Solving -9(x+3)+12=-3(2x+5)-3x Determining The Correct Solution

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Understanding the Problem

Before diving into the solution, it’s crucial to understand the nature of the problem. We are given the equation -9(x+3)+12=-3(2x+5)-3x and need to determine which statement about its solution is true. The options are:

A. The equation has one solution, x=1. B. The equation has one solution, x=0. C. The equation has no solution. D. The equation has infinitely many solutions.

To find the correct answer, we must simplify and solve the equation, then analyze the result.

Step-by-Step Solution

Step 1: Distribute the Constants

The first step in solving the equation is to distribute the constants outside the parentheses to the terms inside. This involves multiplying -9 by both x and 3 on the left side, and multiplying -3 by both 2x and 5 on the right side.

  • Left Side: -9(x+3) becomes -9x - 27
  • Right Side: -3(2x+5) becomes -6x - 15

So, the equation now looks like this:

  • -9x - 27 + 12 = -6x - 15 - 3x

Step 2: Simplify Both Sides

Next, simplify each side of the equation by combining like terms. On the left side, combine -27 and +12. On the right side, combine -6x and -3x.

  • Left Side: -27 + 12 = -15
  • Right Side: -6x - 3x = -9x

Now, the equation is:

  • -9x - 15 = -9x - 15

Step 3: Analyze the Simplified Equation

At this stage, we have the equation -9x - 15 = -9x - 15. Notice that both sides of the equation are identical. This is a crucial observation that helps us determine the nature of the solution.

Step 4: Determine the Solution Type

When both sides of an equation are the same, it means that the equation is an identity. An identity is an equation that is true for all values of the variable. In this case, no matter what value we substitute for x, the equation will always hold true.

Therefore, the equation has infinitely many solutions.

Why Infinitely Many Solutions?

To further illustrate why this equation has infinitely many solutions, let’s try adding 9x to both sides:

  • -9x - 15 + 9x = -9x - 15 + 9x
  • -15 = -15

This resulting equation, -15 = -15, is always true, regardless of the value of x. This confirms that any real number can be a solution to the original equation.

Common Mistakes to Avoid

  1. Incorrect Distribution: A common mistake is to incorrectly distribute the constants. For example, not multiplying the negative sign along with the constant, which would lead to errors in the subsequent steps.
  2. Miscombining Like Terms: Another error is incorrectly combining like terms. Ensure that you are only adding or subtracting terms that have the same variable and exponent.
  3. Premature Conclusion: Sometimes, students might prematurely conclude that there is no solution if variables cancel out. However, if the remaining constants are equal, it indicates infinitely many solutions rather than no solution.

Conclusion

The given equation -9(x+3)+12=-3(2x+5)-3x simplifies to -9x - 15 = -9x - 15, which is an identity. This means the equation is true for all values of x. Therefore, the correct answer is:

D. The equation has infinitely many solutions.

By following this step-by-step solution and understanding the underlying principles, you can confidently solve similar algebraic equations. This methodical approach ensures accuracy and helps in grasping the concept of equations with infinitely many solutions.

In the realm of algebra, understanding the nature of solutions to linear equations is fundamental. When dealing with linear equations, we often encounter three distinct scenarios: equations with a single solution, equations with no solution, and equations with infinitely many solutions. This section delves into the intricacies of linear equations that possess infinitely many solutions, providing a comprehensive analysis to help you identify and solve them effectively. Understanding these equations is crucial for a solid foundation in algebra and its applications.

What are Linear Equations?

Before we dive into infinite solutions, it’s essential to define what a linear equation is. 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 raised to the first power. These equations, when graphed, form a straight line. The general form of a linear equation in one variable is:

  • ax + b = 0

Where a and b are constants, and x is the variable. Linear equations can also involve multiple variables, but the key characteristic remains: no variable is raised to a power greater than one.

Types of Solutions in Linear Equations

Linear equations can have three types of solutions:

  1. One Unique Solution: This is the most common scenario, where the equation has a single value of the variable that satisfies it. For example, the equation 2x + 3 = 7 has one solution, x = 2.
  2. No Solution: This occurs when the equation leads to a contradiction. For example, the equation x + 1 = x + 2 has no solution because subtracting x from both sides leads to 1 = 2, which is false.
  3. Infinitely Many Solutions: This is the focus of our discussion. An equation with infinitely many solutions is true for all values of the variable. It’s also known as an identity.

Identifying Equations with Infinitely Many Solutions

So, how do we identify a linear equation that has infinitely many solutions? These equations exhibit a unique property: after simplification, both sides of the equation become identical. This means that the equation is true regardless of the value substituted for the variable.

Consider the equation:

  • 3(x + 2) = 3x + 6

Let’s simplify it step-by-step:

  1. Distribute the 3 on the left side: 3x + 6 = 3x + 6

  2. Notice that both sides of the equation are the same. This indicates that the equation is an identity.

  3. If we subtract 3x from both sides, we get: 6 = 6, which is always true.

This equation has infinitely many solutions because any value of x will satisfy the original equation. Whether x is 0, 1, -1, or any other real number, the equation holds true.

