Solving The Equation -4+5x-7=10+3x-2x Find The Value Of X

by Admin 58 views

In this comprehensive guide, we will delve into the realm of linear equations and equip you with the skills to confidently solve for the unknown variable, x. We'll dissect the given equation, systematically isolate x, and arrive at the correct solution. By understanding the underlying principles and applying the proper techniques, you'll master the art of solving linear equations.

Understanding the Basics of Linear Equations

Before we dive into the solution, let's establish a solid foundation by defining 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 variable is raised to the power of one, and there are no terms with variables multiplied together. Linear equations can be represented graphically as a straight line, hence the name "linear."

The general form of a linear equation in one variable is:

ax + b = 0

where a and b are constants, and x is the variable. Solving a linear equation involves finding the value of x that satisfies the equation, making the left-hand side equal to the right-hand side.

Step-by-Step Solution to the Equation

Now, let's tackle the equation at hand:

-4 + 5x - 7 = 10 + 3x - 2x

Our goal is to isolate x on one side of the equation. To achieve this, we'll follow a series of algebraic manipulations:

1. Simplify Both Sides

The first step is to simplify both sides of the equation by combining like terms. On the left-hand side, we have two constant terms, -4 and -7. Combining them, we get:

-11 + 5x = 10 + 3x - 2x

On the right-hand side, we have two terms involving x, 3x and -2x. Combining them, we get:

-11 + 5x = 10 + x

2. Isolate the Variable Term

Next, we want to isolate the variable term (the term containing x) on one side of the equation. To do this, we'll subtract x from both sides:

-11 + 5x - x = 10 + x - x

This simplifies to:

-11 + 4x = 10

3. Isolate the Constant Term

Now, we need to isolate the constant term on the other side of the equation. To do this, we'll add 11 to both sides:

-11 + 4x + 11 = 10 + 11

This simplifies to:

4x = 21

4. Solve for x

Finally, to solve for x, we'll divide both sides by the coefficient of x, which is 4:

(4x) / 4 = 21 / 4

This gives us the solution:

x = 21/4

Therefore, the solution for x in the equation -4 + 5x - 7 = 10 + 3x - 2x is x = 21/4.

Verifying the Solution

To ensure our solution is correct, we can substitute x = 21/4 back into the original equation and check if both sides are equal:

-4 + 5(21/4) - 7 = 10 + 3(21/4) - 2(21/4)

Simplifying both sides:

-4 + 105/4 - 7 = 10 + 63/4 - 42/4

Finding a common denominator:

(-16 + 105 - 28) / 4 = (40 + 63 - 42) / 4

Simplifying further:

61/4 = 61/4

Since both sides are equal, our solution x = 21/4 is indeed correct.

Common Mistakes to Avoid

When solving linear equations, it's essential to be mindful of common mistakes that can lead to incorrect solutions. Here are a few pitfalls to avoid:

  • Incorrectly Combining Like Terms: Ensure you only combine terms that have the same variable and exponent. For example, you can combine 5x and 3x, but you cannot combine 5x and 3.
  • Forgetting to Distribute: When an equation involves parentheses, remember to distribute any coefficients or constants outside the parentheses to all terms inside. For example, in the expression 2(x + 3), you need to multiply both x and 3 by 2.
  • Dividing by Zero: Never divide both sides of an equation by zero, as this is an undefined operation and will lead to an incorrect solution.
  • Sign Errors: Pay close attention to the signs of terms when moving them from one side of the equation to the other. Remember that when you move a term across the equals sign, you need to change its sign.

Strategies for Solving Linear Equations

To become proficient in solving linear equations, consider these effective strategies:

  • Simplify First: Before attempting to isolate the variable, simplify both sides of the equation by combining like terms and distributing as needed. This will make the equation easier to work with.
  • Isolate the Variable Term: Move all terms containing the variable to one side of the equation and all constant terms to the other side. This sets the stage for solving for the variable.
  • Perform Inverse Operations: Use inverse operations to undo operations performed on the variable. For example, if the variable is being multiplied by a number, divide both sides by that number. If a number is being added to the variable, subtract that number from both sides.
  • Check Your Solution: After finding a solution, substitute it back into the original equation to verify that it makes the equation true. This step helps prevent errors and ensures accuracy.

Practice Problems

To solidify your understanding of solving linear equations, let's work through a few practice problems:

  1. Solve for y: 3y + 5 = 14
  2. Solve for a: 2(a - 1) = 8
  3. Solve for z: -6 + 4z = 2z + 10

Solutions to Practice Problems

  1. y = 3
  2. a = 5
  3. z = 8

Conclusion

Mastering the art of solving linear equations is a fundamental skill in mathematics. By understanding the principles, applying the proper techniques, and avoiding common mistakes, you can confidently tackle any linear equation that comes your way. Remember to simplify, isolate the variable, perform inverse operations, and always check your solution. With practice and perseverance, you'll become a proficient equation solver. This detailed guide provided a step-by-step solution to the equation -4 + 5x - 7 = 10 + 3x - 2x, demonstrating how to isolate the variable x and arrive at the correct answer, which is x = 21/4. Understanding the process of solving equations is a fundamental skill in mathematics, and this guide aims to make the concept clear and accessible.