Solutions For The Equation 12x + 1 = 3(4x + 1) - 2 A Comprehensive Guide

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How many solutions does the equation 12x + 1 = 3(4x + 1) - 2 have?

Let's explore how to determine the number of solutions for the given equation: 12x + 1 = 3(4x + 1) - 2. This is a fundamental concept in algebra, and understanding how to solve such equations is crucial for various mathematical applications. In this comprehensive guide, we will walk through the step-by-step process of simplifying the equation, identifying the type of equation it is, and ultimately determining whether there are zero, one, or infinitely many solutions.

Understanding Linear Equations and Their Solutions

Before diving into the specifics of our equation, let's establish a solid understanding of linear equations and their possible solutions. 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 often expressed in the form 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. In simpler terms, it's the value (or values) of 'x' that, when plugged into the equation, will make both sides of the equation equal.

Linear equations can have three possible types of solutions:

  1. One Solution: This is the most common scenario. The equation simplifies to a unique value for the variable 'x'. For example, the equation 2x + 3 = 7 has one solution, which is x = 2. When you substitute 2 for x, the equation holds true (2*2 + 3 = 7).

  2. No Solution: In this case, the equation simplifies to a contradiction. This means that after simplifying the equation, you end up with a statement that is mathematically impossible, such as 0 = 5. This indicates that there is no value of 'x' that can make the equation true. The lines represented by the equation are parallel and never intersect.

  3. Infinitely Many Solutions: This occurs when the equation simplifies to an identity. An identity is a statement that is always true, regardless of the value of the variable. For instance, the equation x + 1 = x + 1 is an identity. Any value of 'x' will satisfy this equation. Graphically, this means the equation represents the same line, and every point on the line is a solution.

Step-by-Step Solution to 12x + 1 = 3(4x + 1) - 2

Now, let's apply this knowledge to the equation at hand: 12x + 1 = 3(4x + 1) - 2. We will systematically simplify the equation to determine the number of solutions. This process involves distributing, combining like terms, and isolating the variable 'x'.

Step 1: Distribute on the Right Side

The first step is to distribute the '3' on the right side of the equation. This means multiplying '3' by both terms inside the parentheses: 4x and 1.

12x + 1 = 3 * (4x + 1) - 2
12x + 1 = (3 * 4x) + (3 * 1) - 2
12x + 1 = 12x + 3 - 2

After distributing, the equation becomes:

12x + 1 = 12x + 3 - 2

Step 2: Combine Like Terms

The next step is to combine the like terms on each side of the equation. On the right side, we have two constant terms: '3' and '-2'. We can combine these by subtracting 2 from 3.

12x + 1 = 12x + (3 - 2)
12x + 1 = 12x + 1

After combining like terms, the equation simplifies to:

12x + 1 = 12x + 1

Step 3: Analyze the Simplified Equation

Now, let's carefully examine the simplified equation: 12x + 1 = 12x + 1. Notice that both sides of the equation are exactly the same. This is a crucial observation.

Step 4: Isolate the Variable (If Necessary)

In this case, we could try to isolate the variable 'x' by subtracting '12x' from both sides of the equation:

12x + 1 - 12x = 12x + 1 - 12x
1 = 1

This results in the statement 1 = 1, which is always true. There is no variable 'x' left in the equation.

Determining the Number of Solutions

Based on our step-by-step simplification, we arrived at the equation 1 = 1. This is an identity, meaning it is a statement that is always true, regardless of the value of 'x'. This tells us that the original equation has infinitely many solutions.

To further illustrate this, consider what happens if you were to graph the original equation. If you were to rewrite the equation in the slope-intercept form (y = mx + b), you would find that both sides of the equation represent the exact same line. Therefore, every point on the line is a solution to the equation.

Why Infinitely Many Solutions?

The reason we have infinitely many solutions is that the two expressions in the original equation are equivalent. After simplifying, we saw that both sides of the equation are identical. This means that no matter what value we substitute for 'x', the equation will always hold true.

In contrast, if we had arrived at an equation like 0 = 5, there would be no solution because this is a contradiction. If we had isolated 'x' and found a specific value, such as x = 3, there would be one unique solution.

Conclusion

In summary, by simplifying the equation 12x + 1 = 3(4x + 1) - 2, we determined that it has infinitely many solutions. This is because the equation simplifies to an identity, 1 = 1, which is always true. Understanding how to identify and solve linear equations with different types of solutions is a fundamental skill in algebra and is essential for solving more complex mathematical problems.

Therefore, the correct answer is:

D. infinitely many

This comprehensive analysis provides a clear understanding of how to approach similar problems and determine the number of solutions for linear equations. Remember to always simplify the equation first and then analyze the resulting statement to determine whether there are zero, one, or infinitely many solutions.

Practice Problems

To solidify your understanding, try solving these similar equations and determining the number of solutions:

  1. 5x + 2 = 5x + 7
  2. 2(x - 3) = 2x - 6
  3. 3x + 4 = 10

By working through these practice problems, you'll enhance your ability to identify different types of linear equations and their solutions.

Key Takeaways

  • A linear equation can have one solution, no solution, or infinitely many solutions.
  • To determine the number of solutions, simplify the equation by distributing and combining like terms.
  • If the simplified equation is an identity (e.g., 1 = 1), there are infinitely many solutions.
  • If the simplified equation is a contradiction (e.g., 0 = 5), there is no solution.
  • If the simplified equation leads to a unique value for 'x', there is one solution.

By following these guidelines, you can confidently tackle various algebraic equations and determine the number of solutions they possess.

How many solutions does the equation 12x + 1 = 3(4x + 1) - 2 have?

Solutions for the Equation 12x + 1 = 3(4x + 1) - 2: A Comprehensive Guide