Find The Value Of X. Round To The Nearest Hundredth

Find the Value of x: A Comprehensive Guide to Solving for x and Rounding to the Nearest Hundredth

Finding the value of 'x' is a fundamental concept in algebra and mathematics. It involves manipulating equations to isolate 'x' and determine its numerical value. This process is crucial in various fields, from engineering and physics to finance and computer science. This comprehensive guide will explore different methods for finding the value of 'x,' focusing on techniques that lead to accurate solutions and rounding to the nearest hundredth.

Understanding the Basics: Equations and Variables

Before diving into the methods, let's clarify some essential terms. An equation is a mathematical statement that asserts the equality of two expressions. It typically contains one or more variables, which are symbols (usually letters like x, y, z) representing unknown values. The goal of solving an equation is to determine the value(s) of the variable(s) that make the equation true.

For example, in the equation 2x + 5 = 11, 'x' is the variable, and our goal is to find the value of 'x' that satisfies this equality.

Methods for Solving for x

Various methods exist for solving equations depending on their complexity. Let's explore some common techniques:

1. Solving Linear Equations

Linear equations are equations where the highest power of the variable is 1. These equations are relatively straightforward to solve. The general strategy involves using inverse operations to isolate the variable.

Example: Solve for x in 3x + 7 = 16

1. Subtract 7 from both sides: 3x + 7 - 7 = 16 - 7 => 3x = 9
2. Divide both sides by 3: 3x / 3 = 9 / 3 => x = 3

Therefore, the value of x is 3.

Example (with decimals): Solve for x in 2.5x - 4.2 = 8.8

1. Add 4.2 to both sides: 2.5x - 4.2 + 4.2 = 8.8 + 4.2 => 2.5x = 13
2. Divide both sides by 2.5: 2.5x / 2.5 = 13 / 2.5 => x = 5.2

Therefore, the value of x is 5.2.

2. Solving Quadratic Equations

Quadratic equations are equations where the highest power of the variable is 2. These equations generally have two solutions (roots). Several methods can solve quadratic equations, including:

• Factoring: This method involves expressing the quadratic equation as a product of two linear factors.
• Quadratic Formula: This formula provides a direct solution for any quadratic equation of the form ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / 2a

• Completing the Square: This method involves manipulating the equation to form a perfect square trinomial.

Example (using the quadratic formula): Solve for x in x² + 5x + 6 = 0

Here, a = 1, b = 5, and c = 6. Applying the quadratic formula:

x = [-5 ± √(5² - 4 * 1 * 6)] / (2 * 1) x = [-5 ± √(25 - 24)] / 2 x = [-5 ± √1] / 2 x = (-5 ± 1) / 2

This gives two solutions:

x₁ = (-5 + 1) / 2 = -2 x₂ = (-5 - 1) / 2 = -3

Therefore, the values of x are -2 and -3.

Example (with decimals, using the quadratic formula): Solve for x in 2x² - 3x - 2 = 0

Here, a = 2, b = -3, and c = -2. Applying the quadratic formula:

x = [3 ± √((-3)² - 4 * 2 * -2)] / (2 * 2) x = [3 ± √(9 + 16)] / 4 x = [3 ± √25] / 4 x = (3 ± 5) / 4

This gives two solutions:

x₁ = (3 + 5) / 4 = 2 x₂ = (3 - 5) / 4 = -0.5

Therefore, the values of x are 2 and -0.5.

3. Solving Systems of Equations

Systems of equations involve multiple equations with multiple variables. To solve these systems, methods like substitution, elimination, or matrix methods can be employed.

Example (substitution method): Solve for x and y:

x + y = 5 x - y = 1

From the second equation, we can express x as x = y + 1. Substituting this into the first equation:

(y + 1) + y = 5 2y + 1 = 5 2y = 4 y = 2

Now, substitute y = 2 back into either original equation to solve for x:

x + 2 = 5 x = 3

Therefore, x = 3 and y = 2.

4. Solving Exponential and Logarithmic Equations

These equations involve exponents and logarithms. Techniques for solving these equations often involve using logarithmic properties or exponentiation.

Example (exponential equation): Solve for x in 2ˣ = 16

We can rewrite 16 as 2⁴, so the equation becomes:

2ˣ = 2⁴

Therefore, x = 4.

Example (logarithmic equation): Solve for x in log₂(x) = 3

By definition of logarithms, this equation is equivalent to:

2³ = x

Therefore, x = 8.

Rounding to the Nearest Hundredth

Once you've found the value of x, you may need to round the result to the nearest hundredth. This involves considering the digit in the thousandths place:

• If the digit in the thousandths place is 5 or greater, round up the hundredths digit.
• If the digit in the thousandths place is less than 5, keep the hundredths digit as it is.

Example:

• x = 3.14159... rounded to the nearest hundredth is 3.14
• x = 2.71828... rounded to the nearest hundredth is 2.72
• x = 1.999 rounded to the nearest hundredth is 2.00

Practical Applications and Advanced Techniques

The methods described above are foundational. More complex equations might require advanced techniques, including:

• Numerical methods: For equations that are difficult or impossible to solve analytically, numerical methods (like the Newton-Raphson method) provide approximate solutions.
• Calculus: Calculus provides tools for solving equations involving derivatives and integrals.
• Software and calculators: Software like MATLAB, Mathematica, or even advanced calculators can assist in solving complex equations.

Conclusion: Mastering the Art of Solving for x

Finding the value of x is a fundamental skill with wide-ranging applications. Mastering the various methods discussed in this guide, along with understanding how to round to the nearest hundredth, will empower you to tackle a variety of mathematical problems effectively. Remember to practice regularly and explore more advanced techniques as needed to broaden your mathematical proficiency. The ability to accurately and efficiently solve for x is a cornerstone of success in many quantitative disciplines.