Solving First-Degree Equation X + 3 * (x + 1) = 3 - 4(2x + 5)
In this article, we will delve into the step-by-step process of solving the first-degree equation x + 3 * (x + 1) = 3 - 4(2x + 5). This type of equation is fundamental in algebra, and understanding how to solve it is crucial for more advanced mathematical concepts. We will break down the equation, simplify it, and isolate the variable 'x' to find its value. This comprehensive guide aims to provide a clear and concise explanation, ensuring that readers of all levels can grasp the methodology involved. By the end of this article, you will have a solid understanding of how to solve similar first-degree equations, empowering you to tackle more complex algebraic problems with confidence.
Understanding First-Degree Equations
Before we dive into solving the equation, let's clarify what a first-degree equation actually is. A first-degree equation, also known as a linear equation, is an algebraic equation where the highest power of the variable is 1. These equations can be written in the general form ax + b = c, where a, b, and c are constants, and x is the variable we want to find. The key characteristic of a linear equation is that when graphed, it forms a straight line. This simplicity makes them a cornerstone of algebra and a starting point for understanding more complex equations.
To successfully solve first-degree equations, it is essential to grasp the fundamental principles of algebraic manipulation. This involves applying operations to both sides of the equation to maintain equality while isolating the variable. Common operations include addition, subtraction, multiplication, and division. The goal is always to simplify the equation step by step until the variable is alone on one side, revealing its value. Understanding these principles will not only help in solving the given equation but also in tackling a wide range of algebraic problems.
Furthermore, it is crucial to be familiar with the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which mathematical operations should be performed to ensure accurate simplification. Ignoring the order of operations can lead to incorrect results, highlighting the importance of adhering to this fundamental rule in algebra. By mastering these basics, solving first-degree equations becomes a straightforward and logical process.
Step-by-Step Solution
Now, let's move on to the step-by-step solution of the equation:
x + 3 * (x + 1) = 3 - 4(2x + 5)
Step 1: Distribute the constants
The first step in solving this equation involves distributing the constants outside the parentheses. This means multiplying the numbers outside the parentheses by each term inside the parentheses. This process simplifies the equation by removing the parentheses and combining like terms. By correctly applying the distributive property, we can transform the equation into a more manageable form, making it easier to isolate the variable.
In our equation, we have two instances where we need to apply the distributive property: 3 * (x + 1) and -4(2x + 5). Let's break this down:3 * (x + 1) becomes 3 * x + 3 * 1, which simplifies to 3x + 3.-4(2x + 5) becomes -4 * 2x + -4 * 5, which simplifies to -8x - 20.By applying these distributions, we expand the equation, paving the way for further simplification and ultimately leading us to the solution.
After distributing the constants, the equation looks like this:
x + 3x + 3 = 3 - 8x - 20
Step 2: Combine like terms on each side
The next crucial step is to combine like terms on each side of the equation. Like terms are those that contain the same variable raised to the same power, or constants. Combining them simplifies the equation, making it easier to isolate the variable. This step is essential for organizing the equation and bringing it closer to a solvable form. By grouping and combining like terms, we reduce the complexity of the equation, which helps in avoiding errors in subsequent steps.
On the left side of the equation, we have 'x' and '3x', which are like terms. Combining them gives us 4x. The constant term on the left side is '3'. So, the left side simplifies to 4x + 3.On the right side, we have the constant terms '3' and '-20'. Combining them gives us -17. The term with the variable is '-8x'. Therefore, the right side simplifies to -8x - 17. By meticulously combining like terms, we make the equation more concise and easier to handle, setting the stage for the next steps in the solution process.
Combining like terms, we get:
4x + 3 = -8x - 17
Step 3: Move variable terms to one side
To further isolate the variable, we need to move all variable terms to one side of the equation. This is achieved by adding or subtracting terms from both sides to cancel out the variable term on one side. The goal is to consolidate all terms containing the variable on one side, simplifying the equation and bringing us closer to the solution. This step is a critical part of the process, as it sets up the equation for the final isolation of the variable.
