Problem 54 - Algebra A Combined Function

Question

Twice the sum of a number and six equals three times the sum of the number and four. Find the number.

Step-by-Step Solution

Answer

The number is 0.

Step 1: Understanding the Problem

Let's translate the problem into a mathematical equation. We have two expressions: "twice the sum of a number and six" and "three times the sum of the number and four." Let the unknown number be \( x \). So, the problem states that \( 2(x + 6) = 3(x + 4) \).

Step 2: Simplifying the Equation

Begin by expanding both sides of the equation. Expand \( 2(x + 6) \) to get \( 2x + 12 \), and \( 3(x + 4) \) to get \( 3x + 12 \). So, the equation now is \( 2x + 12 = 3x + 12 \).

Step 3: Isolating the Variable

Subtract \( 2x \) from both sides to remove \( 2x \) from the left side of the equation. The equation now becomes \( 12 = x + 12 \).

Step 4: Solving the Simplified Equation

Subtract 12 from both sides to isolate \( x \). This gives us \( 0 = x \).

Step 5: Conclusion

The solution to the equation is \( x = 0 \). This means the number we're looking for is 0.

Key Concepts

Linear Equations

Linear equations are one of the foundational concepts in algebra. They involve expressions where the highest power of the variable is one. The standard form of a linear equation in one variable is given by the expression \( ax + b = 0 \), where \( a \) and \( b \) are constants, and \( x \) is the variable we need to solve for. In the given exercise, we work with two linear expressions equated to each other: \( 2(x + 6) = 3(x + 4) \). Linear equations can graphically be represented as straight lines on a coordinate plane. They're crucial for understanding more complex mathematical concepts and are a key part of mathematical literacy.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. When dealing with linear equations like \( 2(x + 6) = 3(x + 4) \), our goal is to simplify the equation step by step until we isolate the variable. Here's how you can solve it effectively:

• First, expand both sides of the equation to remove any parentheses: for our problem, \( 2(x + 6) \) becomes \( 2x + 12 \), and \( 3(x + 4) \) becomes \( 3x + 12 \).
• Next, simplify the equation by getting all the \( x \) terms on one side and constant terms on the other. In this exercise, subtracting \( 2x \) from both sides leaves us with \( 12 = x + 12 \).
• Finally, isolate \( x \) by subtracting 12 from both sides, leading to \( x = 0 \).

Solving equations is a systematic process that requires careful application of algebraic principles. Practice strengthens these skills.

Mathematical Problem Solving

Mathematical problem solving involves identifying the problem, translating it into a mathematical form, and applying logical reasoning to find a solution. It requires an understanding of the concepts and the ability to apply appropriate methods. In our given exercise:

• We first translated a verbal problem into a mathematical equation, which involves understanding the phrases "twice the sum" and "three times the sum." This requires strong comprehension skills.
• The step-by-step approach in problem-solving, such as expanding terms and isolating variables, showcases logical progression.
• Finally, verifying the result ensures the solution is accurate and reflects the problem's requirements.

Developing mathematical problem-solving skills is not just about solving equations but also about applying this logical thinking to various problems in mathematics and real-life scenarios. Mastery of these skills aids in deeper learning and understanding of mathematics.