Solving Linear Equations A Step-by-Step Guide To -24x + 37 = -131

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In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of many mathematical concepts and real-world applications. This comprehensive guide will walk you through the process of solving a specific linear equation: -24x + 37 = -131. We will break down each step, ensuring a clear understanding of the underlying principles. Mastering this process will empower you to tackle a wide array of linear equations with confidence.

Understanding Linear Equations

Before diving into the solution, let's establish a firm understanding of what a linear equation is. A linear equation is an algebraic equation in which the highest power of the variable is 1. It represents a straight line when graphed on a coordinate plane. The general form of a linear equation is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for. In our case, the equation -24x + 37 = -131 fits this form, with a = -24, b = 37, and c = -131.

The Goal: Isolating the Variable

The core objective in solving any equation is to isolate the variable. This means manipulating the equation until the variable (in this case, 'x') stands alone on one side of the equals sign. To achieve this, we employ a series of algebraic operations, ensuring that we maintain the balance of the equation. Whatever operation we perform on one side, we must also perform on the other side to preserve the equality.

Step 1: Isolating the Term with the Variable

Our first step is to isolate the term containing the variable, which is -24x. To do this, we need to eliminate the constant term, +37, from the left side of the equation. We can achieve this by subtracting 37 from both sides of the equation. This is based on the addition property of equality, which states that if we subtract the same number from both sides of an equation, the equality remains true.

-24x + 37 - 37 = -131 - 37

Simplifying both sides, we get:

-24x = -168

Now, we have successfully isolated the term with the variable on the left side of the equation.

Step 2: Solving for the Variable

The next step is to isolate the variable 'x' itself. Currently, 'x' is being multiplied by -24. To undo this multiplication, we need to perform the inverse operation, which is division. We will divide both sides of the equation by -24. This is based on the division property of equality, which states that if we divide both sides of an equation by the same non-zero number, the equality remains true.

-24x / -24 = -168 / -24

Simplifying both sides, we get:

x = 7

Therefore, the solution to the equation -24x + 37 = -131 is x = 7.

Verification: Ensuring the Solution is Correct

It's always a good practice to verify our solution to ensure its accuracy. To do this, we substitute the value we found for 'x' (which is 7) back into the original equation and check if both sides of the equation are equal.

-24(7) + 37 = -131

-168 + 37 = -131

-131 = -131

Since both sides of the equation are equal, our solution x = 7 is correct.

Alternative Methods for Solving Linear Equations

While the method described above is a standard approach, there can be alternative ways to solve linear equations, especially as equations become more complex. Understanding these alternative methods can provide valuable flexibility in problem-solving.

Method 1: Distributive Property and Combining Like Terms

This method is particularly useful when the equation involves parentheses or multiple terms with the variable. The first step is to apply the distributive property to eliminate parentheses, if any. Then, we combine like terms on each side of the equation to simplify it. Finally, we proceed with the steps of isolating the variable as described earlier.

Example:

Consider the equation 2(x + 3) - 5x = 11

  1. Apply the distributive property: 2x + 6 - 5x = 11
  2. Combine like terms: -3x + 6 = 11
  3. Subtract 6 from both sides: -3x = 5
  4. Divide both sides by -3: x = -5/3

Method 2: Clearing Fractions or Decimals

When the equation involves fractions or decimals, it can be helpful to clear them before proceeding with the solution. To clear fractions, we multiply both sides of the equation by the least common multiple (LCM) of the denominators. To clear decimals, we multiply both sides by a power of 10 that will eliminate the decimal points.

Example with Fractions:

Consider the equation (1/2)x + (2/3) = (5/6)

  1. Find the LCM of the denominators (2, 3, and 6), which is 6.
  2. Multiply both sides by 6: 6[(1/2)x + (2/3)] = 6(5/6)
  3. Distribute and simplify: 3x + 4 = 5
  4. Subtract 4 from both sides: 3x = 1
  5. Divide both sides by 3: x = 1/3

Example with Decimals:

Consider the equation 0.2x + 1.5 = 2.1

  1. Multiply both sides by 10 to clear the decimals: 10(0.2x + 1.5) = 10(2.1)
  2. Distribute and simplify: 2x + 15 = 21
  3. Subtract 15 from both sides: 2x = 6
  4. Divide both sides by 2: x = 3

Method 3: Using a Graphing Calculator or Software

For more complex equations, or when a visual representation is helpful, a graphing calculator or software can be a powerful tool. By graphing the equation (or the two expressions on either side of the equation), the solution can be found by identifying the point(s) where the graph intersects the x-axis (for equations in the form f(x) = 0) or the point(s) where the graphs of the two expressions intersect.

