Solving 5x - 5 = 3x - 9 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. It's the bedrock upon which more complex mathematical concepts are built. Linear equations, in particular, are the simplest type of equations to solve, and mastering them is crucial for anyone venturing into algebra and beyond. In this comprehensive guide, we will dissect the linear equation 5x - 5 = 3x - 9, providing a step-by-step solution and offering insights into the underlying principles. This exploration will empower you to tackle similar problems with confidence and clarity.
Before diving into the specifics, let's first understand 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. These equations are called "linear" because when plotted on a graph, they form a straight line. The general form of a linear equation is ax + b = 0, where x is the variable, and a and b are constants. Our equation, 5x - 5 = 3x - 9, fits this description, making it a linear equation in one variable.
Why is solving linear equations so important? The answer lies in their ubiquitous applications across various fields. From physics and engineering to economics and computer science, linear equations are used to model and solve real-world problems. They help us determine unknown quantities, make predictions, and understand relationships between variables. Whether you're calculating the trajectory of a projectile, balancing a chemical equation, or analyzing financial data, a solid grasp of linear equations is indispensable. Therefore, investing time in mastering this skill will undoubtedly pay dividends in your academic and professional pursuits.
In the following sections, we will meticulously walk through the process of solving 5x - 5 = 3x - 9. We will start by isolating the variable terms on one side of the equation and the constant terms on the other. Then, we will simplify the equation and solve for x. Along the way, we will highlight key concepts and techniques, ensuring a thorough understanding of each step. By the end of this guide, you will not only know the solution to this specific equation but also possess the knowledge and skills to solve a wide range of linear equations.
Step-by-Step Solution to 5x - 5 = 3x - 9
Now, let's embark on the journey of solving our equation: 5x - 5 = 3x - 9. We'll break down the process into manageable steps, ensuring clarity and understanding at each stage. Remember, the goal is to isolate the variable x on one side of the equation, thereby determining its value.
Step 1: Grouping Like Terms
The first crucial step in solving any linear equation is to group like terms. Like terms are those that contain the same variable raised to the same power (in this case, x) or are constants. In our equation, the terms 5x and 3x are like terms, as are the constants -5 and -9. To group these terms, we need to move them to the same side of the equation. The fundamental principle here is that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain equality. This is the golden rule of equation solving.
To move the 3x term from the right side to the left side, we subtract 3x from both sides of the equation. This operation cancels out the 3x term on the right side, leaving us with only the constant. Simultaneously, on the left side, we combine the 5x term with the -3x term. This can be represented mathematically as follows:
5x - 5 - 3x = 3x - 9 - 3x
Simplifying this, we get:
2x - 5 = -9
Now, we need to move the constant term -5 from the left side to the right side. To do this, we add 5 to both sides of the equation. This cancels out the -5 on the left side, leaving us with only the term containing the variable. On the right side, we combine the -9 with the 5. This can be represented as:
2x - 5 + 5 = -9 + 5
Simplifying this, we get:
2x = -4
Step 2: Isolating the Variable
After grouping like terms, the next step is to isolate the variable x. This means getting x by itself on one side of the equation. In our case, x is currently being multiplied by 2. To isolate x, we need to perform the inverse operation, which is division. We divide both sides of the equation by 2. This cancels out the multiplication by 2 on the left side, leaving us with x alone. Mathematically, this looks like:
(2x) / 2 = (-4) / 2
Simplifying this, we get:
x = -2
And there we have it! We have successfully solved the equation 5x - 5 = 3x - 9 and found that x equals -2. This value is the solution to the equation, meaning that if we substitute -2 for x in the original equation, both sides of the equation will be equal.
Step 3: Verification of the Solution
In mathematics, it's always a good practice to verify your solution. This ensures that you haven't made any errors in your calculations and that your answer is indeed correct. To verify our solution, we substitute x = -2 back into the original equation: 5x - 5 = 3x - 9. If both sides of the equation are equal after the substitution, then our solution is correct.
Substituting x = -2 into the left side of the equation, we get:
5(-2) - 5 = -10 - 5 = -15
Now, substituting x = -2 into the right side of the equation, we get:
3(-2) - 9 = -6 - 9 = -15
As we can see, both sides of the equation are equal to -15. This confirms that our solution, x = -2, is indeed correct. Verification is a crucial step in problem-solving, especially in high-stakes situations like exams or practical applications. It provides assurance that your answer is accurate and allows you to move forward with confidence.
Alternative Methods for Solving Linear Equations
While the step-by-step method we used above is a standard approach for solving linear equations, there are other valid methods that can be employed. Understanding these alternative methods can provide you with a broader perspective and allow you to choose the most efficient method for a given problem. In this section, we will explore two such methods: the transposition method and the graphical method.
1. Transposition Method
The transposition method is a shortcut technique that streamlines the process of moving terms across the equals sign. Instead of explicitly adding or subtracting terms from both sides, we simply "transpose" the term to the other side and change its sign. This method is based on the same principles as the step-by-step method but offers a more concise way of representing the operations.