The Algebraic Process: A Detailed Explanation

To solidify your understanding, let’s dissect the algebraic process involved in identifying infinitely many solutions. Here’s a detailed breakdown:

  1. Distribute: If the equation contains parentheses, begin by distributing any constants across the terms inside the parentheses. This step eliminates the parentheses and simplifies the equation.

  2. Combine Like Terms: Next, combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. Combining them simplifies the equation further.

  3. Isolate the Variable (if possible): Attempt to isolate the variable on one side of the equation. This usually involves adding or subtracting terms to both sides. However, for equations with infinitely many solutions, you’ll notice that the variable terms cancel out.

  4. Analyze the Result: After simplifying, if you end up with an equation where both sides are identical (e.g., 5 = 5, -2 = -2), the equation has infinitely many solutions. This indicates that the equation is an identity and is true for all values of the variable.

Let’s illustrate this process with another example:

  • -2(4x - 1) + 8x = 2

  1. Distribute: -8x + 2 + 8x = 2

  2. Combine Like Terms: The -8x and +8x terms cancel each other out, leaving: 2 = 2

  3. Analyze the Result: Since 2 = 2 is always true, the equation has infinitely many solutions.

Real-World Implications and Applications

The concept of equations with infinitely many solutions might seem abstract, but it has real-world implications. In various fields, such as engineering and economics, mathematical models often involve equations that represent relationships between different variables. Understanding the nature of solutions helps in interpreting these models effectively.

For instance, in linear programming, if the constraints of a problem result in an equation with infinitely many solutions, it implies that there are multiple optimal solutions. This flexibility can be advantageous in decision-making processes.

Strategies for Teaching and Learning

Teaching and learning about linear equations with infinitely many solutions can be enhanced through specific strategies:

  1. Visual Aids: Use visual aids, such as graphs, to illustrate the concept. An equation with infinitely many solutions can be represented as a single line overlapping itself.

  2. Step-by-Step Examples: Provide numerous step-by-step examples, as demonstrated above, to help students understand the algebraic process.

  3. Hands-On Activities: Incorporate hands-on activities, such as using algebra tiles or manipulatives, to make the concept more concrete.

  4. Problem-Solving Practice: Encourage students to practice solving a variety of problems to build fluency in identifying and solving equations with infinitely many solutions.

  5. Conceptual Understanding: Emphasize the conceptual understanding of why these equations have infinitely many solutions. Help students grasp that the equation is essentially a statement of equality that is true regardless of the variable's value.

Potential Challenges and How to Overcome Them

Students often face specific challenges when learning about equations with infinitely many solutions:

  1. Confusion with No Solution: The most common challenge is differentiating between equations with no solution and those with infinitely many solutions. Emphasize that a no-solution equation leads to a contradiction (e.g., 1 = 2), while an equation with infinitely many solutions simplifies to a true statement (e.g., 5 = 5).

  2. Premature Conclusion: Some students may prematurely conclude that there is no solution when the variable terms cancel out. Teach them to look for the resulting equality. If the equality is true, it indicates infinitely many solutions.

  3. Abstract Concept: The concept of infinitely many solutions can be abstract. Use real-world examples and visual aids to make the concept more relatable.

Conclusion: Mastering Equations with Infinite Solutions

In summary, linear equations with infinitely many solutions are a fascinating aspect of algebra. They represent equations that are true for all values of the variable and are characterized by both sides of the equation becoming identical after simplification. By understanding the algebraic process and practicing with various examples, you can master the identification and solution of these equations. This skill is not only crucial for algebraic proficiency but also for interpreting mathematical models in real-world contexts. By teaching effective strategies and addressing common challenges, educators can empower students to confidently tackle equations with infinitely many solutions and enhance their overall mathematical understanding.

In mathematics, solving equations efficiently and accurately often hinges on mastering two fundamental concepts: the distributive property and combining like terms. These techniques are indispensable tools in simplifying equations, especially when dealing with complex expressions. Understanding and applying these principles correctly allows for the systematic reduction of equations to their simplest forms, making the solution process more manageable. This section will delve into each concept, providing a clear understanding of how they work and why they are crucial in algebra.

The Distributive Property: Unpacking Expressions

The distributive property is a cornerstone of algebra, providing a method to simplify expressions that involve multiplication over addition or subtraction. This property states that for any numbers a, b, and c:

  • a(b + c) = a b + a c
  • a(b - c) = a b - a c

In simpler terms, the distributive property allows you to multiply a single term by each term inside a set of parentheses. This is particularly useful when dealing with equations that have expressions enclosed in parentheses, as it enables you to eliminate the parentheses and simplify the equation.

How the Distributive Property Works

To understand how the distributive property works, let’s consider an example:

  • 3(x + 4)

To apply the distributive property, you multiply the term outside the parentheses (3) by each term inside the parentheses (x and 4):

  • 3 * x = 3x
  • 3 * 4 = 12

So, 3(x + 4) simplifies to 3x + 12. This process effectively removes the parentheses and expands the expression into a more manageable form.

