Solving Linear Equations: A Step-by-Step Guide

by Alex Johnson 47 views

Introduction

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by variables raised to the first power, appear in various contexts, from basic algebra to advanced calculus and real-world applications. Mastering the techniques to solve these equations is crucial for mathematical proficiency. This comprehensive guide will walk you through the process of solving the linear equation −38x+2=−14-\frac{3}{8}x + 2 = -\frac{1}{4}, providing a clear and step-by-step approach applicable to a wide range of similar problems. We'll explore the underlying principles and strategies, ensuring you gain a solid understanding of how to tackle linear equations effectively. By the end of this guide, you'll be equipped with the knowledge and confidence to solve linear equations with ease. The principles you'll learn here are building blocks for more advanced mathematical concepts, making this a valuable skill for anyone pursuing studies in mathematics, science, or engineering.

Understanding Linear Equations

Before diving into the solution, it’s essential to 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 they represent a straight line when graphed on a coordinate plane. The general form of a linear equation in one variable is ax + b = c, where a, b, and c are constants, and x is the variable. The goal of solving a linear equation is to isolate the variable on one side of the equation, thereby determining its value. This involves performing algebraic operations on both sides of the equation to maintain equality while simplifying it. Understanding the structure of linear equations helps in identifying the appropriate steps to solve them. In our example, −38x+2=−14-\frac{3}{8}x + 2 = -\frac{1}{4}, we have a fractional coefficient for the variable x and constants on both sides of the equation. This form is common, and the techniques we'll use to solve it are applicable to many similar equations. Recognizing the components of the equation—the variable term, the constants, and the equality sign—is the first step in formulating a solution strategy. Linear equations are ubiquitous in mathematics and its applications, making their comprehension and solution a critical skill.

Step 1: Eliminating Fractions

To simplify the equation −38x+2=−14-\frac{3}{8}x + 2 = -\frac{1}{4}, the first step is to eliminate the fractions. Fractions can often make equations look more complex than they are, and removing them can significantly simplify the solving process. The most effective way to eliminate fractions is to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 8 and 4. The LCM of 8 and 4 is 8. By multiplying both sides of the equation by 8, we will clear the fractions and obtain an equation with integer coefficients. This step is crucial because it transforms the equation into a more manageable form. When multiplying, it's essential to distribute the multiplication across all terms on both sides of the equation. This ensures that the equality is maintained and that the resulting equation is equivalent to the original one. The process of eliminating fractions is a standard technique in solving equations, not just linear ones, and mastering it will make solving a wide range of algebraic problems easier. This step sets the stage for the subsequent steps, where we will isolate the variable and find its value.

Calculation:

  • Multiply both sides by 8: 8(−38x+2)=8(−14)8(-\frac{3}{8}x + 2) = 8(-\frac{1}{4})
  • Distribute the 8: 8(−38x)+8(2)=8(−14)8(-\frac{3}{8}x) + 8(2) = 8(-\frac{1}{4})
  • Simplify: −3x+16=−2-3x + 16 = -2

Step 2: Isolating the Variable Term

After eliminating the fractions, the equation is now in a simpler form: −3x+16=−2-3x + 16 = -2. The next step in solving linear equations is to isolate the term containing the variable, which in this case is -3x. To do this, we need to eliminate the constant term on the same side of the equation. Here, the constant term is +16. The principle of maintaining equality dictates that any operation performed on one side of the equation must also be performed on the other side. Therefore, to eliminate +16, we subtract 16 from both sides of the equation. This is a fundamental technique in algebra, ensuring that the equation remains balanced and that the solution remains accurate. Subtracting 16 from both sides will move the constant term to the right side of the equation, leaving the variable term isolated on the left side. This isolation is a crucial step towards solving for the variable x. By isolating the variable term, we are one step closer to determining the value of x that satisfies the equation. The process of isolating terms is a common strategy in algebra and is applicable to solving various types of equations.

Calculation:

  • Subtract 16 from both sides: −3x+16−16=−2−16-3x + 16 - 16 = -2 - 16
  • Simplify: −3x=−18-3x = -18

Step 3: Solving for the Variable

With the variable term isolated, the equation is now −3x=−18-3x = -18. The final step in solving linear equations is to solve for the variable x. This involves eliminating the coefficient of x, which in this case is -3. To do this, we divide both sides of the equation by -3. This operation maintains the equality of the equation while isolating x on one side. Division is the inverse operation of multiplication, so dividing by the coefficient will undo the multiplication and leave x by itself. It's important to remember that dividing by a negative number will change the sign of the result. This step is the culmination of the previous steps, where we systematically simplified the equation to isolate the variable. Once we divide both sides by the coefficient, we will have the value of x that satisfies the original equation. This value is the solution to the linear equation. The process of dividing by the coefficient is a standard technique in solving equations and is a crucial skill in algebra.

Calculation:

  • Divide both sides by -3: −3x−3=−18−3\frac{-3x}{-3} = \frac{-18}{-3}
  • Simplify: x=6x = 6

Solution

By following the steps outlined above, we have successfully solved the linear equation −38x+2=−14-\frac{3}{8}x + 2 = -\frac{1}{4}. The solution we found is x = 6. This means that when x is replaced with 6 in the original equation, the equation holds true. To verify the solution, we can substitute x = 6 back into the original equation and check if both sides are equal. This process of verification is a good practice to ensure the accuracy of the solution, especially in more complex equations. The solution x = 6 represents the value of x that makes the equation a true statement. In the context of a linear equation representing a line, this solution can be interpreted as the x-coordinate of a specific point on the line. Understanding the solution in this way can provide a deeper insight into the nature of linear equations. The ability to solve linear equations and interpret their solutions is a fundamental skill in mathematics and has wide-ranging applications in various fields.

Verification:

  • Substitute x = 6 into the original equation: −38(6)+2=−14-\frac{3}{8}(6) + 2 = -\frac{1}{4}
  • Simplify: −188+2=−14-\frac{18}{8} + 2 = -\frac{1}{4}
  • Convert 2 to a fraction with a denominator of 8: −188+168=−14-\frac{18}{8} + \frac{16}{8} = -\frac{1}{4}
  • Simplify: −28=−14-\frac{2}{8} = -\frac{1}{4}
  • Further simplification: −14=−14-\frac{1}{4} = -\frac{1}{4}

The solution is verified.

Conclusion

In conclusion, we have successfully solved the linear equation −38x+2=−14-\frac{3}{8}x + 2 = -\frac{1}{4} by systematically eliminating fractions, isolating the variable term, and solving for the variable. The solution we found is x = 6, which was verified by substituting it back into the original equation. This step-by-step approach demonstrates the fundamental principles of solving linear equations, which are applicable to a wide range of similar problems. Mastering these techniques is crucial for mathematical proficiency and has applications in various fields. Linear equations are a cornerstone of algebra and serve as a foundation for more advanced mathematical concepts. Understanding how to solve them effectively equips you with a valuable skill for problem-solving and critical thinking. The process involves not just algebraic manipulation but also a conceptual understanding of what it means to solve an equation—to find the value(s) of the variable(s) that make the equation true. With practice, solving linear equations becomes second nature, allowing you to tackle more complex mathematical challenges with confidence. Remember, the key is to approach each equation systematically, applying the principles of equality and inverse operations to isolate the variable.

For further learning and to solidify your understanding of linear equations, explore resources such as Khan Academy's Algebra I section, which provides comprehensive lessons and practice exercises.