Solving 2x-y=7 For Y Explained Step-by-Step

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In the realm of algebra, manipulating equations to isolate specific variables is a fundamental skill. When presented with a two-variable equation like 2x - y = 7, our goal is to express y in terms of x. This means rearranging the equation so that y is alone on one side, and the other side contains an expression involving x. This process, often referred to as solving for y, allows us to understand how the value of y changes as the value of x varies. In this comprehensive guide, we will explore the step-by-step method of solving the equation 2x - y = 7 for y, while also delving into the underlying principles and the significance of this algebraic manipulation.

The Importance of Solving for y

Before we dive into the solution, let's first understand why this process is crucial. Solving for y in terms of x allows us to:

  • Graph the equation: Equations in the form y = mx + b are easily graphed, where m represents the slope and b represents the y-intercept. Solving for y puts the equation in this slope-intercept form.
  • Identify the relationship between variables: By isolating y, we can directly see how changes in x affect the value of y. This helps us understand the relationship between the two variables.
  • Solve systems of equations: When dealing with multiple equations, solving for y in each equation allows us to use substitution or elimination methods to find the solutions.
  • Model real-world scenarios: Many real-world situations can be modeled using equations. Solving for a specific variable allows us to make predictions and understand the behavior of the model.

Step-by-Step Solution

Now, let's break down the process of solving the equation 2x - y = 7 for y:

Step 1: Isolate the term containing y

Our first goal is to get the term containing y (which is -y) by itself on one side of the equation. To do this, we need to eliminate the 2x term from the left side. We can achieve this by subtracting 2x from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to maintain equality.

2x - y - 2x = 7 - 2x

This simplifies to:

-y = 7 - 2x

Step 2: Eliminate the negative sign

We now have -y on the left side, but we want y by itself. To eliminate the negative sign, we can multiply both sides of the equation by -1. This will change the sign of each term on both sides.

-1 * (-y) = -1 * (7 - 2x)

This simplifies to:

y = -7 + 2x

Step 3: Rearrange the terms (Optional)

While the equation y = -7 + 2x is technically solved for y, it's often preferred to write the term with x first. We can rearrange the terms on the right side using the commutative property of addition, which states that the order in which numbers are added does not change the sum.

y = 2x - 7

The Solution

Therefore, the equation 2x - y = 7 solved for y in terms of x is:

y = 2x - 7

This equation is now in slope-intercept form (y = mx + b), where the slope (m) is 2 and the y-intercept (b) is -7. This form makes it easy to graph the equation and understand its properties.

Checking the Solution

It's always a good practice to check our solution to ensure it's correct. We can do this by substituting the expression we found for y back into the original equation and verifying that it holds true.

Original equation:

2x - y = 7

Substitute y = 2x - 7:

2x - (2x - 7) = 7

Distribute the negative sign:

2x - 2x + 7 = 7

Simplify:

7 = 7

Since the equation holds true, our solution y = 2x - 7 is correct.

Understanding the Implications

Now that we have successfully solved for y, let's discuss the implications of this result. The equation y = 2x - 7 tells us that for every increase of 1 in the value of x, the value of y increases by 2. The -7 represents the y-intercept, which is the point where the line crosses the y-axis (when x is 0). Understanding these relationships is crucial for interpreting the equation and its graphical representation.

Common Mistakes to Avoid

While solving for y is a straightforward process, there are some common mistakes students often make. Being aware of these pitfalls can help you avoid them:

  • Forgetting to distribute the negative sign: When multiplying both sides of the equation by -1, remember to distribute the negative sign to all terms on both sides.
  • Incorrectly combining like terms: Ensure you are only combining terms that have the same variable and exponent.
  • Not performing the same operation on both sides: Remember, whatever you do to one side of the equation, you must do to the other side to maintain equality.
  • Skipping steps: It's often helpful to write out each step explicitly, especially when you're first learning the process. This reduces the chances of making errors.