In our equation, we have '4x' on the left side and '-8x' on the right side. To move the '-8x' term to the left side, we add '8x' to both sides of the equation. This ensures that the equation remains balanced while consolidating the variable terms. Adding '8x' to both sides cancels out the '-8x' on the right side, leaving us with only constant terms on that side.
The operation looks like this:4x + 3 + 8x = -8x - 17 + 8xThis simplifies to:12x + 3 = -17By moving the variable terms to one side, we have effectively simplified the equation, making it easier to solve for 'x' in the subsequent steps. This strategic manipulation of the equation is a fundamental technique in algebra.
Adding 8x to both sides:
12x + 3 = -17
Step 4: Move constant terms to the other side
After consolidating the variable terms on one side, the next step is to move all constant terms to the other side. This is done by adding or subtracting constants from both sides of the equation, similar to how we moved variable terms. The objective is to isolate the variable term by ensuring that all constants are on the opposite side of the equation. This step is essential for simplifying the equation further and getting closer to the final solution.
In our current equation, 12x + 3 = -17, we have the constant term '+3' on the same side as the variable term '12x'. To move '+3' to the right side, we subtract '3' from both sides of the equation. This maintains the balance of the equation while effectively isolating the variable term. Subtracting '3' from both sides cancels out the '+3' on the left side, leaving us with just the variable term.
The operation looks like this:12x + 3 - 3 = -17 - 3This simplifies to:12x = -20By moving the constant terms to the other side, we have successfully isolated the variable term, making it straightforward to solve for 'x' in the final step. This strategic manipulation is a key technique in solving algebraic equations.
Subtracting 3 from both sides:
12x = -20
Step 5: Isolate the variable
The final step in solving for 'x' is to isolate the variable completely. This means getting 'x' by itself on one side of the equation. To achieve this, we typically divide both sides of the equation by the coefficient of the variable. The coefficient is the number that is multiplied by the variable. This step is the culmination of all the previous steps, and it directly leads to the solution of the equation.
In our equation, 12x = -20, the coefficient of 'x' is '12'. To isolate 'x', we need to divide both sides of the equation by '12'. This operation will cancel out the '12' on the left side, leaving 'x' by itself. Dividing both sides by the same number ensures that the equation remains balanced and that we arrive at the correct solution.
The operation looks like this:(12x) / 12 = -20 / 12This simplifies to:x = -20/12We can further simplify the fraction -20/12 by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us x = -5/3. By isolating the variable, we have successfully solved the equation for 'x'. The value of 'x' that satisfies the equation is -5/3.
Divide both sides by 12:
x = -20/12
Step 6: Simplify the fraction
The final touch in solving for 'x' is to simplify the fraction if possible. Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator have no common factors other than 1. This makes the solution cleaner and easier to understand. Simplifying fractions is a fundamental skill in mathematics, and it's important to always present solutions in their simplest form.
In our case, we have the fraction x = -20/12. Both the numerator (-20) and the denominator (12) have common factors. To simplify, we need to find the greatest common divisor (GCD) of 20 and 12. The GCD is the largest number that divides both numbers without leaving a remainder. The GCD of 20 and 12 is 4.
To simplify the fraction, we divide both the numerator and the denominator by the GCD, which is 4. This operation looks like this:x = (-20 ÷ 4) / (12 ÷ 4)This simplifies to:x = -5/3By simplifying the fraction, we have presented the solution in its most concise form. The value of 'x' that satisfies the equation is -5/3, and this simplified fraction is the clearest way to express the solution.
Simplifying the fraction -20/12 by dividing both numerator and denominator by 4, we get:
x = -5/3
Final Answer
Therefore, the solution to the equation x + 3 * (x + 1) = 3 - 4(2x + 5) is:
x = -5/3
This comprehensive step-by-step solution demonstrates the process of solving a first-degree equation. By following these steps, you can confidently tackle similar algebraic problems.
