Solving X = ⅘ (x + 10) A Step-by-Step Guide

Solve the equation x = ⅘ (x + 10)

This article provides a detailed, step-by-step guide to solving the algebraic equation x = ⅘ (x + 10). Understanding how to solve linear equations like this is a fundamental skill in algebra, with applications in various fields, including physics, engineering, and economics. We will break down each step, explaining the reasoning and mathematical principles involved, making it easy to follow even if you're new to algebra. By the end of this guide, you'll not only be able to solve this specific equation but also have a solid foundation for tackling similar problems. Let's dive in and unravel the solution together.

Understanding the Equation

Before we jump into the solution, let's first understand the equation itself: x = ⅘ (x + 10). This is a linear equation in one variable, x. The goal is to isolate x on one side of the equation to find its value. The equation states that x is equal to four-fifths of the quantity (x + 10). The presence of the fraction ⅘ might seem intimidating, but we will address it systematically using algebraic techniques. Mastering the basics of algebra allows us to manipulate equations and solve for unknowns. In this particular case, we need to eliminate the fraction and then simplify the equation to get x by itself. Remember, every step we take must maintain the equality of both sides, so we'll perform the same operations on both sides to keep the equation balanced. This principle is crucial in solving any algebraic equation.

Step-by-Step Solution

To solve the equation x = ⅘ (x + 10), we'll follow these steps:

1. Eliminate the Fraction: The first step is to get rid of the fraction ⅘. To do this, we multiply both sides of the equation by the denominator of the fraction, which is 5. This maintains the equality of the equation and simplifies it by removing the fraction.

   5 * x = 5 * [⅘ (x + 10)]

   This simplifies to:

   5x = 4(x + 10)

   Eliminating fractions in equations is a key technique for simplification. By multiplying both sides by the denominator, we transform the equation into a more manageable form. This step makes it easier to proceed with the rest of the solution.

2. Distribute: Next, we need to distribute the 4 on the right side of the equation. This involves multiplying the 4 by both terms inside the parentheses: x and 10.

   5x = 4 * x + 4 * 10

   This simplifies to:

   5x = 4x + 40

   The distributive property is a fundamental algebraic principle that allows us to multiply a single term by a group of terms inside parentheses. This step is essential for expanding the equation and preparing it for further simplification.

3. Isolate x Terms: Now, we want to get all the x terms on one side of the equation. To do this, we subtract 4x from both sides. This maintains the equality of the equation and groups the x terms together.

   5x - 4x = 4x + 40 - 4x

   This simplifies to:

   x = 40

   Isolating the variable is the core strategy in solving equations. By performing the same operation on both sides, we can move terms around until the variable we're solving for is alone on one side.

The Solution

After completing these steps, we arrive at the solution: x = 40. This means that the value of x that satisfies the original equation is 40. We can verify this solution by substituting x = 40 back into the original equation.

Verification

To verify our solution, we substitute x = 40 into the original equation:

40 = ⅘ (40 + 10)

First, we simplify the expression inside the parentheses:

40 = ⅘ (50)

Next, we multiply ⅘ by 50:

40 = (4/5) * 50

40 = 4 * (50/5)

40 = 4 * 10

40 = 40

Since both sides of the equation are equal, our solution x = 40 is correct. Verifying your solution is a crucial step in problem-solving. It ensures that you haven't made any errors in the process and that your answer is accurate. By substituting the solution back into the original equation, you can confirm that it satisfies the equation.

Alternative Methods

While we've solved the equation step-by-step, there are alternative approaches you could take. For instance, instead of multiplying by 5 initially, you could distribute the ⅘ first:

x = ⅘x + ⅘(10)

x = ⅘x + 8

Then, subtract ⅘x from both sides:

x - ⅘x = 8

⅕x = 8

Finally, multiply both sides by 5:

5 * (⅕x) = 5 * 8

x = 40

This alternative method also leads to the same solution. Understanding different methods for solving the same problem can enhance your problem-solving skills. It allows you to choose the approach that you find most efficient and comfortable.

Common Mistakes to Avoid

When solving equations like this, there are several common mistakes to watch out for:

• Incorrect Distribution: Make sure to distribute the number outside the parentheses to every term inside. For example, in the step 5x = 4(x + 10), it's crucial to multiply 4 by both x and 10.
• Combining Unlike Terms: You can only combine terms that are