Solving 3x + 4 = 19 Using Substitution A Step-by-Step Guide

Introduction: The Power of Substitution in Equation Solving

In the realm of mathematics, equations serve as fundamental tools for modeling and solving real-world problems. Among the various methods available for tackling equations, substitution stands out as a versatile and intuitive technique. This article delves into the process of solving the equation 3x + 4 = 19 using the substitution method, providing a step-by-step guide and highlighting the underlying principles. We will meticulously examine each potential solution—x = 5, x = 4, x = 6, and x = 3—to pinpoint the correct value that satisfies the equation. Through this exploration, we aim to illuminate the efficacy of substitution in verifying solutions and solidifying understanding of algebraic concepts. To solve this equation effectively, it is crucial to grasp the fundamentals of substitution. Substitution, in the context of equation solving, involves replacing a variable with a given value to check if the equation holds true. This method is particularly useful when presented with multiple potential solutions, as it provides a direct way to verify each one. In the equation 3x + 4 = 19, our goal is to find the value of 'x' that makes the equation a true statement. We are given four possible solutions: x = 5, x = 4, x = 6, and x = 3. By substituting each of these values into the equation, we can determine which one satisfies the condition. This process not only helps us find the solution but also reinforces our understanding of equation solving and algebraic manipulation. The beauty of substitution lies in its simplicity and directness. It transforms an algebraic problem into a simple arithmetic calculation, making it easier to identify the correct solution. Understanding and mastering this technique is essential for anyone delving into algebra and beyond, as it forms the basis for more complex problem-solving strategies. So, let's embark on this journey of substitution and discover the solution to the equation 3x + 4 = 19.

Understanding the Equation: 3x + 4 = 19

Before we dive into the substitution method, let's first dissect the equation 3x + 4 = 19. This is a linear equation in one variable, where 'x' represents the unknown value we are trying to find. The equation states that three times the value of 'x', plus 4, equals 19. Our task is to isolate 'x' and determine which of the given options satisfies this condition. To truly understand this equation, it is important to recognize the operations involved and their order. The equation involves multiplication (3 times x), addition (adding 4), and equality (the result equals 19). The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates that we perform multiplication before addition. However, in this case, we are working backward to solve for 'x', so we will address the addition first when using algebraic manipulation to isolate 'x'. Understanding the structure of the equation is crucial for choosing the right approach to solve it. In this case, we are employing the substitution method, which is particularly effective when we have a set of potential solutions to test. By substituting each value for 'x', we can directly observe whether it makes the equation true. This method is not only a practical way to find the solution but also a valuable exercise in understanding the relationship between variables and constants in an equation. Furthermore, understanding the equation involves recognizing that there is only one value of 'x' that will make the equation true. This is a fundamental property of linear equations in one variable. Our goal is to identify that specific value from the given options. This understanding sets the stage for the substitution process, where we systematically test each option until we find the one that satisfies the equation. The equation 3x + 4 = 19 is a simple yet powerful example of how mathematical expressions can represent relationships and how we can use algebraic techniques to uncover hidden values.

Step-by-Step Substitution: Testing Each Solution

Now, let's embark on the substitution process, systematically testing each potential solution for the equation 3x + 4 = 19. This involves replacing 'x' with each given value—5, 4, 6, and 3—and evaluating whether the resulting equation holds true. This methodical approach allows us to pinpoint the correct solution while reinforcing our understanding of equation solving.

Testing x = 5

First, we substitute x = 5 into the equation:

3(5) + 4 = 19

15 + 4 = 19

19 = 19

In this case, the equation holds true. When x is 5, the left side of the equation equals the right side. This indicates that x = 5 is indeed a solution to the equation. However, to be thorough, we must test the other options as well to ensure we haven't made any errors and to further solidify our understanding of the process.

Testing x = 4

Next, we substitute x = 4 into the equation:

3(4) + 4 = 19

12 + 4 = 19

16 = 19

In this case, the equation does not hold true. When x is 4, the left side of the equation (16) does not equal the right side (19). Therefore, x = 4 is not a solution to the equation.

Testing x = 6

Now, let's substitute x = 6 into the equation:

3(6) + 4 = 19

18 + 4 = 19

22 = 19

Again, the equation does not hold true. When x is 6, the left side of the equation (22) is not equal to the right side (19). Thus, x = 6 is not a solution.

Testing x = 3

Finally, we substitute x = 3 into the equation:

3(3) + 4 = 19

9 + 4 = 19

13 = 19

As with the previous two attempts, the equation does not hold true. When x is 3, the left side of the equation (13) does not equal the right side (19). Therefore, x = 3 is not a solution.

Through this step-by-step substitution process, we have systematically evaluated each potential solution. Only x = 5 satisfies the equation 3x + 4 = 19. This exercise demonstrates the power of substitution in verifying solutions and reinforces the importance of meticulous calculation in algebra.

