91Ó°ÊÓ

Solving equations is like unraveling a mystery. We dig through layers of mathematical operations to discover the value of the unknown variable. In our exercise, the goal is to find what number, when substituted for the variable \( w \), makes the equation true. To do this, we follow a systematic approach:

• Identify operations involving the variable.
• Use inverse operations to isolate the variable on one side of the equation.
• Simplify the equation until the answer is clear.

Let's take the given equation \( \frac{4}{9}w - 11 = 1 \). Our first step is to get rid of any constants on the same side as the variable. We can do this by using inverse operations like addition or subtraction. After the constants are cleared, our next goal is to handle any coefficients, which is nicely wrapped up in our next concept: algebraic manipulation.

Algebraic manipulation refers to the use of algebraic operations to transform equations and expressions for a desired output. Here, we use these skills to isolate the variable \( w \). By algebraically manipulating the equation, we aim to eliminate the fractional coefficient that is tied to our variable.
First, we handle this by adding 11 to both sides:

• This neutralizes the \(-11\) attached to the \(w\) term, resulting in \( \frac{4}{9}w = 12 \).

Next, we address the coefficient \( \frac{4}{9} \) by multiplying both sides of the equation by its reciprocal, \( \frac{9}{4} \). This effectively makes the \( \frac{4}{9} \) disappear, as it cancels itself out:

• This leaves us with \(w = \frac{9}{4} \cdot 12\).
• Upon simplifying, \(w\) equates to 27.

The use of algebraic manipulation helps break down complex equations into more manageable parts, ultimately simplifying the path to the correct answer.

After solving an equation, checking your solution is a crucial step. It confirms that the value calculated indeed satisfies the original equation. In our case, once we found \( w = 27 \), we return to the original equation to validate our result:
Substitute \( w = 27 \) back into the equation:

• \(\frac{4}{9}(27) - 11 = 1\)
• This simplifies to \(12 - 11 = 1\)
• 1 equals 1, thereby confirming the accuracy of our solution.

Checking the solution not only gives peace of mind but also ensures there are no calculation errors. It is a best practice in solving algebraic problems.

Fractional coefficients can initially seem daunting, as they add an extra layer of complexity to equations. However, understanding how to manipulate these fractions can make solving equations straightforward.
For the equation \(\frac{4}{9} w - 11 = 1\), the fractional coefficient \(\frac{4}{9}\) is multiplied by the variable \(w\). To solve for \(w\), we need to "clear" this fraction:

• Multiply both sides by the reciprocal \(\frac{9}{4}\).
• This action essentially cancels the fraction on the side with \(w\), simplifying your equation.
• Remember that fractions multiply by reciprocals result into unity, i.e., \(\frac{a}{b} \cdot \frac{b}{a} = 1\).

Understanding and implementing this technique turns a seemingly complex equation into a more familiar linear one, making it much easier to solve.