Solving -9(x+3)+12=-3(2x+5)-3x Determining The Nature Of Solutions
In this article, we will embark on a detailed exploration of the equation -9(x+3)+12=-3(2x+5)-3x, aiming to determine the nature of its solutions. Understanding how to solve linear equations is a fundamental skill in algebra, and this particular equation provides an excellent opportunity to delve into the intricacies of the process. We will carefully examine each step involved in simplifying the equation, isolating the variable, and ultimately arriving at a conclusion about the solution set.
To begin, let's restate the equation: -9(x+3)+12=-3(2x+5)-3x. Our initial task is to simplify both sides of the equation by applying the distributive property. This involves multiplying the constants outside the parentheses by each term inside the parentheses. On the left side, we have -9 multiplied by both x and 3, and on the right side, we have -3 multiplied by both 2x and 5. Performing these multiplications, we get:
-9x - 27 + 12 = -6x - 15 - 3x
Now that we have eliminated the parentheses, our next step is to combine like terms on each side of the equation. On the left side, we can combine the constant terms -27 and +12. On the right side, we can combine the terms involving x, namely -6x and -3x. After combining like terms, the equation becomes:
-9x - 15 = -9x - 15
At this point, we observe a significant characteristic of the equation. Both sides are identical! This immediately suggests that the equation might have infinitely many solutions. To confirm this, let's proceed with the standard algebraic technique of isolating the variable. We can add 9x to both sides of the equation:
-9x - 15 + 9x = -9x - 15 + 9x
This simplifies to:
-15 = -15
This final statement is a true identity. It means that the equation holds true regardless of the value of x. No matter what number we substitute for x, the equation will always be satisfied. This definitively confirms that the equation has infinitely many solutions.
Therefore, the correct answer is not A (x=1), not B (x=0), and not C (no solution). The correct answer is D: the equation has infinitely many solutions. This equation serves as a great example of a special case in linear equations where the solution set encompasses all real numbers.
Delving deeper into the world of linear equations, it's crucial to understand the different types of solutions that can arise. A linear equation, in its simplest form, represents a straight line when graphed on a coordinate plane. The solution to a linear equation is the value (or values) of the variable that make the equation a true statement. However, not all linear equations behave the same way; they can have one solution, no solution, or infinitely many solutions.
One Solution: This is the most common scenario. An equation with one solution has a single, unique value for the variable that satisfies the equation. Graphically, this corresponds to two lines intersecting at a single point. The x-coordinate of this point represents the solution to the equation.
No Solution: Sometimes, when simplifying an equation, we arrive at a contradiction, such as 5 = 7. This indicates that there is no value of the variable that can make the equation true. In this case, the equation has no solution. Graphically, this represents two parallel lines that never intersect.
Infinitely Many Solutions: As we saw in the given example, an equation can also have infinitely many solutions. This occurs when the equation simplifies to a true identity, such as -15 = -15. This means that any value of the variable will satisfy the equation. Graphically, this represents two lines that are actually the same line, overlapping each other completely.
Understanding these different types of solutions is essential for solving linear equations and interpreting their results. By carefully simplifying the equation and analyzing the outcome, we can accurately determine the nature of the solution set.
To further solidify our understanding, let's revisit the step-by-step solution of the equation -9(x+3)+12=-3(2x+5)-3x with even greater detail.
Step 1: Distribute
The first step is to eliminate the parentheses by applying the distributive property. This involves multiplying the constant term outside the parentheses by each term inside the parentheses. On the left side, we multiply -9 by both x and 3, and on the right side, we multiply -3 by both 2x and 5:
-9 * x = -9x
-9 * 3 = -27
-3 * 2x = -6x
-3 * 5 = -15
This gives us the equation:
-9x - 27 + 12 = -6x - 15 - 3x
Step 2: Combine Like Terms
Next, we combine like terms on each side of the equation. On the left side, we can combine the constant terms -27 and +12. On the right side, we can combine the terms involving x, namely -6x and -3x:
-27 + 12 = -15
-6x - 3x = -9x
This simplifies the equation to:
-9x - 15 = -9x - 15
Step 3: Isolate the Variable (Attempt)
To isolate the variable x, we can add 9x to both sides of the equation:
-9x - 15 + 9x = -9x - 15 + 9x
This simplifies to:
-15 = -15
Step 4: Interpret the Result
The equation -15 = -15 is a true identity. This means that the equation is true for any value of x. Therefore, the equation has infinitely many solutions.
While the concept of an equation having infinitely many solutions might seem abstract, it has practical applications in various real-world scenarios. These situations often involve modeling relationships where the variables are interdependent, and changes in one variable automatically lead to corresponding changes in another, maintaining the equation's balance.
Financial Planning: Consider a scenario where you're trying to allocate a fixed budget between two categories, say, savings and spending. If the amount allocated to savings increases, the amount available for spending decreases proportionally, and vice versa. This relationship can be represented by an equation with infinitely many solutions, as there are multiple combinations of savings and spending that satisfy the budget constraint.
Mixture Problems: In chemistry or cooking, you might encounter problems where you need to mix two solutions or ingredients with different concentrations to achieve a desired concentration. The relationship between the amounts of each solution and the final concentration can sometimes be represented by an equation with infinitely many solutions, indicating that there are multiple ways to combine the ingredients to reach the target.
Geometric Relationships: In geometry, relationships between angles or sides of certain shapes can lead to equations with infinite solutions. For example, the angles in a triangle must add up to 180 degrees. If you have an equation representing the relationship between the angles, there might be multiple combinations of angles that satisfy the condition.
These are just a few examples of how equations with infinitely many solutions can arise in real-world situations. Recognizing these scenarios and understanding how to interpret the solutions is crucial for effective problem-solving.
In conclusion, the equation -9(x+3)+12=-3(2x+5)-3x exemplifies a linear equation with infinitely many solutions. By systematically simplifying the equation and arriving at a true identity, we can confidently determine that any value of x will satisfy the equation. This exploration has provided a comprehensive understanding of the nature of solutions in linear equations, including the concepts of one solution, no solution, and infinitely many solutions.
Furthermore, we've delved into the step-by-step solution process, highlighting the importance of distribution, combining like terms, and interpreting the final result. We've also explored real-world applications of equations with infinite solutions, demonstrating their relevance in various fields.
By mastering these concepts and techniques, you'll be well-equipped to tackle a wide range of algebraic problems and confidently analyze the nature of their solutions. The ability to solve linear equations is a fundamental skill that will serve you well in mathematics and beyond.