Let's revisit our equation: 5x - 5 = 3x - 9. Using the transposition method, we can move the 3x term from the right side to the left side by changing its sign to -3x. Similarly, we can move the -5 term from the left side to the right side by changing its sign to +5. This gives us:
5x - 3x = -9 + 5
Notice how this directly leads us to the simplified equation we obtained in Step 1 of our previous method. From here, we can simplify and solve for x as before:
2x = -4 x = -2
The transposition method is a valuable tool for solving linear equations quickly and efficiently. However, it's essential to understand the underlying principles of adding and subtracting from both sides to avoid making errors. This method is particularly useful for mental calculations and for simplifying complex equations.
2. Graphical Method
Another way to solve linear equations is through graphical representation. This method involves plotting the equation on a coordinate plane and finding the point where the line intersects the x-axis. The x-coordinate of this point represents the solution to the equation. While the graphical method may not be the most precise for all equations, it provides a visual understanding of the solution and can be particularly helpful for understanding systems of linear equations.
To solve our equation 5x - 5 = 3x - 9 graphically, we can first rewrite it in the form y = mx + b, which is the slope-intercept form of a linear equation. To do this, we can rearrange the equation as follows:
5x - 5 = 3x - 9 2x = -4 x = -2
Now, let's consider two separate linear functions:
- y = 5x - 5
- y = 3x - 9
To solve the original equation 5x - 5 = 3x - 9, we need to find the x-value where these two lines intersect. At the intersection point, the y-values of both functions will be equal, which means the equation 5x - 5 = 3x - 9 holds true.
When you plot these two lines on a graph, you'll find that they intersect at the point where x = -2 and y = -15. This confirms our algebraic solution that x = -2. The graphical method provides a visual confirmation of the solution and helps to understand the relationship between the two sides of the equation as linear functions.
Common Mistakes to Avoid
Solving linear equations may seem straightforward, but it's easy to make mistakes if you're not careful. Identifying and avoiding these common pitfalls can significantly improve your accuracy and efficiency. In this section, we'll discuss some of the most frequent errors students make when solving linear equations and how to prevent them.
1. Incorrectly Combining Like Terms
One of the most common mistakes is incorrectly combining like terms. This often happens when students try to combine terms that are not actually like terms, or when they make errors in the arithmetic of combining them. For example, trying to add 5x and -5 is a mistake because they are not like terms. Only terms with the same variable raised to the same power can be combined.
To avoid this mistake, always double-check that you are only combining like terms. Pay close attention to the signs of the terms and perform the arithmetic carefully. It can be helpful to rearrange the equation so that like terms are grouped together before you start combining them. This visual organization can help prevent errors.
2. Forgetting to Distribute
Another frequent error occurs when dealing with equations that involve parentheses. Students sometimes forget to distribute a number or sign to all terms inside the parentheses. For instance, if you have an expression like 2(x + 3), you must multiply both x and 3 by 2. The correct expansion is 2x + 6, not 2x + 3.
To avoid this, always make sure to distribute any number or sign outside parentheses to every term inside. A helpful strategy is to draw arrows from the term outside the parentheses to each term inside, as a visual reminder of the distribution. This simple technique can greatly reduce the chances of making this mistake.
3. Performing Operations on Only One Side
A fundamental principle of solving equations is that whatever operation you perform on one side, you must also perform on the other side to maintain equality. A common mistake is forgetting to do this, leading to an unbalanced equation and an incorrect solution. For example, if you subtract 3x from the left side of the equation, you must also subtract 3x from the right side.
To prevent this, make it a habit to write down each operation you perform on both sides of the equation. This provides a clear record of your steps and helps you ensure that you are maintaining balance. Using different colors for each side of the equation can also serve as a visual reminder to perform operations consistently.
4. Sign Errors
Sign errors are particularly common when solving equations, especially when dealing with negative numbers. A simple mistake in adding, subtracting, multiplying, or dividing signed numbers can lead to an incorrect solution. For example, incorrectly calculating -5 - 9 as -4 instead of -14 is a sign error that can derail the entire solution.
To minimize sign errors, pay extra attention when working with negative numbers. Double-check your arithmetic and use a number line if needed to visualize the operations. It can also be helpful to rewrite subtraction as addition of a negative number (e.g., a - b = a + (-b)) to simplify the calculations and reduce the risk of sign errors.
5. Not Verifying the Solution
As we discussed earlier, verifying your solution is a crucial step in problem-solving. Many students skip this step, assuming their answer is correct without checking. However, even if you've followed all the steps correctly, there's always a chance of a minor error that went unnoticed. Verifying your solution by substituting it back into the original equation will catch these errors and ensure accuracy.
Make it a standard practice to verify your solution every time you solve an equation. This simple step can save you from submitting incorrect answers and build your confidence in your problem-solving abilities. Verification provides peace of mind and reinforces the importance of accuracy in mathematics.
