Solutions For 3(x+10)+6=3(x+12) An Equation Analysis

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Figuring out how many solutions an equation has is a fundamental concept in algebra. In this article, we will explore the equation 3(x+10)+6=3(x+12) and determine whether it has zero, one, two, or infinitely many solutions. Understanding the nature of solutions to linear equations is crucial for problem-solving in various mathematical and real-world contexts. By systematically simplifying and analyzing the equation, we can uncover its properties and classify its solution set. This article aims to provide a detailed explanation, making the process clear and understandable for anyone studying algebra.

Understanding Linear Equations

Before diving into the specific equation, it's essential to understand the basics of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called "linear" because they describe a straight line when graphed on a coordinate plane. The general form of a linear equation in one variable is ax + b = c, where a, b, and c are constants, and x is the variable. The solutions to a linear equation are the values of the variable that make the equation true. Solving linear equations involves isolating the variable on one side of the equation by performing operations that maintain equality.

In the context of solutions, there are three possibilities for a linear equation:

  1. One Solution: The equation has a unique value for the variable that satisfies it. For example, the equation 2x + 3 = 7 has one solution, x = 2.
  2. No Solution: The equation leads to a contradiction, meaning there is no value for the variable that can make the equation true. For instance, the equation x + 1 = x + 2 has no solution.
  3. Infinitely Many Solutions: The equation is an identity, meaning it is true for all values of the variable. An example of this is the equation x + 1 = x + 1, which is always true regardless of the value of x.

Identifying which of these cases applies to a given equation requires careful algebraic manipulation and analysis. This foundational knowledge sets the stage for effectively determining the nature of the solutions for the equation 3(x+10)+6=3(x+12).

Step-by-Step Solution

To determine the number of solutions for the equation 3(x+10)+6=3(x+12), we'll follow a step-by-step approach, applying algebraic principles to simplify and analyze the equation. This process will involve distributing, combining like terms, and isolating the variable to understand the equation's true nature. Let’s start by distributing the constants on both sides of the equation.

Step 1: Distribute

The first step in solving the equation is to distribute the constants outside the parentheses to the terms inside. This means multiplying 3 by both x and 10 on the left side, and also multiplying 3 by x and 12 on the right side. This is a crucial step in simplifying the equation and revealing its underlying structure. By applying the distributive property, we can eliminate the parentheses and make the equation easier to manipulate.

  • Original Equation: 3(x+10)+6=3(x+12)
  • Distribute: 3 * x + 3 * 10 + 6 = 3 * x + 3 * 12
  • Simplified: 3x + 30 + 6 = 3x + 36

Step 2: Combine Like Terms

After distributing, the next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power, or constants. In this case, we can combine the constants 30 and 6 on the left side of the equation. This simplification helps to further clarify the equation and move closer to isolating the variable.

  • Simplified Equation: 3x + 30 + 6 = 3x + 36
  • Combine Like Terms: 3x + 36 = 3x + 36

Step 3: Analyze the Result

At this point, the equation has been simplified to 3x + 36 = 3x + 36. This is a significant result, as both sides of the equation are now exactly the same. This type of equation is known as an identity, which means it is true for any value of x. No matter what number you substitute for x, the left side will always equal the right side. This indicates that the equation has infinitely many solutions.

Determining the Number of Solutions

From our step-by-step solution, we've arrived at the equation 3x + 36 = 3x + 36. This simplified form provides a clear indication of the nature of the solutions. When an equation simplifies to an identity, it means that the equation is true for all values of the variable. In other words, there are infinitely many values of x that will satisfy the equation. This outcome is different from equations that have a unique solution or no solution at all.

Understanding Infinite Solutions

An equation with infinite solutions essentially represents a situation where both sides of the equation are equivalent, regardless of the variable's value. Graphically, if this equation were plotted, both sides would represent the same line, overlapping each other completely. This overlap signifies that every point on the line is a solution to the equation. In algebraic terms, any value substituted for x will make the equation true, confirming the presence of infinitely many solutions.

Contrasting with Other Solution Types

To further understand the concept of infinite solutions, it's helpful to contrast it with equations that have one solution or no solution. An equation with one solution has a specific value for the variable that satisfies the equation, and no other value will work. For example, the equation 2x = 4 has only one solution: x = 2. On the other hand, an equation with no solution leads to a contradiction when simplified, such as x + 1 = x + 2. In this case, no value of x can make the equation true.

Final Answer

Given the simplified equation 3x + 36 = 3x + 36, it is evident that the original equation 3(x+10)+6=3(x+12) has infinitely many solutions. This is because the equation is an identity, true for all values of x.

Conclusion

In conclusion, determining the number of solutions for an equation involves simplifying the equation and analyzing the result. For the equation 3(x+10)+6=3(x+12), through the steps of distributing and combining like terms, we found that it simplifies to 3x + 36 = 3x + 36, which is an identity. This means the equation has infinitely many solutions because any value of x will satisfy the equation. Understanding how to identify equations with infinite solutions is a key concept in algebra. This skill is not only essential for academic success but also for problem-solving in various real-world scenarios. The ability to manipulate and simplify equations allows for a deeper understanding of mathematical relationships and the nature of solutions. By mastering these techniques, students can confidently tackle more complex algebraic problems and appreciate the elegance and logic of mathematics.