solve for . simplify your answer as much as possible

2 min read 17-09-2024

solve for . simplify your answer as much as possible

In the world of mathematics, particularly in algebra, solving for variables is a fundamental skill. In this article, we'll discuss how to solve algebraic equations, along with methods to simplify your answers as much as possible. We'll provide practical examples and refer to insights gathered from Stack Overflow to enhance your understanding.

What Does It Mean to Solve for a Variable?

When asked to "solve for ( x )" in an equation, the goal is to isolate ( x ) on one side of the equation. Simplifying the expression that includes ( x ) is also crucial.

Example Problem

Let's consider a basic algebraic equation:

[ 2x + 4 = 12 ]

Step-by-Step Solution

Isolate the Variable: Start by subtracting 4 from both sides:

[ 2x + 4 - 4 = 12 - 4 ] This simplifies to:

[ 2x = 8 ]
Divide to Solve for ( x ): Next, divide both sides by 2:

[ \frac{2x}{2} = \frac{8}{2} ] Therefore, we find:

[ x = 4 ]

Verification

To verify, we can substitute ( x ) back into the original equation:

[ 2(4) + 4 = 12 ]

This simplifies to:

[ 8 + 4 = 12 ]

Since both sides are equal, our solution is confirmed.

Simplifying More Complex Expressions

Stack Overflow users often face more complicated equations. Here’s a slightly advanced problem based on user queries:

Problem

[ 3(x - 2) + 4 = 2(x + 3) ]

Solution Steps

Distribute: Distributing the numbers gives us:

[ 3x - 6 + 4 = 2x + 6 ] This simplifies to:

[ 3x - 2 = 2x + 6 ]
Isolate ( x ): Subtract ( 2x ) from both sides:

[ 3x - 2x - 2 = 6 ] Which simplifies to:

[ x - 2 = 6 ]
Solve for ( x ): Finally, add 2 to both sides:

[ x = 8 ]

Checking the Solution

Substituting back into the original equation:

[ 3(8 - 2) + 4 = 2(8 + 3) ]

This simplifies to:

[ 3(6) + 4 = 2(11) ]

Which gives:

[ 18 + 4 = 22 ]

Both sides equal 22, confirming our solution.

Best Practices for Simplifying Answers

Combine Like Terms: Always look for terms that can be combined, as seen in the previous examples.
Factor When Possible: This can help simplify complex equations before solving.
Use the Distributive Property Wisely: This often helps to eliminate parentheses and simplify the equation.

Conclusion

Solving for a variable and simplifying the answer is a vital skill in algebra. With practice, problems that seem complex at first can become manageable. By applying systematic approaches—like isolating variables, combining like terms, and checking solutions—you can enhance your algebraic proficiency.

Feel free to dive into your problems with confidence, knowing that you have a clear strategy to isolate and simplify variables!

References

Stack Overflow Community Contributions (Various authors)