Solving For X In The Equation 3x - 4y = 65 A Step-by-Step Guide
Solve for x in the equation 3x - 4y = 65 when y = 4.
In this article, we will walk through the process of solving for the value of x in the linear equation 3x - 4y = 65, given that y = 4. This is a fundamental concept in algebra, and mastering it is crucial for more advanced mathematical problem-solving. We will break down the steps, explain the reasoning behind each one, and provide a clear, concise solution. This guide is designed to help anyone, regardless of their mathematical background, understand how to approach and solve similar problems.
Understanding the Problem
Before diving into the solution, let's clearly understand the problem. We are given a linear equation with two variables, x and y: 3x - 4y = 65. We are also given a specific value for y, which is y = 4. Our goal is to find the value of x that satisfies the equation when y is 4. Essentially, we're trying to find the point on the line represented by the equation where the y-coordinate is 4. This involves substituting the known value of y into the equation and then using algebraic manipulation to isolate x on one side of the equation. This process is a cornerstone of solving algebraic equations and is widely applicable in various mathematical and real-world scenarios.
Step-by-Step Solution
Now, let's proceed with the step-by-step solution to find the value of x:
Step 1: Substitute the value of y
The first step is to substitute the given value of y, which is 4, into the equation 3x - 4y = 65. This replaces the variable y with its numerical value, allowing us to simplify the equation and solve for x. By substituting y = 4, the equation becomes:
3x - 4(4) = 65
This substitution is a fundamental technique in algebra, allowing us to reduce the complexity of an equation by replacing variables with known values. It transforms the equation from one with two unknowns to one with a single unknown, which we can then solve directly.
Step 2: Simplify the equation
Next, we simplify the equation by performing the multiplication: -4(4) = -16. This reduces the equation to a simpler form, making it easier to isolate x. The equation now becomes:
3x - 16 = 65
Simplifying equations is a crucial step in solving for unknowns. By performing basic arithmetic operations, we can reduce the number of terms and make the equation more manageable. This step often involves combining like terms, distributing coefficients, or, as in this case, performing multiplication.
Step 3: Isolate the term with x
To isolate the term with x, we need to get rid of the constant term on the left side of the equation. In this case, we have -16 being subtracted from 3x. To undo this subtraction, we add 16 to both sides of the equation. This maintains the equality of the equation while moving the constant term to the right side. The equation becomes:
3x - 16 + 16 = 65 + 16
Adding 16 to both sides, we get:
3x = 81
The principle of adding the same value to both sides of an equation is a fundamental algebraic operation. It allows us to manipulate the equation without changing its solution. This step is crucial for isolating the variable we are trying to solve for.
Step 4: Solve for x
Finally, to solve for x, we need to isolate x by dividing both sides of the equation by the coefficient of x, which is 3. This undoes the multiplication of x by 3, leaving x by itself on the left side of the equation. The equation becomes:
3x / 3 = 81 / 3
Performing the division, we find:
x = 27
Dividing both sides of an equation by the same non-zero value is another essential algebraic operation. It allows us to isolate the variable and find its value. This final step completes the solution process, giving us the value of x that satisfies the original equation.
The Answer
Therefore, the value of x in the equation 3x - 4y = 65, when y = 4, is x = 27. This corresponds to option D in the given choices.
Why This Matters
Understanding how to solve linear equations is a fundamental skill in mathematics. It forms the basis for more advanced concepts, such as solving systems of equations, graphing linear functions, and working with inequalities. The ability to manipulate equations and isolate variables is crucial in various fields, including science, engineering, economics, and computer science. Moreover, the problem-solving skills developed through solving algebraic equations are transferable to many other areas of life.
Common Mistakes to Avoid
When solving linear equations, several common mistakes can lead to incorrect answers. Here are a few to watch out for:
- Incorrect Substitution: Make sure you substitute the value of y correctly into the equation. A simple mistake in substitution can throw off the entire solution.
- Arithmetic Errors: Pay close attention to arithmetic operations, especially when dealing with negative numbers. Double-check your addition, subtraction, multiplication, and division.
- Incorrect Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Perform multiplication and division before addition and subtraction.
- Not Applying Operations to Both Sides: Always apply the same operation to both sides of the equation to maintain equality. If you add a number to one side, you must add it to the other side as well.
- Forgetting to Simplify: Simplify the equation at each step to make it easier to work with. This reduces the chances of making errors.
By being mindful of these common mistakes, you can increase your accuracy and confidence in solving linear equations.
Practice Problems
To solidify your understanding, try solving these practice problems:
- Solve for x in the equation 2x + 5y = 30, when y = 2.
- Find the value of a in the equation 4a - 3b = 18, when b = -2.
- What is the value of m in the equation m/2 + n = 10, when n = 4?
Working through these practice problems will reinforce the steps and techniques discussed in this guide and help you develop your problem-solving skills.
Conclusion
In conclusion, solving for x in the equation 3x - 4y = 65, when y = 4, involves a series of straightforward algebraic steps: substitution, simplification, isolating the term with x, and solving for x. By following these steps carefully and avoiding common mistakes, you can confidently solve similar linear equations. Remember, practice is key to mastering these skills. The more you practice, the more comfortable and proficient you will become in solving algebraic problems.
This guide has provided a comprehensive explanation of the solution process and highlighted the importance of understanding linear equations in mathematics. With a solid grasp of these fundamentals, you'll be well-equipped to tackle more complex mathematical challenges.