Solving Linear Equations Find The Value Of X And Justify The Solution

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In the realm of mathematics, solving linear equations stands as a fundamental skill. It's the bedrock upon which more advanced mathematical concepts are built. A linear equation, at its core, is an algebraic equation where each term is either a constant or the product of a constant and a single variable. These equations, when graphed, produce a straight line, hence the name "linear." The ultimate goal in solving a linear equation is to isolate the variable, typically represented by 'x', on one side of the equation. This isolation reveals the value of 'x' that satisfies the equation, making the equation a true statement.

Unraveling the Linear Equation: A Step-by-Step Approach

Let's delve into the process of solving the linear equation: -2x = -1/4 + 1/2x + 5. This equation presents a classic example of a linear equation with the variable 'x' appearing on both sides. Our mission is to manipulate this equation using algebraic principles until we have 'x' isolated and its value revealed.

  1. Combining Like Terms: The first step towards solving this equation involves simplifying both sides by combining like terms. On the right side, we have a constant term (-1/4) and another constant term (5). Let's add these together: -1/4 + 5 = -1/4 + 20/4 = 19/4. This simplifies the right side of the equation to 1/2x + 19/4. Now, our equation looks like this: -2x = 1/2x + 19/4.

  2. Isolating the Variable Term: To gather all the 'x' terms on one side, we need to eliminate the 1/2x term from the right side. We achieve this by subtracting 1/2x from both sides of the equation. This maintains the equation's balance. Subtracting 1/2x from both sides gives us: -2x - 1/2x = 1/2x + 19/4 - 1/2x. Simplifying this, we get -5/2x = 19/4. We now have all the 'x' terms on the left side.

  3. Isolating the Variable: Our next goal is to isolate 'x' completely. Currently, 'x' is being multiplied by -5/2. To undo this multiplication, we multiply both sides of the equation by the reciprocal of -5/2, which is -2/5. This gives us: (-5/2x) * (-2/5) = (19/4) * (-2/5). On the left side, the -5/2 and -2/5 cancel out, leaving us with 'x'. On the right side, we multiply the fractions: (19/4) * (-2/5) = -38/20. Simplifying this fraction by dividing both the numerator and denominator by 2, we get -19/10. Therefore, we have x = -19/10.

Verifying the Solution: The Importance of Justification

Finding a solution is only half the battle. We must verify that our solution is correct. This is done by substituting the value we found for 'x' back into the original equation. If both sides of the equation are equal after the substitution, our solution is verified.

Let's substitute x = -19/10 into the original equation: -2x = -1/4 + 1/2x + 5.

Substituting, we get: -2(-19/10) = -1/4 + 1/2(-19/10) + 5.

Simplifying the left side: -2 * (-19/10) = 38/10 = 19/5.

Simplifying the right side: -1/4 + 1/2(-19/10) + 5 = -1/4 - 19/20 + 5. To add these terms, we need a common denominator, which is 20. So, we rewrite the terms: -5/20 - 19/20 + 100/20 = 76/20 = 19/5.

Comparing both sides: We see that both sides of the equation simplify to 19/5. This confirms that our solution, x = -19/10, is correct. The left-hand side (LHS) equals the right-hand side (RHS), thus justifying our solution.

Why Justification Matters: Building Mathematical Confidence

Justifying solutions is not merely a formality; it's a cornerstone of mathematical understanding. It solidifies the connection between the steps taken and the final answer. It transforms the process from a rote application of rules to a logical journey where each step is validated. This process of verification builds confidence in one's mathematical abilities and fosters a deeper understanding of the concepts involved. In the real world, verification is equally critical. Engineers, scientists, and financial analysts, among others, rely on verifying their calculations to ensure accuracy and prevent costly errors. Therefore, mastering the art of justification is a valuable skill that extends far beyond the classroom.

Exploring Different Types of Linear Equations

Linear equations come in various forms, each with its own nuances. Some equations may have fractions or decimals, while others may involve parentheses or multiple variables. However, the core principles of solving them remain the same: simplify, isolate the variable, and verify. Understanding the different types of linear equations helps to broaden problem-solving capabilities.

  1. Equations with Fractions: When dealing with fractions, the first step is often to eliminate them by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. This clears the fractions and makes the equation easier to solve. For instance, in the equation x/2 + 1/3 = 5/6, the LCM of 2, 3, and 6 is 6. Multiplying both sides by 6 eliminates the fractions.

  2. Equations with Decimals: Equations with decimals can be solved directly, but it's often easier to eliminate the decimals by multiplying both sides of the equation by a power of 10. The power of 10 is chosen based on the maximum number of decimal places in any term. For example, in the equation 0.25x + 1.5 = 3, we can multiply both sides by 100 to eliminate the decimals.

  3. Equations with Parentheses: Parentheses indicate a need to distribute. The distributive property states that a(b + c) = ab + ac. Apply this property to remove parentheses before proceeding with isolating the variable. For example, in the equation 2(x + 3) = 10, distribute the 2 to get 2x + 6 = 10.

  4. Equations with Multiple Variables: While the equation we solved had only one variable, linear equations can also involve multiple variables. Solving for one variable in terms of others is a common task in algebra and is essential for solving systems of equations.

Real-World Applications of Linear Equations

Linear equations are not confined to the abstract world of mathematics textbooks; they are powerful tools for modeling and solving real-world problems. Their applications span a wide range of fields, from science and engineering to economics and finance. Understanding how to apply linear equations in practical situations enhances their value and relevance.

  1. Physics: Linear equations are used extensively in physics to describe motion, forces, and energy. For example, the equation d = vt (distance equals velocity times time) is a linear equation that relates distance, velocity, and time. Similarly, Ohm's law (V = IR) is a linear equation that relates voltage, current, and resistance in an electrical circuit.

  2. Engineering: Engineers use linear equations to design structures, analyze circuits, and model systems. For example, linear equations can be used to calculate the forces acting on a bridge or the flow of fluids in a pipe.

  3. Economics: Linear equations are used in economics to model supply and demand, cost and revenue, and other economic relationships. For example, a linear demand curve shows the relationship between the price of a product and the quantity demanded.

  4. Finance: Financial analysts use linear equations to calculate interest, depreciation, and investment returns. For example, simple interest is calculated using a linear equation.

  5. Everyday Life: Linear equations also have everyday applications. They can be used to calculate the cost of a taxi ride, the amount of paint needed for a room, or the calories in a meal.

Mastering Linear Equations: A Path to Mathematical Proficiency

Solving linear equations is a foundational skill in mathematics. It's a gateway to more advanced topics such as systems of equations, inequalities, and calculus. Mastering linear equations requires practice, patience, and a willingness to learn from mistakes. The more you practice, the more comfortable and confident you will become in your ability to solve them. Remember, each equation is a puzzle waiting to be solved, and with the right approach, you can unlock its solution.

The value of x that is the solution to the equation -2x = -1/4 + 1/2x + 5 is -19/10. This solution has been rigorously justified by substituting it back into the original equation and verifying that both sides are equal. This process of solving and justifying linear equations is a fundamental skill that opens doors to a deeper understanding of mathematics and its applications in the world around us.

By following the steps outlined in this comprehensive guide, you can confidently tackle linear equations of all types. Remember to simplify, isolate the variable, verify your solution, and explore the real-world applications of these powerful tools. With dedication and practice, you can master linear equations and unlock your mathematical potential.