Solve For X. Round To The Nearest Hundredth

Solve for x: A Comprehensive Guide to Solving for Unknowns, with Rounding to the Nearest Hundredth

Solving for 'x' is a fundamental concept in algebra, representing the process of finding the value of an unknown variable within an equation. This seemingly simple task underlies much of mathematics and its applications in science, engineering, and everyday life. This article will provide a comprehensive guide to solving for 'x' in various equation types, including detailed explanations, examples, and a focus on rounding to the nearest hundredth. We'll cover everything from basic linear equations to more complex scenarios, ensuring you gain a solid understanding of this essential skill.

Understanding the Basics: What Does "Solve for x" Mean?

When we say "solve for x," we're essentially asking: "What value of 'x' makes this equation true?" An equation is a statement that two mathematical expressions are equal. For example, 2x + 5 = 11 is an equation. Solving this equation means finding the value of 'x' that makes the left side (2x + 5) equal to the right side (11).

The core principle behind solving for 'x' is maintaining the balance of the equation. Whatever operation you perform on one side, you must perform on the other side to keep the equation equivalent. This ensures that the solution you find is valid.

Solving Linear Equations: The Foundation

Linear equations are equations where the highest power of the variable 'x' is 1. They are the simplest type of equation to solve and form the basis for understanding more complex equation types.

Steps to Solve a Linear Equation:

1. Simplify both sides: Combine like terms on each side of the equation. This involves adding or subtracting similar terms (e.g., combining all the 'x' terms and all the constant terms).

2. Isolate the 'x' term: Use addition or subtraction to move all terms without 'x' to one side of the equation.

3. Solve for 'x': Use multiplication or division to isolate 'x' completely. This means getting 'x' by itself on one side of the equation.

Example 1: Solve for x: 3x + 7 = 16

1. Simplify: The equation is already simplified.

2. Isolate 'x': Subtract 7 from both sides: 3x + 7 - 7 = 16 - 7, which simplifies to 3x = 9.

3. Solve for 'x': Divide both sides by 3: 3x / 3 = 9 / 3, resulting in x = 3.

Example 2: Solve for x: 5x - 2 = 2x + 10

1. Simplify: This equation requires moving the 'x' terms to one side and the constant terms to the other. Let's subtract 2x from both sides: 5x - 2x - 2 = 2x - 2x + 10, simplifying to 3x - 2 = 10.

2. Isolate 'x': Add 2 to both sides: 3x - 2 + 2 = 10 + 2, resulting in 3x = 12.

3. Solve for 'x': Divide both sides by 3: 3x / 3 = 12 / 3, giving x = 4.

Solving Quadratic Equations: Stepping Up the Complexity

Quadratic equations are equations where the highest power of the variable 'x' is 2. They are generally of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants. Solving quadratic equations often involves more steps and potentially multiple solutions for 'x'.

Methods for Solving Quadratic Equations:

• Factoring: This method involves expressing the quadratic equation as a product of two linear factors. For example, x² + 5x + 6 = 0 can be factored as (x + 2)(x + 3) = 0. The solutions are then x = -2 and x = -3. Factoring is not always possible for all quadratic equations.

• Quadratic Formula: This is a general formula that can be used to solve any quadratic equation:

  x = (-b ± √(b² - 4ac)) / 2a

  This formula will yield two solutions for 'x', represented by the ± symbol (plus or minus).

• Completing the Square: This is a less commonly used method but involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Example 3: Solve for x: x² - 4x + 3 = 0 (using factoring)

This equation can be factored as (x - 1)(x - 3) = 0. Therefore, the solutions are x = 1 and x = 3.

Example 4: Solve for x: 2x² + 5x - 3 = 0 (using the quadratic formula)

Here, a = 2, b = 5, and c = -3. Substituting these values into the quadratic formula gives:

x = (-5 ± √(5² - 4 * 2 * -3)) / (2 * 2)

x = (-5 ± √(49)) / 4

x = (-5 ± 7) / 4

This gives two solutions: x = (2) / 4 = 0.5 and x = (-12) / 4 = -3.

Solving Equations with Fractions and Decimals

Equations involving fractions or decimals require careful handling to maintain the balance. A common strategy is to eliminate fractions by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions. Similarly, multiplying both sides by a power of 10 can eliminate decimals.

Example 5: Solve for x: (1/2)x + 3 = 7

Multiply both sides by 2 (the LCD): 2 * ((1/2)x + 3) = 7 * 2, which simplifies to x + 6 = 14. Subtracting 6 from both sides gives x = 8.

Example 6: Solve for x: 0.25x - 1.5 = 2

Multiply both sides by 100 to eliminate the decimals: 100 * (0.25x - 1.5) = 2 * 100, which simplifies to 25x - 150 = 200. Adding 150 to both sides and then dividing by 25 gives x = 14.

Rounding to the Nearest Hundredth

When solving for 'x', the answer is often not a whole number. Rounding to the nearest hundredth means expressing the answer with two decimal places. This involves looking at the third decimal place:

• If the third decimal place is 5 or greater, round the second decimal place up.
• If the third decimal place is less than 5, keep the second decimal place as it is.

Example 7: Suppose solving an equation yields x = 3.14159. Rounding to the nearest hundredth gives x ≈ 3.14.

Example 8: Suppose solving an equation yields x = 2.786. Rounding to the nearest hundredth gives x ≈ 2.79.

Solving Systems of Equations

A system of equations involves two or more equations with two or more variables. Solving a system means finding values for all the variables that satisfy all the equations simultaneously. Common methods include substitution and elimination.

Substitution: Solve one equation for one variable in terms of the other, and substitute that expression into the other equation.

Elimination: Multiply equations by constants to make the coefficients of one variable opposite, then add the equations to eliminate that variable.

Solving Exponential and Logarithmic Equations

These involve equations with exponents or logarithms. Techniques for solving these often involve using logarithmic properties or exponential properties to simplify the equations.

Dealing with Extraneous Solutions

Sometimes, when solving equations (particularly those involving radicals or absolute values), you might obtain solutions that don't actually satisfy the original equation. These are called extraneous solutions and must be discarded. Always check your solutions by plugging them back into the original equation.

Frequently Asked Questions (FAQ)

Q: What if I get a negative value for 'x'?

A: Negative values are perfectly valid solutions for 'x'.

Q: What if I get a fraction as a solution?

A: Fractions are also valid solutions. You might need to convert them to decimal form and round to the nearest hundredth if required.

Q: What if I make a mistake during the solving process?

A: Carefully review your steps. Double-check your arithmetic and ensure you're applying the rules of algebra correctly. It's often helpful to work through the problem step-by-step, writing each step clearly.

Q: Are there online calculators or tools to help solve for x?

A: Yes, many online calculators and software programs can solve for 'x' in various types of equations. However, it's crucial to understand the underlying principles yourself to apply these methods effectively in different contexts.

Conclusion

Solving for 'x' is a fundamental skill in algebra and a gateway to understanding more advanced mathematical concepts. Mastering this skill requires practice and a solid understanding of the fundamental principles of equation manipulation. By diligently following the steps outlined in this article and working through various examples, you can build confidence and proficiency in solving equations of varying complexity, accurately rounding your solutions to the nearest hundredth as needed. Remember that practice is key – the more you solve, the better you'll become!