91Ó°ÊÓ

In solving linear equations, you often encounter expressions within parentheses, especially with a negative sign before them. Distributing the negative sign properly is crucial. When you have an expression like \( -(\frac{3}{4}x - 9) \), you need to apply the negative sign to both terms inside the parentheses. This changes the sign of each term inside:
\( -(\frac{3}{4}x - 9) = -\frac{3}{4}x + 9 \).

Remember: The negative sign flips the plus to a minus and vice versa.
This step ensures that you correctly simplify and combine your terms later on.

After distributing the negative sign, the next crucial step is to combine like terms. This means adding or subtracting terms that have the same variable part. For example:
\[ x - \frac{3}{4}x + 9 = \frac{73}{8} \]
Here, \ x \ and \ -\frac{3}{4}x \ are like terms because both involve the variable \ x \. Combine them to form one term:
\[ \frac{4}{4}x - \frac{3}{4}x = \frac{1}{4}x \]
This simplification helps in managing and reducing the number of terms, making the equation easier to solve.

After combining like terms, the goal is to isolate the variable. This means you want the variable on one side of the equation and the constant on the other. In the example:
\[ \frac{1}{4}x + 9 = \frac{73}{8} \]
First, subtract 9 from both sides to remove the constant term from the left:
\[ \frac{1}{4}x + 9 - 9 = \frac{73}{8} - 9 \]
This simplifies to:
\[ \frac{1}{4}x = \frac{1}{8} \]
To get \ x \ alone, multiply both sides by 4:
\[ x = 4 \times \frac{1}{8} \]
Finally, simplifying gives:
\[ x = \frac{1}{2} \].

Isolating the variable step by step ensures accuracy and helps avoid mistakes.

Checking your solution is the last but crucial step to ensure accuracy. Substitute your solution back into the original equation and verify both sides are equal. Using the solution \ x = \frac{1}{2} \ in:
\[ \frac{1}{2} - (\frac{3}{4} \times \frac{1}{2} - 9) = \frac{73}{8} \]
Simplify the terms inside the parentheses:
\[ \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \]
Then substitute and simplify:
\[ \frac{1}{2} - (\frac{3}{8} - 9) = \frac{73}{8}\]

Combine terms:
\[ \frac{1}{2} - \frac{3}{8} + 9 = \frac{73}{8} \]
Convert \ \frac{1}{2} \ to an eighth:
\[ \frac{1}{2} = \frac{4}{8} \]
Combine and verify both sides:
\[ \frac{4}{8} - \frac{3}{8} + 9 = \frac{73}{8} \]
This confirms that the solution is correct, as both sides equal \ \frac{73}{8} \.
Checking solutions helps you validate your work and solidify understanding.