The Correct Solution: x = 5

After meticulously testing each potential solution through substitution, we have arrived at the definitive answer. The correct solution to the equation 3x + 4 = 19 is x = 5. This conclusion is supported by our earlier calculation, where substituting x = 5 into the equation resulted in a true statement: 3(5) + 4 = 19, which simplifies to 19 = 19. This result confirms that 5 is the value of 'x' that satisfies the equation.

Our exploration has not only identified the correct solution but also demonstrated the effectiveness of the substitution method. By replacing the variable 'x' with each given value and evaluating the equation, we were able to directly verify whether it held true. This approach is particularly useful when presented with multiple potential solutions, as it provides a straightforward way to check each one.

The other options—x = 4, x = 6, and x = 3—were all found to be incorrect. Substituting these values into the equation resulted in false statements, indicating that they do not satisfy the condition 3x + 4 = 19. This process of elimination further reinforces the correctness of our solution, x = 5.

Understanding why x = 5 is the solution is crucial. It highlights the fundamental concept of an equation: a statement of equality between two expressions. In this case, the expression 3x + 4 must equal 19. Only when x is 5 does this equality hold true. This understanding is not only important for solving this specific equation but also for grasping the broader principles of algebra and equation solving.

In conclusion, through the rigorous application of the substitution method, we have confidently identified x = 5 as the solution to the equation 3x + 4 = 19. This exercise underscores the power of substitution in verifying solutions and reinforces the importance of systematic problem-solving in mathematics.

Importance of Verification: Ensuring Accuracy

In the realm of mathematics, the importance of verification cannot be overstated. It is the cornerstone of accuracy and ensures that the solutions we obtain are indeed correct. Verification involves revisiting our steps, checking our calculations, and confirming that our final answer satisfies the original problem. In the context of equation solving, verification is particularly crucial, as it helps us avoid errors and build confidence in our results. The substitution method, as we have seen, is not only a way to find solutions but also an inherent verification tool. By substituting the obtained solution back into the original equation, we can directly check if it makes the equation true. This process provides a tangible confirmation of our answer's validity. In the case of the equation 3x + 4 = 19, we verified that x = 5 is the correct solution by substituting it back into the equation and observing that it resulted in a true statement (19 = 19). This verification step eliminates any doubt and strengthens our understanding of the solution. Beyond the substitution method, there are other ways to verify solutions. For instance, we can use estimation to approximate the solution and see if our calculated answer is within a reasonable range. We can also use graphical methods to visualize the equation and its solution. Furthermore, in more complex problems, we can use computer algebra systems or calculators to verify our results. The importance of verification extends beyond just getting the right answer. It also fosters a habit of critical thinking and attention to detail. By consistently verifying our work, we develop a deeper understanding of the underlying concepts and improve our problem-solving skills. Verification also helps us identify and correct errors, which is an essential part of the learning process. Mistakes are inevitable, but by verifying our work, we can catch them early and learn from them. In essence, verification is not just a final step in problem-solving; it is an integral part of the entire process. It ensures accuracy, builds confidence, and promotes a deeper understanding of mathematics. By making verification a habit, we become more effective and confident problem solvers.

Conclusion: Mastering Substitution for Equation Solving

In conclusion, we have successfully navigated the process of solving the equation 3x + 4 = 19 using the substitution method. This journey has not only led us to the correct solution, x = 5, but has also illuminated the power and versatility of substitution in equation solving. Through meticulous step-by-step testing of each potential solution, we have demonstrated how substitution allows us to directly verify whether a given value satisfies an equation. This approach is particularly valuable when dealing with multiple-choice scenarios or when we want to ensure the accuracy of our results. The substitution method is a fundamental tool in algebra and beyond. It is a simple yet powerful technique that can be applied to a wide range of equations and problems. By mastering substitution, we gain a deeper understanding of algebraic concepts and develop our problem-solving skills. Furthermore, our exploration has underscored the importance of verification in mathematics. By substituting the solution back into the original equation, we have confirmed its validity and eliminated any doubt. This practice of verification is crucial for ensuring accuracy and building confidence in our answers. It also fosters a habit of critical thinking and attention to detail, which are essential qualities for any successful problem solver. In addition to substitution, there are other methods for solving equations, such as algebraic manipulation and graphical techniques. Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation and the context of the problem. However, substitution remains a cornerstone of equation solving, and its mastery is essential for anyone delving into mathematics. As we move forward in our mathematical journey, the principles and techniques we have explored in this article will serve as valuable building blocks for tackling more complex problems. The ability to solve equations is a fundamental skill that has applications in various fields, from science and engineering to finance and economics. By mastering substitution and other equation-solving methods, we empower ourselves to tackle real-world challenges and make informed decisions.