How to Solve Equations like x/2 5 - 3x 2

Mathematics is a fundamental subject that often translates into real-world applications and problem-solving scenarios. One common type of problem in algebra is solving linear equations, such as?x/2 5 - 3x 2. This article provides a detailed step-by-step guide on solving such equations, with clear explanations and useful tips for beginners.

Solving the Equation Step-by-Step

Let us consider the following equation:

Equation: (dfrac{x}{2} 5 - 3x 2)

Step 1: Simplify the Right-Hand Side of the Equation

First, simplify the right-hand side of the equation:

[dfrac{x}{2} 5 - 3x 2][dfrac{x}{2} 7 - 3x]

Step 2: Isolate the Variable Term on One Side of the Equation

Next, we want to get all the terms involving the variable (x) on one side and the constant terms on the other side. Add (3x) to both sides of the equation:

[dfrac{x}{2} 3x 7 - 3x 3x][dfrac{x}{2} 3x 7]

Step 3: Combine Like Terms

Since the term (dfrac{x}{2}) can be rewritten as (dfrac{1}{2}x) or (dfrac{x}{2}), we need to combine the terms involving (x). To do this, express (3x) with a common denominator of 2:

[dfrac{x}{2} dfrac{6x}{2} 7][dfrac{7x}{2} 7]

Step 4: Solve for (x)

Now, we have the simplified equation (dfrac{7x}{2} 7). To solve for (x), multiply both sides of the equation by 2:

[dfrac{7x}{2} cdot 2 7 cdot 2][7x 14]

Finally, divide both sides by 7:

[x dfrac{14}{7}][x 2]

Explanation and Tips for Solving Equations

1. **Understanding the Equation:** Clearly understand the equation given. This involves looking at it from left to right, identifying variables and constants, and the operations being performed.

2. **Algebraic Manipulation:** Perform algebraic operations, like adding or subtracting terms, to isolate the variable on one side of the equation. Remember to do the same operation to both sides of the equation to maintain balance.

3. **Combining Like Terms:** Combine terms with the same variable or constants to simplify the equation. This often involves finding a common denominator or performing arithmetic operations.

4. **Solving for the Variable:** Once the variable term is isolated on one side, solve for the variable by performing the necessary arithmetic operations, such as multiplication or division.

5. **Checking the Solution:** After finding the value of the variable, substitute the value back into the original equation to ensure it satisfies the equation. This step is crucial to confirm the solution is correct.

Examples of Similar Problems

Let's look at a similar problem to reinforce our understanding of the steps involved:

Problem: (dfrac{y}{3} 4 - 2y 1)

Solution:

Simplify the right-hand side: (dfrac{y}{3} 4 - 2y 1) (dfrac{y}{3} 5 - 2y) Isolate the variable term: Add (2y) to both sides: (dfrac{y}{3} 2y 5 - 2y 2y) (dfrac{y}{3} dfrac{6y}{3} 5) (dfrac{7y}{3} 5) Solve for (y): Multiply both sides by 3: (dfrac{7y}{3} cdot 3 5 cdot 3) (7y 15) Divide both sides by 7: (y dfrac{15}{7})

Conclusion

Solving equations like (dfrac{x}{2} 5 - 3x 2) involves a series of algebraic steps, including simplification, manipulation, and solving for the variable. By following a structured approach, you can effectively solve such equations and gain confidence in algebraic problem-solving. The key lies in understanding the equation, performing accurate operations, and verifying your solution.