Solving Equations: A Step-by-Step Guide

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In mathematics, solving an equation means finding the value(s) of the unknown variable(s) that make the equation true. This often involves isolating the variable on one side of the equation by performing the same operations on both sides. Let's break down the process of solving the equation x−2−17=−7\frac{x}{-2} - 17 = -7 step by step.

1. Understanding the Equation

The equation we are given is x−2−17=−7\frac{x}{-2} - 17 = -7. This is a linear equation in one variable, x. Our goal is to find the value of x that satisfies this equation. To do this, we will use the principles of algebraic manipulation to isolate x on one side of the equation.

Key Concepts: Before we dive into the steps, let’s review some fundamental concepts that will guide our solution:

  • Equation: An equation is a mathematical statement that asserts the equality of two expressions. It contains an equals sign (=).
  • Variable: A variable is a symbol (usually a letter) that represents an unknown quantity. In our equation, x is the variable.
  • Isolating the Variable: This means getting the variable alone on one side of the equation. To do this, we perform inverse operations.
  • Inverse Operations: These are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division.

2. Isolating the Term with the Variable

Our first goal is to isolate the term that contains the variable x. In this case, the term is x−2\frac{x}{-2}. To isolate this term, we need to get rid of the -17 on the left side of the equation. Since -17 is being subtracted, we will use the inverse operation, which is addition. We add 17 to both sides of the equation to maintain the equality:

x−2−17+17=−7+17\frac{x}{-2} - 17 + 17 = -7 + 17

This simplifies to:

x−2=10\frac{x}{-2} = 10

Explanation: Adding 17 to both sides cancels out the -17 on the left side, leaving us with the term containing x alone. This step brings us closer to isolating x itself.

3. Isolating the Variable

Now that we have x−2=10\frac{x}{-2} = 10, we need to isolate x. The variable is currently being divided by -2. To undo this division, we will use the inverse operation, which is multiplication. We multiply both sides of the equation by -2:

(−2)⋅x−2=10⋅(−2)(-2) \cdot \frac{x}{-2} = 10 \cdot (-2)

This simplifies to:

x=−20x = -20

Explanation: Multiplying both sides by -2 cancels out the division by -2 on the left side, leaving x alone. On the right side, 10 multiplied by -2 gives us -20. Thus, we have found the value of x.

4. Verifying the Solution

It's always a good practice to verify our solution by substituting the value we found for x back into the original equation. This ensures that our solution is correct. Let's substitute x = -20 into the original equation:

−20−2−17=−7\frac{-20}{-2} - 17 = -7

Simplifying the left side, we get:

10−17=−710 - 17 = -7

−7=−7-7 = -7

Explanation: The equation holds true when we substitute x = -20. This confirms that our solution is correct. Verification is a crucial step in solving equations, as it helps catch any potential errors in the process.

5. Importance of Showing Steps

Showing each step in solving an equation is crucial for several reasons:

  • Clarity: Writing out each step makes the solution process clear and easy to follow. This is especially important when dealing with more complex equations.
  • Error Detection: By showing each step, it becomes easier to identify any mistakes made along the way. If an error is made, it can be pinpointed and corrected more efficiently.
  • Understanding: Each step represents a logical progression in the solution. Showing these steps helps in understanding the underlying principles of equation solving.
  • Communication: In an educational setting, showing steps is essential for demonstrating your understanding of the concepts and for receiving partial credit even if the final answer is incorrect.

6. Common Mistakes to Avoid

When solving equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them:

  • Not Performing Operations on Both Sides: Remember, whatever operation you perform on one side of the equation, you must perform on the other side to maintain equality. For example, if you add a number to the left side, you must add the same number to the right side.
  • Incorrect Order of Operations: Follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions. This ensures that you perform operations in the correct sequence.
  • Sign Errors: Pay close attention to signs (positive and negative) when performing operations. A simple sign error can lead to an incorrect solution.
  • Forgetting to Distribute: If you have a term multiplied by an expression in parentheses, remember to distribute the term to each element inside the parentheses.
  • Not Verifying the Solution: As mentioned earlier, always verify your solution by substituting it back into the original equation. This can help catch errors that you might have missed.

7. Conclusion

In summary, solving the equation x−2−17=−7\frac{x}{-2} - 17 = -7 involves isolating the variable x by performing inverse operations on both sides of the equation. We first added 17 to both sides to isolate the term with x, and then we multiplied both sides by -2 to isolate x. This gave us the solution x = -20. We then verified our solution by substituting it back into the original equation. Remember to show your steps clearly, be mindful of common mistakes, and always verify your solution. With practice, solving linear equations will become second nature.

Solve the equation: x−2−17=−7\frac{x}{-2} - 17 = -7

Solving Equations A Step-by-Step Guide with Examples