Real-World Applications of Linear Equations
Linear equations are not just abstract mathematical concepts; they are powerful tools that can be used to model and solve real-world problems. From calculating costs and distances to predicting trends and making decisions, linear equations play a vital role in various fields. Understanding these applications can make learning linear equations more engaging and demonstrate their practical relevance.
1. Calculating Costs and Expenses
One of the most common applications of linear equations is in calculating costs and expenses. For example, if you're planning a road trip, you can use a linear equation to estimate the total cost of gasoline. Let's say the cost of gasoline is $3 per gallon, and your car gets 25 miles per gallon. You can create a linear equation to model the total cost (C) based on the distance you plan to travel (d).
The equation would be:
C = (3/25)d
This equation tells you that for every mile you drive, the cost increases by $3/25 or $0.12. If you plan to drive 500 miles, you can substitute d = 500 into the equation to find the total cost:
C = (3/25) * 500 = $60
Linear equations are also used in business to calculate costs, revenue, and profit. For instance, a company can use a linear equation to determine the break-even point, which is the point at which total revenue equals total costs. These applications highlight the importance of linear equations in financial planning and decision-making.
2. Determining Distances and Travel Time
Linear equations are also useful for determining distances and travel time. The fundamental relationship between distance, speed, and time is expressed by the formula:
Distance = Speed * Time
This is a linear equation where distance is directly proportional to both speed and time. If you know the speed and time, you can easily calculate the distance, or if you know the distance and speed, you can calculate the time.
For example, if you're traveling at a constant speed of 60 miles per hour, you can use the equation to determine how far you'll travel in 3 hours:
Distance = 60 * 3 = 180 miles
This linear relationship is crucial in navigation, transportation planning, and logistics. Whether you're planning a road trip, scheduling deliveries, or calculating flight times, linear equations provide a simple and effective way to estimate distances and travel times.
3. Predicting Trends and Making Projections
Linear equations can be used to model trends and make projections based on historical data. For example, if you're tracking sales data for a product and you observe a linear trend, you can use a linear equation to predict future sales. This is particularly useful in business and economics for forecasting demand, planning inventory, and setting sales targets.
Suppose a company's sales have been increasing by $10,000 per year. You can create a linear equation to model the sales (S) as a function of time (t) in years:
S = 10,000t + Initial Sales
If the initial sales were $50,000, the equation becomes:
S = 10,000t + 50,000
Using this equation, you can project sales for future years. For example, to predict sales in 5 years, you would substitute t = 5 into the equation:
S = 10,000 * 5 + 50,000 = $100,000
This application of linear equations is widely used in market analysis, financial forecasting, and strategic planning. Linear models provide a simple yet powerful tool for making informed decisions based on observed trends.
4. Solving Mixture Problems
Mixture problems, which involve combining different quantities with varying concentrations or values, can often be solved using linear equations. These types of problems are common in chemistry, finance, and everyday situations like cooking. For example, you might need to determine how much of a certain solution to mix with another to achieve a desired concentration.
Suppose you have a 20% saline solution and a 50% saline solution, and you want to create 100 milliliters of a 30% saline solution. You can use a system of linear equations to determine how much of each solution to mix. Let x be the amount of the 20% solution and y be the amount of the 50% solution. The equations would be:
x + y = 100 (total volume) 0.20x + 0.50y = 0.30 * 100 (total saline content)
Solving this system of equations will give you the amounts of each solution needed to create the desired mixture. Mixture problems illustrate the versatility of linear equations in solving practical problems that involve combining different quantities.
Conclusion
In this comprehensive guide, we have explored the process of solving the linear equation 5x - 5 = 3x - 9. We began by understanding the definition and importance of linear equations, highlighting their ubiquitous applications across various fields. We then meticulously dissected the equation, providing a step-by-step solution that involved grouping like terms, isolating the variable, and verifying the solution. We also delved into alternative methods for solving linear equations, such as the transposition method and the graphical method, offering a broader perspective on problem-solving techniques.
Furthermore, we addressed common mistakes that students often make when solving linear equations, such as incorrectly combining like terms, forgetting to distribute, performing operations on only one side, sign errors, and neglecting to verify the solution. By identifying these pitfalls and providing strategies to avoid them, we aim to empower you to solve linear equations with greater accuracy and confidence. Lastly, we explored the real-world applications of linear equations, demonstrating their practical relevance in areas such as calculating costs, determining distances, predicting trends, and solving mixture problems.
Mastering linear equations is not just an academic exercise; it's a fundamental skill that will serve you well in various aspects of life. Whether you're pursuing a career in science, technology, engineering, mathematics, or any other field that requires analytical thinking and problem-solving abilities, a solid understanding of linear equations is indispensable. By internalizing the concepts and techniques presented in this guide, you will be well-equipped to tackle a wide range of mathematical challenges and apply your knowledge to real-world situations.
As you continue your mathematical journey, remember that practice is the key to mastery. The more you practice solving linear equations and other mathematical problems, the more proficient you will become. Don't be discouraged by challenges or mistakes; view them as opportunities to learn and grow. Embrace the power of mathematics, and you will unlock a world of possibilities.
