Solving For X In The Equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x)
In this article, we will delve into the process of finding the value of x in the given equation: 2.5(6x - 4) = 10 + 4(1.5 + 0.5x). This is a typical algebraic problem that requires us to apply the principles of distribution, simplification, and isolation of the variable. By systematically working through each step, we will arrive at the solution for x. Understanding how to solve such equations is crucial for various applications in mathematics and real-world scenarios.
Understanding the Equation
Before we begin, let's take a closer look at the equation: 2.5(6x - 4) = 10 + 4(1.5 + 0.5x). This equation involves both multiplication and addition operations, as well as the variable x. Our goal is to isolate x on one side of the equation to determine its value. The left side of the equation, 2.5(6x - 4), indicates that we need to multiply 2.5 by both terms inside the parentheses, which are 6x and -4. Similarly, on the right side, 4(1.5 + 0.5x) implies multiplying 4 by both 1.5 and 0.5x. The constants 10, -4, 1.5, and the coefficients 2.5, 6, and 0.5 play significant roles in determining the final value of x. Careful attention to these details is essential for accurate calculations.
Step-by-Step Solution
1. Distribute the Constants
The first step in solving the equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x) is to distribute the constants outside the parentheses to the terms inside. On the left side, we multiply 2.5 by both 6x and -4:
2.5 * 6x = 15x
2.5 * -4 = -10
So, the left side of the equation becomes 15x - 10. On the right side, we multiply 4 by both 1.5 and 0.5x:
4 * 1.5 = 6
4 * 0.5x = 2x
Thus, the right side of the equation becomes 10 + 6 + 2x. Now our equation looks like this:
15x - 10 = 10 + 6 + 2x
This distribution step simplifies the equation and prepares it for further simplification and solving.
2. Simplify Both Sides
After distributing the constants, our equation is 15x - 10 = 10 + 6 + 2x. The next step involves simplifying both sides of the equation by combining like terms. On the left side, there are no like terms to combine, so 15x - 10 remains as it is. On the right side, we can combine the constants 10 and 6:
10 + 6 = 16
So, the right side simplifies to 16 + 2x. Now, our equation is:
15x - 10 = 16 + 2x
This simplification makes the equation easier to work with and brings us closer to isolating the variable x.
3. Isolate the Variable Term
Now that we have simplified the equation to 15x - 10 = 16 + 2x, we need to isolate the variable term (x) on one side of the equation. A common approach is to move all terms containing x to one side and all constant terms to the other side. To do this, we can subtract 2x from both sides of the equation:
15x - 10 - 2x = 16 + 2x - 2x
This simplifies to:
13x - 10 = 16
Next, we want to isolate the 13x term, so we add 10 to both sides of the equation:
13x - 10 + 10 = 16 + 10
This simplifies to:
13x = 26
Now, we have successfully isolated the variable term on one side of the equation.
4. Solve for x
With the variable term isolated, our equation is now 13x = 26. To solve for x, we need to divide both sides of the equation by the coefficient of x, which is 13:
13x / 13 = 26 / 13
This simplifies to:
x = 2
Therefore, the value of x that satisfies the equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x) is 2. This final step completes the solution process, providing us with the numerical value of x.
Verification
To ensure the correctness of our solution, it is essential to verify whether the calculated value of x satisfies the original equation. We substitute x = 2 back into the equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x):
Left side:
2. 5(6 * 2 - 4) = 2.5(12 - 4) = 2.5 * 8 = 20
Right side:
3. + 4(1.5 + 0.5 * 2) = 10 + 4(1.5 + 1) = 10 + 4 * 2.5 = 10 + 10 = 20
Since both sides of the equation are equal when x = 2, our solution is correct. This verification step confirms that we have accurately solved for x.
Conclusion
In this article, we have successfully found the value of x in the equation 2.5(6x - 4) = 10 + 4(1.5 + 0.5x). By systematically applying the principles of distribution, simplification, isolation of the variable, and verification, we determined that x = 2. This step-by-step approach is fundamental in solving algebraic equations and forms the basis for more complex mathematical problem-solving. Understanding these techniques is invaluable for students and anyone working with mathematical concepts in various fields.
Key takeaways from this solution include:
- Distributing constants to simplify equations.
- Combining like terms to reduce complexity.
- Isolating the variable term to solve for the unknown.
- Verifying the solution to ensure accuracy.
These skills are not only crucial for solving algebraic equations but also for building a strong foundation in mathematics and its applications.
