Is K=4 A Solution Of 4k + 7 = 7k - 1 Detailed Explanation

Introduction

In mathematics, one of the fundamental skills is the ability to solve equations. Solving an equation means finding the value(s) of the variable(s) that make the equation true. To verify whether a particular value is a solution, we substitute the value into the equation and check if both sides of the equation are equal. This article delves into the process of determining whether $k = 4$ is a solution to the equation $4k + 7 = 7k - 1$. We will break down the steps, substitute the value, and simplify the equation to arrive at a definitive conclusion. Understanding this process is crucial for anyone studying algebra and beyond, as it forms the basis for more complex problem-solving strategies.

To begin, we must first understand what an equation represents. An equation is a mathematical statement that asserts the equality of two expressions. These expressions are connected by an equals sign (=). In our case, the equation is $4k + 7 = 7k - 1$. This equation involves the variable $k$, and our task is to determine if the value $k = 4$ satisfies this equation. The left side of the equation is $4k + 7$, and the right side is $7k - 1$. To verify whether $k = 4$ is a solution, we will substitute 4 for $k$ in both expressions and simplify. If both sides yield the same numerical value after simplification, then $k = 4$ is indeed a solution. If the values differ, then $k = 4$ is not a solution. This process is a core concept in algebra and is essential for solving linear equations and more complex mathematical problems. Let’s proceed step by step to evaluate the equation.

Substituting $k=4$ into the Equation

The initial step in determining whether $k = 4$ is a solution to the equation $4k + 7 = 7k - 1$ is to substitute the value of $k$ into both sides of the equation. This means replacing every instance of $k$ with the number 4. On the left-hand side (LHS) of the equation, we have $4k + 7$. Substituting $k = 4$, we get $4(4) + 7$. Similarly, on the right-hand side (RHS) of the equation, we have $7k - 1$. Substituting $k = 4$, we get $7(4) - 1$. The next step involves simplifying each side of the equation separately to see if they yield the same result. This substitution process is a fundamental technique in algebra, allowing us to evaluate expressions for specific variable values and ultimately solve equations.

After substituting $k = 4$ into the equation, we now have two numerical expressions to evaluate. The LHS is $4(4) + 7$, and the RHS is $7(4) - 1$. The next step involves performing the arithmetic operations to simplify each expression. This requires following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In both expressions, we first perform the multiplication and then the addition or subtraction. This systematic approach ensures we arrive at the correct simplified values for both sides of the equation. Accurately simplifying these expressions is crucial for determining whether $k = 4$ is a solution, as it will reveal if the two sides of the equation are equal when $k$ is 4.

Simplifying the Left-Hand Side (LHS)

To simplify the left-hand side (LHS) of the equation, which is $4(4) + 7$, we need to follow the order of operations. According to the order of operations, multiplication should be performed before addition. Therefore, we first multiply 4 by 4, which gives us 16. The expression then becomes $16 + 7$. Now, we perform the addition. Adding 16 and 7, we get 23. Thus, the simplified value of the left-hand side of the equation when $k = 4$ is 23. This step is crucial because it provides us with a single numerical value to compare with the simplified value of the right-hand side. If the two values are equal, it indicates that $k = 4$ is a solution to the equation. The process of simplifying expressions is a fundamental skill in algebra and is essential for solving equations and understanding mathematical concepts.

Simplifying the left-hand side not only provides a numerical value but also helps in understanding the structure of the equation. By breaking down the expression into its individual operations, we can clearly see how the value of $k$ influences the final result. In this case, the multiplication of 4 by 4 is the first step, followed by the addition of 7. Each operation contributes to the overall value of the expression, and by performing them in the correct order, we arrive at the accurate simplified value. The result, 23, now serves as a benchmark against which we will compare the simplified value of the right-hand side. This comparison will ultimately determine whether $k = 4$ satisfies the equation. Understanding these fundamental steps is key to mastering algebraic manipulations and problem-solving.

Simplifying the Right-Hand Side (RHS)

Now, let’s simplify the right-hand side (RHS) of the equation, which is $7(4) - 1$. Similar to the left-hand side, we need to follow the order of operations. This means performing the multiplication before the subtraction. First, we multiply 7 by 4, which results in 28. The expression then becomes $28 - 1$. Next, we perform the subtraction. Subtracting 1 from 28 gives us 27. Therefore, the simplified value of the right-hand side of the equation when $k = 4$ is 27. This step is equally important as simplifying the left-hand side because it provides us with the second numerical value needed for comparison. If the simplified value of the RHS matches the simplified value of the LHS, then $k = 4$ is a solution. Otherwise, it is not. The process of simplifying expressions is a cornerstone of algebra and is critical for solving equations effectively.

Simplifying the right-hand side mirrors the process we used for the left-hand side, emphasizing the importance of adhering to the order of operations. By first multiplying 7 by 4 and then subtracting 1, we systematically reduce the expression to its simplest form. This methodical approach is essential for ensuring accuracy in algebraic manipulations. The resulting value, 27, now allows us to make a direct comparison with the simplified value of the left-hand side. This comparison will definitively answer the question of whether $k = 4$ is a solution to the original equation. The consistency in applying the order of operations across both sides of the equation highlights the fundamental principles of algebraic simplification and problem-solving.

Comparing LHS and RHS

After simplifying both sides of the equation, we have determined that the left-hand side (LHS), $4(4) + 7$, simplifies to 23, and the right-hand side (RHS), $7(4) - 1$, simplifies to 27. Now, we must compare these two values to determine if they are equal. The LHS is 23, and the RHS is 27. Clearly, 23 is not equal to 27. Since the left-hand side and the right-hand side do not yield the same value when $k = 4$, we can conclude that $k = 4$ is not a solution to the equation $4k + 7 = 7k - 1$. This comparison is the final step in verifying a potential solution and is a critical aspect of solving equations in algebra.

Comparing the simplified values of the LHS and RHS is a straightforward but essential step. It directly reveals whether the chosen value for the variable satisfies the equation's condition of equality. In this case, the discrepancy between 23 and 27 decisively indicates that $k = 4$ does not make the equation true. This process underscores the importance of accurate simplification and comparison in solving algebraic equations. The result not only answers the specific question but also reinforces the method for verifying solutions, a skill applicable to a wide range of mathematical problems. Understanding this verification process is vital for students learning algebra, as it provides a means to check their work and ensure the accuracy of their solutions.

Conclusion

In conclusion, to determine whether $k = 4$ is a solution of the equation $4k + 7 = 7k - 1$, we substituted $k = 4$ into both sides of the equation. We then simplified the left-hand side (LHS) to 23 and the right-hand side (RHS) to 27. Since 23 is not equal to 27, we can definitively state that $k = 4$ is not a solution of the given equation. This process demonstrates the fundamental steps involved in verifying potential solutions to algebraic equations. By substituting the value, simplifying each side, and comparing the results, we can determine whether the value satisfies the equation. This method is a crucial skill in algebra and is applicable to various types of equations and mathematical problems. The ability to accurately solve and verify equations is essential for further studies in mathematics and related fields.

The process we’ve undertaken highlights the systematic approach required to solve and verify algebraic equations. It emphasizes the importance of careful substitution, accurate simplification, and precise comparison. These steps are not only crucial for solving this specific problem but also form a general strategy applicable to more complex equations and mathematical scenarios. The conclusion that $k = 4$ is not a solution reinforces the idea that not all values will satisfy a given equation, and it’s essential to verify potential solutions to ensure accuracy. Mastering these fundamental algebraic techniques is a building block for more advanced mathematical concepts and problem-solving skills.