Common Mistakes to Avoid
When solving first-degree equations, it's easy to make mistakes if you're not careful. Recognizing these common mistakes can help you avoid them and ensure you arrive at the correct solution. One frequent error is neglecting the order of operations (PEMDAS/BODMAS). For instance, multiplying before distributing or adding before multiplying can lead to incorrect simplifications. Always remember to address parentheses first, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right).
Another common mistake is incorrectly distributing constants over parentheses. Ensure that you multiply the constant by every term inside the parentheses. For example, in the expression -2(x + 3), it's crucial to distribute the -2 to both 'x' and '+3', resulting in -2x - 6, not -2x + 6. Similarly, sign errors are prevalent, especially when dealing with negative numbers. Pay close attention to the signs when adding, subtracting, multiplying, and dividing terms. A small mistake with a sign can drastically alter the final result.
Combining like terms incorrectly is another area where errors often occur. Make sure you are only combining terms that have the same variable raised to the same power. For instance, you can combine 3x and 5x to get 8x, but you cannot combine 3x and 5x². Finally, failing to perform the same operation on both sides of the equation is a fundamental error that violates the balance of the equation. If you add, subtract, multiply, or divide on one side, you must do the same on the other side to maintain equality. By being mindful of these common pitfalls, you can enhance your accuracy and proficiency in solving first-degree equations.
Practice Problems
To solidify your understanding of solving first-degree equations, working through practice problems is essential. Practice helps reinforce the steps and techniques discussed, making you more comfortable and confident in tackling different types of equations. It also allows you to identify any areas where you may need further clarification or practice. The more you practice, the more fluent you will become in algebraic manipulation and problem-solving.
Here are a few practice problems you can try:
- 2(x - 1) + 3x = 13
- 5x - 3(x + 2) = 4
- 7 - 2(3x - 1) = 2x
- 4(2x + 3) - x = 15
- 9x + 2(x - 4) = 3
For each problem, follow the step-by-step process outlined earlier in this article: distribute constants, combine like terms, move variable terms to one side, move constant terms to the other side, isolate the variable, and simplify the fraction if necessary. After solving each equation, check your solution by substituting the value of 'x' back into the original equation to ensure it holds true. This verification step is crucial for confirming the accuracy of your solution and identifying any errors.
By consistently practicing with a variety of problems, you will develop a strong foundation in solving first-degree equations, which is a fundamental skill for more advanced topics in algebra and mathematics. Regular practice not only improves your accuracy but also enhances your problem-solving speed and efficiency.
Conclusion
In conclusion, solving first-degree equations is a fundamental skill in algebra that forms the basis for more advanced mathematical concepts. Throughout this article, we have provided a step-by-step guide on how to solve the equation x + 3 * (x + 1) = 3 - 4(2x + 5), emphasizing the importance of each step and the underlying principles of algebraic manipulation. We started by distributing constants, combining like terms, moving variable terms to one side, moving constant terms to the other side, isolating the variable, and finally, simplifying the fraction to arrive at the solution x = -5/3.
We also highlighted common mistakes to avoid, such as neglecting the order of operations, incorrectly distributing constants, making sign errors, combining unlike terms, and failing to perform the same operation on both sides of the equation. Recognizing these pitfalls is crucial for maintaining accuracy and avoiding errors in your problem-solving process. Furthermore, we stressed the significance of practice and provided several practice problems to help you solidify your understanding and build confidence in solving first-degree equations.
By mastering the techniques and principles discussed in this article, you will not only be able to solve first-degree equations effectively but also lay a strong foundation for tackling more complex algebraic problems. Remember, consistent practice and attention to detail are key to success in mathematics. With dedication and the right approach, you can confidently navigate the world of algebra and beyond.