Solving 3x - 7y = 10 For X: A Step-by-Step Guide

by Alex Johnson 49 views

Have you ever stumbled upon a linear equation and wondered how to isolate a specific variable? It's a common task in algebra, and in this article, we'll break down the process of expressing the linear equation 3x - 7y = 10 in terms of x. We'll walk through each step, making it clear and easy to understand, so you can confidently tackle similar problems in the future.

Understanding Linear Equations

Before we dive into the solution, let's quickly recap 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. The variables are only raised to the power of one. A simple example is y = 2x + 1, which represents a straight line when graphed. Our equation, 3x - 7y = 10, also fits this description.

Why is it important to be able to manipulate these equations? Well, being able to express a linear equation in terms of a specific variable is crucial for various applications, from graphing lines and solving systems of equations to more advanced mathematical concepts. It gives us the flexibility to view the relationship between variables from different perspectives. This skill forms a foundational part of mathematical literacy and is valuable in many fields, including engineering, economics, and computer science. So, understanding how to solve for a variable isn't just an academic exercise; it's a practical skill with real-world implications.

Why Expressing in Terms of x Matters

Expressing an equation in terms of x means we want to isolate x on one side of the equation, with everything else on the other side. Think of it as rewriting the equation to have the form x = (some expression involving y). This is particularly useful when:

  • Graphing the equation: If you want to plot the equation on a graph, having x isolated can make it easier to substitute different values of y and find the corresponding x values.
  • Solving systems of equations: When you have multiple equations, expressing one equation in terms of x can help you substitute and solve for the variables.
  • Understanding relationships: Isolating x can sometimes provide a clearer understanding of how x changes in relation to y.

Now that we understand why this is important, let's get started with solving our equation.

Step-by-Step Solution: Isolating x

Our goal is to rewrite the equation 3x - 7y = 10 so that x is by itself on one side. Here’s how we do it:

Step 1: Isolate the Term with x

The first thing we want to do is get the term with x (which is 3x) by itself on the left side of the equation. To do this, we need to get rid of the -7y term. Remember the golden rule of algebra: what you do to one side, you must do to the other. So, we'll add 7y to both sides of the equation:

3x - 7y + 7y = 10 + 7y

This simplifies to:

3x = 10 + 7y

Great! We've now isolated the term with x.

Step 2: Solve for x

Now that we have 3x = 10 + 7y, we need to get x completely by itself. To do this, we'll divide both sides of the equation by 3. This is because 3x means "3 times x", and the opposite operation of multiplication is division:

(3x) / 3 = (10 + 7y) / 3

This simplifies to:

x = (10 + 7y) / 3

And there you have it! We've successfully expressed the equation in terms of x.

Step 3: Understanding the Solution

Our final equation is x = (10 + 7y) / 3. This tells us how to find the value of x for any given value of y. For example:

  • If y = 0, then x = (10 + 7(0)) / 3 = 10/3
  • If y = 1, then x = (10 + 7(1)) / 3 = 17/3
  • If y = -1, then x = (10 + 7(-1)) / 3 = 3/3 = 1

You can see that by substituting different values for y, we can easily find the corresponding x values. This is the power of expressing an equation in terms of a specific variable. It allows us to analyze the relationship between the variables and make calculations more easily.

Common Mistakes to Avoid

When solving for x, it's easy to make a few common mistakes. Here are some things to watch out for:

  • Forgetting to apply operations to both sides: Remember, whatever you do to one side of the equation, you must do to the other. This is crucial for maintaining the equality.
  • Incorrectly applying the order of operations: Make sure you follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions.
  • Dividing only part of the expression: When dividing both sides by a number, make sure you divide the entire expression on the other side, not just a part of it. For example, in our case, we divided the entire expression (10 + 7y) by 3, not just 7y.

By keeping these common pitfalls in mind, you can avoid mistakes and solve equations with greater confidence.

Practice Problems

Now that we've walked through the solution, let's test your understanding with a couple of practice problems:

  1. Express the equation 2x + 5y = 15 in terms of x.
  2. Express the equation 4x - 3y = 9 in terms of x.

Try solving these on your own, using the steps we've discussed. The more you practice, the more comfortable you'll become with manipulating linear equations. Remember, mathematics is a skill that improves with consistent effort and application.

Solutions to Practice Problems

Here are the solutions to the practice problems:

  1. 2x + 5y = 15 in terms of x:

    • Subtract 5y from both sides: 2x = 15 - 5y
    • Divide both sides by 2: x = (15 - 5y) / 2

  2. 4x - 3y = 9 in terms of x:

    • Add 3y to both sides: 4x = 9 + 3y
    • Divide both sides by 4: x = (9 + 3y) / 4

How did you do? If you got the correct answers, congratulations! You're well on your way to mastering this skill. If you struggled with either problem, review the steps and try again. Remember, persistence is key to success in mathematics.

Real-World Applications

Expressing equations in terms of a specific variable isn't just a math textbook exercise; it has numerous applications in the real world. Here are a few examples:

  • Physics: In physics, you often need to rearrange equations to solve for a specific variable, such as velocity, time, or distance. For example, the equation d = vt (distance = velocity × time) can be rearranged to solve for v (velocity = d/t) or t (time = d/v).
  • Economics: Economists use equations to model various economic relationships. Being able to express these equations in terms of different variables is essential for analyzing and predicting economic trends.
  • Computer Science: In programming, you often need to manipulate equations to perform calculations or solve problems. For example, in game development, you might need to rearrange equations to calculate the trajectory of a projectile.
  • Everyday Life: Even in everyday situations, you might find yourself using this skill. For example, if you're planning a road trip and want to know how long it will take to reach your destination, you might need to rearrange the equation time = distance / speed.

These are just a few examples, but they illustrate the wide range of applications for this mathematical skill. By mastering the ability to express equations in terms of a specific variable, you'll be well-equipped to tackle a variety of challenges in both academic and real-world settings.

Conclusion

Expressing the linear equation 3x - 7y = 10 in terms of x involves a few simple steps: isolating the term with x and then dividing to solve for x. The result, x = (10 + 7y) / 3, allows us to easily find x for any given value of y. By understanding the process and practicing regularly, you can confidently solve similar equations. Remember to avoid common mistakes and apply this skill in various real-world scenarios. With this knowledge, you're one step closer to mastering algebra!

To further expand your knowledge on linear equations, consider visiting a trusted resource like Khan Academy's Linear Equations section.