values in equations and solve a wide range of problems in mathematics and other fields.

Final Answer

In conclusion, the algebraic equation representing the phrase "The sum of -12y and 13y is -5" is -12y + 13y = -5. After simplifying this equation by combining like terms, we arrive at y = -5. Therefore, the solution for y is -5. This step-by-step solution demonstrates the process of translating a verbal phrase into an algebraic equation, simplifying the equation, and solving for the unknown variable. These are fundamental skills in algebra that are essential for success in mathematics and related fields. The ability to translate phrases into equations allows us to represent real-world scenarios mathematically. Simplifying equations makes them easier to work with and solve. Solving for a variable provides us with the value of the unknown, which can be used to answer questions and solve problems. By mastering these skills, you will be well-equipped to tackle a wide range of algebraic challenges. Remember to always break down problems into smaller, manageable steps, and to carefully apply the rules of algebra to arrive at the correct solution. With practice and perseverance, you can develop confidence and proficiency in solving algebraic equations.