Common Mistakes to Avoid

Solving linear equations is a systematic process, but it's easy to make mistakes if not careful. Here are some common errors to watch out for:

1. Incorrectly Applying the Order of Operations

Always follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. This ensures that calculations are performed in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

2. Forgetting to Distribute

When dealing with parentheses, remember to distribute the term outside the parentheses to every term inside. A common mistake is to only multiply the first term inside the parentheses.

3. Not Performing the Same Operation on Both Sides

The fundamental principle of solving equations is maintaining balance. Any operation performed on one side of the equation must be performed on the other side to preserve the equality. Failing to do so will lead to an incorrect solution.

4. Sign Errors

Be particularly careful with signs, especially when dealing with negative numbers. A small sign error can significantly alter the result. Double-check your work, especially when adding, subtracting, multiplying, or dividing negative numbers.

5. Incorrectly Combining Like Terms

Only combine terms that have the same variable raised to the same power. For example, you can combine 3x and 5x, but you cannot combine 3x and 5x². Also, be mindful of the signs when combining terms.

6. Dividing by Zero

Division by zero is undefined and is a major mathematical error. Always make sure that you are not dividing by zero at any point in the solution process.

7. Not Verifying the Solution

As mentioned earlier, verifying the solution is a crucial step to ensure accuracy. Substitute the solution back into the original equation and check if both sides are equal. This will help catch any errors made during the solution process.

Real-World Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have wide-ranging applications in various real-world scenarios. Understanding these applications can make learning linear equations more engaging and meaningful.

1. Budgeting and Finance

Linear equations are frequently used in budgeting and personal finance. For example, you can use a linear equation to model your monthly expenses, calculate loan payments, or determine the break-even point for a business venture.

Example:

Suppose you have a monthly budget of $2000. Your fixed expenses (rent, utilities, etc.) are $1200, and you want to allocate the remaining amount for discretionary spending (groceries, entertainment, etc.). If you want to save $300 each month, you can set up a linear equation to determine how much you can spend each day.

Let 'x' be the amount you can spend each day. The equation would be:

1200 + 300 + 30x = 2000

Solving for 'x' will tell you the maximum amount you can spend each day.

2. Physics and Engineering

Many physical phenomena can be modeled using linear equations. For example, the relationship between distance, speed, and time is often expressed as a linear equation. Engineers use linear equations to design structures, analyze circuits, and model various systems.

Example:

The equation d = rt (distance = rate × time) is a linear equation that relates distance, rate (speed), and time. If you know the rate and time, you can use this equation to calculate the distance traveled.

3. Chemistry

Linear equations are used in chemistry to calculate concentrations, balance chemical equations, and determine reaction rates.

Example:

The equation M1V1 = M2V2 (where M is molarity and V is volume) is a linear equation used in dilution calculations. If you have a solution of a certain concentration and volume, you can use this equation to calculate the volume of solvent needed to dilute it to a desired concentration.

4. Economics

Linear equations are fundamental in economics for modeling supply and demand, cost functions, and revenue projections.

Example:

The supply and demand curves in economics are often represented as linear equations. The point where these curves intersect represents the market equilibrium, where the quantity supplied equals the quantity demanded.

5. Computer Science

Linear equations are used in computer graphics, linear programming, and algorithm analysis.

Example:

In computer graphics, linear equations are used to perform transformations such as scaling, rotation, and translation of objects in a 2D or 3D space.

Conclusion

In conclusion, solving the linear equation -24x + 37 = -131 demonstrates the core principles of algebraic manipulation. By systematically isolating the variable, we arrive at the solution x = 7. Remember to always verify your solution to ensure accuracy. Furthermore, exploring alternative methods and understanding common mistakes will enhance your problem-solving skills. Linear equations are a fundamental tool in mathematics with a wide range of real-world applications, making their mastery essential for academic and practical success. By understanding the steps involved, practicing regularly, and recognizing the real-world applications, you can build a strong foundation in solving linear equations and confidently tackle more complex mathematical challenges.