Examples of Distributive Property

Let’s explore a few more examples to solidify the understanding of the distributive property:

  1. 5(2x - 3):

    • 5 * 2x = 10x
    • 5 * (-3) = -15
    • Therefore, 5(2x - 3) simplifies to 10x - 15.

  2. -2(x + 6):

    • -2 * x = -2x
    • -2 * 6 = -12
    • Therefore, -2(x + 6) simplifies to -2x - 12.

  3. 4(3x + 1) - 2x:

    • First, distribute the 4: 4 * 3x = 12x
    • 4 * 1 = 4
    • So, 4(3x + 1) becomes 12x + 4
    • The expression is now: 12x + 4 - 2x

Importance of the Distributive Property

The distributive property is not just a mathematical trick; it's a fundamental tool that enables us to simplify complex equations and expressions. Without it, many algebraic problems would be significantly more challenging to solve. By removing parentheses and expanding expressions, the distributive property allows us to manipulate equations and isolate variables, which is essential for finding solutions.

Combining Like Terms: Simplifying Expressions

After applying the distributive property, the next crucial step in simplifying equations is combining like terms. This process involves identifying and combining terms that have the same variable raised to the same power. For example, 3x and 5x are like terms because they both contain the variable x raised to the first power. Similarly, constants like 7 and -2 are like terms because they are both constant values.

How to Combine Like Terms

To combine like terms, you simply add or subtract their coefficients (the numbers in front of the variables). Here’s how it works:

  1. Identify the like terms in the expression.
  2. Add or subtract the coefficients of the like terms.
  3. Write the simplified term.

Let’s consider an example:

  • 4x + 2x - 3 + 5

In this expression, 4x and 2x are like terms, and -3 and 5 are like terms. To combine them:

  1. Combine 4x and 2x: 4x + 2x = 6x
  2. Combine -3 and 5: -3 + 5 = 2

So, the simplified expression is 6x + 2.

Examples of Combining Like Terms

Let’s look at a few more examples:

  1. 7y - 3y + 2 + 1:

    • Combine 7y and -3y: 7y - 3y = 4y
    • Combine 2 and 1: 2 + 1 = 3
    • The simplified expression is 4y + 3.

  2. 5x + 3 - 2x + 4:

    • Combine 5x and -2x: 5x - 2x = 3x
    • Combine 3 and 4: 3 + 4 = 7
    • The simplified expression is 3x + 7.

  3. -2a + 6 + 4a - 1:

    • Combine -2a and 4a: -2a + 4a = 2a
    • Combine 6 and -1: 6 - 1 = 5
    • The simplified expression is 2a + 5.

The Significance of Combining Like Terms

Combining like terms is essential for simplifying expressions and making equations easier to solve. By reducing the number of terms in an equation, you create a clearer, more manageable problem. This process not only simplifies calculations but also helps in visualizing the structure of the equation, making it easier to identify the next steps in the solution process.

Combining Distributive Property and Combining Like Terms

In many algebraic problems, you'll need to use both the distributive property and combining like terms to simplify equations. The typical approach involves first distributing, then combining like terms. This two-step process is a powerful way to tackle complex expressions.

Example: Combining Both Techniques

Let’s consider an example that requires both techniques:

  • 3(2x - 1) + 4x + 5

  1. Apply the Distributive Property:

    • Distribute the 3: 3 * 2x = 6x
    • 3 * (-1) = -3
    • The expression becomes: 6x - 3 + 4x + 5

  2. Combine Like Terms:

    • Combine 6x and 4x: 6x + 4x = 10x
    • Combine -3 and 5: -3 + 5 = 2
    • The simplified expression is 10x + 2.

By applying the distributive property and then combining like terms, we've successfully simplified the expression to its simplest form.

Step-by-Step Approach for Complex Equations

When faced with complex equations, follow this step-by-step approach to ensure accurate simplification:

  1. Distribute: Apply the distributive property to remove any parentheses.
  2. Combine Like Terms on Each Side: Simplify each side of the equation by combining like terms.
  3. Isolate the Variable: Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the equation.
  4. Solve for the Variable: Perform the necessary operations to find the value of the variable.
  5. Check Your Solution: Substitute the solution back into the original equation to verify its correctness.

Common Mistakes to Avoid

When using the distributive property and combining like terms, it’s crucial to avoid common mistakes:

  1. Incorrect Distribution: Ensure you multiply the term outside the parentheses by every term inside the parentheses, including signs.
  2. Sign Errors: Pay close attention to signs when distributing negative numbers or combining like terms with negative coefficients.
  3. Misidentifying Like Terms: Only combine terms that have the same variable raised to the same power. For instance, 2x and 2x2 are not like terms.
  4. Skipping Steps: Avoid skipping steps, especially when dealing with complex expressions. Write down each step to minimize errors.

Conclusion: Mastering Simplification Techniques

The distributive property and combining like terms are essential techniques in algebra, serving as the foundation for solving complex equations. By understanding and applying these principles correctly, you can simplify expressions, reduce equations to their simplest forms, and solve problems more efficiently. Mastering these skills is crucial for success in algebra and higher-level mathematics. Consistent practice and attention to detail will help you become proficient in using these powerful tools.