Practice Makes Perfect

Like any mathematical skill, solving for y requires practice. The more you practice, the more comfortable and confident you'll become. Try solving various two-variable equations for y to solidify your understanding. You can find practice problems in textbooks, online resources, and worksheets.

Conclusion

Solving for y in terms of x is a fundamental algebraic skill with numerous applications. By following a step-by-step approach and understanding the underlying principles, you can confidently manipulate equations and gain valuable insights into the relationships between variables. Remember to check your solutions and practice regularly to master this essential skill. In the case of the equation 2x - y = 7, the correct solution is y = 2x - 7, which represents a linear equation with a slope of 2 and a y-intercept of -7.

Understanding why certain answer choices are incorrect is just as important as knowing the correct solution. By examining the common errors that lead to incorrect answers, we can further solidify our understanding of the problem-solving process and avoid similar mistakes in the future. Let's analyze each of the incorrect answer choices provided in the original question:

  • A. y = 5x

    This answer choice represents a linear equation with a slope of 5 and a y-intercept of 0. While it is a valid linear equation, it does not result from correctly solving 2x - y = 7 for y. A student might arrive at this answer if they incorrectly added 2x and 3x or made a mistake in the sign manipulation.

  • B. y = 2x + 7

    This answer choice is close to the correct solution but has a crucial sign error. A student might arrive at this answer if they added 2x to both sides of the equation but forgot to multiply by -1 to isolate y. This highlights the importance of carefully tracking signs throughout the solution process.

  • C. x = (y + 7) / 2

    This answer choice is not solving for y in terms of x; instead, it solves for x in terms of y. While mathematically correct, it does not answer the specific question asked. This underscores the importance of reading the question carefully and ensuring the answer matches the desired format.

  • E. y = 14x

    This answer choice is significantly different from the correct solution and likely results from a more fundamental misunderstanding of the algebraic manipulations involved. A student might arrive at this answer through a series of incorrect steps, such as multiplying terms that should be added or misapplying the distributive property.

Key Takeaways from Analyzing Incorrect Choices

Analyzing the incorrect answer choices provides valuable insights into common errors and misconceptions. Here are some key takeaways:

  • Sign errors are common: Pay close attention to signs throughout the solution process, especially when multiplying or dividing by negative numbers.
  • Read the question carefully: Ensure you are answering the specific question asked and providing the answer in the correct format.
  • Understand the order of operations: Follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions.
  • Double-check your work: Take the time to review each step of your solution to catch any errors.
  • Practice recognizing equation forms: Familiarize yourself with common equation forms, such as slope-intercept form, to help you identify potential errors and verify your solutions.

Conclusion

By carefully analyzing the incorrect answer choices, we gain a deeper understanding of the problem-solving process and the common pitfalls to avoid. This not only helps us improve our accuracy but also strengthens our conceptual understanding of algebra. In the case of solving 2x - y = 7 for y, the correct solution is y = 2x - 7. Recognizing why the other options are incorrect reinforces our understanding of the steps involved in isolating y and highlights the importance of attention to detail in algebraic manipulations.

In conclusion, solving the two-variable equation 2x - y = 7 for y in terms of x is a fundamental algebraic exercise that demonstrates the principles of equation manipulation. The correct solution, y = 2x - 7, highlights the importance of isolating the desired variable while maintaining the equation's balance. The step-by-step process involves isolating the y term, addressing the negative sign, and rearranging the equation into slope-intercept form. By understanding the underlying principles and potential pitfalls, such as sign errors and misapplication of operations, students can confidently approach similar problems and develop a strong foundation in algebraic problem-solving. Furthermore, analyzing incorrect answer choices provides valuable insights into common mistakes, reinforcing the importance of careful attention to detail and a thorough understanding of algebraic concepts. The ability to solve equations for specific variables is a crucial skill in mathematics and has wide-ranging applications in various fields, making it an essential topic for students to master.