Solving Linear Equations Find The Value Of X In 16x - 4 + 5x = -67
Linear equations are fundamental to mathematics and are encountered in various real-world applications. Understanding how to solve them is a crucial skill for anyone studying mathematics, science, or engineering. This comprehensive guide will walk you through the process of solving the linear equation 16x - 4 + 5x = -67, providing a detailed explanation of each step. We'll not only find the correct solution but also discuss the underlying principles and concepts involved. Mastering this process will empower you to tackle a wide range of linear equations with confidence.
Understanding Linear Equations
Before we dive into the solution, it's essential to understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called “linear” because when plotted on a graph, they form a straight line. The general form of a linear equation in one variable (usually denoted as x) is ax + b = c, where a, b, and c are constants, and a is not equal to zero. The goal when solving a linear equation is to isolate the variable x on one side of the equation to determine its value.
In our specific equation, 16x - 4 + 5x = -67, we have a linear equation with one variable, x. Our task is to find the value of x that satisfies this equation. We will achieve this by simplifying the equation and applying algebraic operations to both sides until x is isolated. This process involves combining like terms, adding or subtracting constants, and multiplying or dividing to maintain the balance of the equation. By carefully following these steps, we can confidently arrive at the correct solution.
Step 1: Combine Like Terms
The first step in solving the equation 16x - 4 + 5x = -67 is to simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power. In this equation, we have two terms that contain the variable x: 16x and 5x. These are like terms and can be combined.
To combine these terms, we simply add their coefficients (the numbers in front of the variable). In this case, we have 16x + 5x. Adding the coefficients, we get 16 + 5 = 21. Therefore, 16x + 5x = 21x. Now, we can rewrite the equation as:
21x - 4 = -67
This simplification makes the equation easier to work with. We have reduced the number of terms on the left side of the equation, bringing us closer to isolating the variable x. By combining like terms, we maintain the equality of the equation while making it more manageable. This step is crucial in solving any linear equation, as it streamlines the process and reduces the likelihood of errors. Remember, the key is to identify terms that share the same variable and exponent, then combine their coefficients accordingly. This fundamental technique is applicable across various algebraic problems and is an essential skill in mathematical problem-solving.
Step 2: Isolate the Variable Term
After combining like terms, our equation is now 21x - 4 = -67. The next step in solving for x is to isolate the term containing the variable, which in this case is 21x. To do this, we need to eliminate the constant term that is on the same side of the equation, which is -4.
To eliminate -4, we perform the inverse operation, which is adding 4. We must add 4 to both sides of the equation to maintain the balance and ensure that the equality remains true. This is a fundamental principle in solving equations: whatever operation you perform on one side, you must also perform on the other side. So, we add 4 to both sides:
21x - 4 + 4 = -67 + 4
On the left side, -4 + 4 cancels out, leaving us with just 21x. On the right side, -67 + 4 equals -63. Thus, our equation now becomes:
21x = -63
By adding 4 to both sides, we have successfully isolated the variable term 21x on the left side of the equation. This step is crucial because it brings us closer to solving for x by removing the constant term that was interfering with the variable. Isolating the variable term is a common technique used in solving various types of equations, not just linear equations. It simplifies the equation and allows us to focus on the next step, which is to isolate the variable itself.
Step 3: Solve for x
Now that we have isolated the variable term, our equation is 21x = -63. The final step in solving for x is to isolate x itself. To do this, we need to eliminate the coefficient that is multiplying x, which is 21.
To eliminate the coefficient 21, we perform the inverse operation, which is division. Since 21 is multiplying x, we divide both sides of the equation by 21. Again, it is crucial to perform the same operation on both sides to maintain the balance of the equation. So, we divide both sides by 21:
(21x) / 21 = -63 / 21
On the left side, 21 in the numerator and denominator cancels out, leaving us with just x. On the right side, -63 / 21 equals -3. Therefore, the solution to the equation is:
x = -3
We have successfully solved for x by dividing both sides of the equation by the coefficient of x. This step is the culmination of the previous steps, where we combined like terms and isolated the variable term. By dividing, we have effectively undone the multiplication, revealing the value of x. The solution, x = -3, means that if we substitute -3 for x in the original equation, the equation will hold true. This step demonstrates the power of inverse operations in solving equations and is a fundamental technique in algebra.
Verification of the Solution
To ensure that our solution is correct, it's always a good practice to verify it. This involves substituting the value we found for x back into the original equation and checking if it holds true. Our original equation was 16x - 4 + 5x = -67, and we found that x = -3. Let's substitute -3 for x in the equation:
16(-3) - 4 + 5(-3) = -67
Now, we perform the operations:
-48 - 4 - 15 = -67
Combine the terms on the left side:
-67 = -67
As we can see, the left side of the equation equals the right side, which means our solution is correct. The equation holds true when x = -3. Verification is a crucial step in problem-solving because it helps us catch any errors we might have made during the process. It gives us confidence in our answer and ensures that we have solved the problem correctly. By verifying our solution, we can be sure that our understanding of the equation and the steps we took to solve it are accurate.
Conclusion
In this comprehensive guide, we have walked through the process of solving the linear equation 16x - 4 + 5x = -67. We started by understanding the concept of linear equations and the goal of isolating the variable. We then followed a step-by-step approach, which included combining like terms, isolating the variable term, and finally solving for x. Our solution was x = -3, and we verified this solution by substituting it back into the original equation.
Solving linear equations is a fundamental skill in mathematics, and the techniques we have discussed here can be applied to a wide range of problems. The key is to understand the underlying principles and to follow a systematic approach. By practicing these steps, you can build your confidence and proficiency in solving linear equations. Remember to always verify your solution to ensure accuracy. With a solid understanding of these concepts, you'll be well-equipped to tackle more complex mathematical problems in the future.
The correct answer is D. x = -3
