2(4x-3)-8=4+2x ?

Solving equations is a fundamental skill in mathematics with wide-ranging applications in various fields. In this 1000-word blog post, we will tackle the equation 2(4x-3)-8=4+2x step by step, exploring the algebraic techniques and principles involved. We’ll also discuss the importance of equation-solving in real-life scenarios, from engineering to finance, and how this mathematical skill can empower problem-solving and decision-making.

2(4x-3)-8=4+2x ?

The equation 2(4x-3)-8=4+2x simplifies to 8x – 16 = 4 + 2x. After rearranging and solving, it results in 6x = 20. Dividing by 6 gives x = 20/6, which simplifies to x = 10/3. So, the solution to the equation is x = 10/3.

Understanding the Equation

Let’s begin by dissecting the given equation: 2(4x-3)-8=4+2x.

This equation contains variables (in this case, “x”), constants (numbers without variables), and mathematical operations (addition, subtraction, multiplication).

Step 1: Simplifying the Equation

Our first step is to simplify the equation by applying the distributive property, which means multiplying the 2 outside the parentheses by each term inside the parentheses:

2(4x-3) = 8x – 6

Now, our equation becomes:

8x – 6 – 8 = 4 + 2x

Step 2: Combining Like Terms

Next, we’ll simplify further by combining like terms. We gather all the terms with “x” on one side of the equation and constants on the other side.

8x – 2x = 4 + 6 – 8

This simplifies to:

6x = 2

Step 3: Isolating the Variable

To solve for “x,” we need to isolate it on one side of the equation. To do this, we’ll divide both sides by the coefficient of “x,” which is 6:

(6x)/6 = 2/6

Simplifying further:

x = 1/3

Step 4: Checking the Solution

Now that we’ve found a solution for “x,” it’s crucial to verify it by substituting it back into the original equation to ensure it satisfies the equation:

2(4(1/3) – 3) – 8 = 4 + 2(1/3)

Let’s break it down:

1. Inside the parentheses: 4(1/3) – 3 simplifies to (4/3) – 3.
2. Multiplying by 2 outside the parentheses: 2[(4/3) – 3].
3. Continuing simplification: 2[(4/3) – 9/3] = 2[(4 – 9)/3] = 2(-5/3).

On the right side:

1. 2(1/3) = 2/3.

Now, the equation becomes:

2(-5/3) – 8 = 4 + 2/3

Simplify both sides:

-10/3 – 8 = 4 + 2/3

Combining constants:

-10/3 – 24/3 = 4 + 2/3

-34/3 = 4 + 2/3

Now, let’s find a common denominator on the right side:

-34/3 = 12/3 + 2/3

Simplify:

-34/3 = 14/3

The equation is not balanced; the left side is not equal to the right side. This discrepancy means there is no solution to the original equation.

Understanding the No Solution Scenario

In this case, the equation “2(4x-3)-8=4+2x” leads to an interesting result: there is no solution that satisfies the equation. This can happen in algebra when the variable terms cancel out during simplification, leaving contradictory statements.

In practical terms, this no-solution scenario could represent situations where the conditions set by the equation are impossible to fulfill. It’s essential to recognize such cases, as they can indicate errors in modeling real-world problems or highlight constraints that need to be considered.

Real-World Applications

Equation-solving is not limited to algebra classrooms; it plays a crucial role in various real-world applications:

1. Engineering: Engineers use equations to design structures, solve mechanical problems, and optimize systems. Solving equations helps ensure the safety and efficiency of engineering projects.

2. Finance: Financial professionals use equations to model investment scenarios, calculate interest rates, and manage portfolios. Accurate calculations are essential for making informed financial decisions.

3. Physics: Equations are the language of physics, enabling scientists to describe and predict physical phenomena. Solving equations is at the core of understanding the natural world.

4. Medicine: Medical researchers use equations to model the behavior of biological systems, analyze medical data, and develop treatment strategies.

5. Economics: Economists use equations to study economic trends, forecast future conditions, and analyze the impact of policies. Equations are essential for making informed economic decisions.

Conclusion

Solving equations is a fundamental mathematical skill with a profound impact on various aspects of our lives. While the equation “2(4x-3)-8=4+2x” resulted in no solution, it illustrates the importance of equation-solving as a tool for problem-solving, decision-making, and modeling real-world scenarios.

In mathematics, finding solutions is not just about solving for “x”; it’s about exploring the relationships between variables, recognizing constraints, and understanding when certain scenarios are mathematically impossible. Equation-solving empowers us to tackle complex problems and make informed decisions in a wide range of fields, from engineering and finance to physics and medicine.