Two More Than 4 Times A Number Is -18

Two More Than 4 Times a Number is -18: A Deep Dive into Solving Algebraic Equations

This seemingly simple statement, "Two more than 4 times a number is -18," hides a powerful lesson in algebraic problem-solving. This article will not only guide you through solving this specific equation but also delve into the broader concepts of translating word problems into algebraic expressions, understanding the order of operations (PEMDAS/BODMAS), and building a strong foundation in algebra. We'll explore different methods of solving the equation, discuss potential pitfalls, and even look at how to apply these skills to more complex problems.

Understanding the Problem: From Words to Algebra

The first step in tackling any word problem is to translate the words into a mathematical expression. Let's break down the statement piece by piece:

• "a number": This represents our unknown value, which we typically denote with a variable, such as x.
• "4 times a number": This translates directly to 4x or 4x.
• "Two more than 4 times a number": This means we add 2 to the expression we just created, resulting in 4x + 2.
• "is -18": This indicates that the entire expression equals -18.

Therefore, the complete algebraic equation representing the statement is:

4x + 2 = -18

Solving the Equation: Unveiling the Mystery Number

Now that we have our equation, we can use various algebraic techniques to solve for x. We'll explore two common methods:

Method 1: The Subtraction and Division Method

This method involves systematically isolating the variable x by performing inverse operations.

1. Subtract 2 from both sides: This eliminates the constant term on the left side, leaving only the term with x.

   4x + 2 - 2 = -18 - 2

   4x = -20

2. Divide both sides by 4: This isolates x and reveals its value.

   4x / 4 = -20 / 4

   x = -5

Therefore, the number is -5.

Method 2: The Reverse Order of Operations Method (PEMDAS/BODMAS in Reverse)

This method uses the order of operations (PEMDAS/BODMAS) in reverse. PEMDAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS is the equivalent British acronym. To solve the equation, we reverse this order:

1. Subtraction: We begin by subtracting 2 from both sides, as in Method 1. This is the reverse of the last step in PEMDAS/BODMAS (addition).

   4x + 2 - 2 = -18 - 2

   4x = -20

2. Division: Next, we divide both sides by 4, the reverse of multiplication.

   4x / 4 = -20 / 4

   x = -5

Again, we find that the number is -5.

Verification: Checking Your Solution

It's crucial to verify your solution by substituting the value of x back into the original equation:

4x + 2 = -18

4(-5) + 2 = -20 + 2 = -18

Since the equation holds true, our solution, x = -5, is correct.

Expanding Your Horizons: More Complex Problems

The principles illustrated here can be applied to more complex algebraic equations. Consider these examples:

Example 1: Three less than twice a number is 11.

1. Translate into an equation: 2x - 3 = 11
2. Solve: Add 3 to both sides: 2x = 14; Divide both sides by 2: x = 7
3. Verify: 2(7) - 3 = 14 - 3 = 11 (Correct!)

Example 2: The sum of a number and its square is 30.

1. Translate into an equation: x + x² = 30
2. Solve: This requires rearranging the equation into a quadratic equation (x² + x - 30 = 0) and then solving using factoring, the quadratic formula, or completing the square. This yields solutions x = 5 and x = -6.
3. Verify: 5 + 5² = 30 (Correct!), and -6 + (-6)² = 30 (Correct!)

Example 3: Involving Fractions: One-half of a number increased by 5 is 12.

1. Translate into an equation: (1/2)x + 5 = 12
2. Solve: Subtract 5 from both sides: (1/2)x = 7; Multiply both sides by 2: x = 14
3. Verify: (1/2)(14) + 5 = 7 + 5 = 12 (Correct!)

Common Mistakes to Avoid

• Incorrect Order of Operations: Remember to follow PEMDAS/BODMAS diligently. Failing to do so can lead to inaccurate results.
• Errors in Simplification: Carefully check your work at each step to avoid arithmetic errors.
• Forgetting to Verify: Always substitute your solution back into the original equation to ensure accuracy.

Building a Strong Foundation in Algebra

Mastering algebraic equation solving is fundamental to success in many areas, including:

• Higher-Level Mathematics: Algebra forms the basis for calculus, linear algebra, and other advanced mathematical disciplines.
• Science and Engineering: Many scientific and engineering problems rely on algebraic modeling and problem-solving techniques.
• Computer Science: Programming often involves writing algorithms and solving equations to manage data and control program flow.
• Financial Modeling: Budgeting, investment analysis, and financial forecasting all require proficiency in algebraic concepts.

By consistently practicing and applying these principles, you'll build a strong foundation in algebra that will serve you well in your academic and professional pursuits.

Conclusion: Unlocking the Power of Algebraic Problem Solving

The seemingly simple problem, "Two more than 4 times a number is -18," serves as an excellent introduction to the power of algebraic problem-solving. By breaking down word problems into equations, applying systematic solution methods, and verifying our results, we can confidently tackle even more complex challenges. Remember to practice regularly, pay attention to detail, and never underestimate the importance of verification. The skills you gain will be invaluable in numerous aspects of your life. So keep practicing, keep learning, and keep unlocking the power of algebra!