Solving -(4/5)x = 80 A Step By Step Guide
In this article, we will delve into the process of solving the equation -(4/5)x = 80 for the variable x in a single step. This type of equation is a fundamental concept in algebra, and understanding how to solve it is crucial for mastering more complex mathematical problems. We will explore the underlying principles and demonstrate the correct method to arrive at the solution efficiently.
Understanding the Basics of Algebraic Equations
Before we dive into the specific problem, let's refresh our understanding of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions often contain variables, which are symbols (usually letters) that represent unknown values. The goal of solving an equation is to isolate the variable on one side of the equation, thereby determining its value.
In the equation -(4/5)x = 80, x is the variable we aim to solve for. The expression on the left side, -(4/5)x, represents a fraction multiplied by the variable. The expression on the right side, 80, is a constant. Our objective is to manipulate the equation in a way that x stands alone on one side, revealing its value.
The Concept of Inverse Operations
The key to solving algebraic equations lies in the concept of inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. To isolate a variable, we perform the inverse operation on both sides of the equation. This maintains the equality while moving us closer to the solution.
In our equation, -(4/5)x = 80, the variable x is being multiplied by the fraction -(4/5). To isolate x, we need to perform the inverse operation of multiplication, which is division. However, instead of dividing by a fraction, it is often more convenient to multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 2/3 is 3/2. When we multiply a fraction by its reciprocal, the result is always 1.
Applying the Correct Method
Now, let's apply this knowledge to solve the equation -(4/5)x = 80. The coefficient of x is -(4/5). To isolate x, we need to multiply both sides of the equation by the reciprocal of -(4/5), which is -(5/4). This is because multiplying -(4/5) by -(5/4) results in 1, effectively isolating x.
Multiplying both sides of the equation by -(5/4), we get:
-(4/5)x * -(5/4) = 80 * -(5/4)
On the left side, -(4/5) and -(5/4) cancel out, leaving us with x:
x = 80 * -(5/4)
On the right side, we multiply 80 by -(5/4). To do this, we can first divide 80 by 4, which gives us 20, and then multiply by -5:
x = 20 * -5
This simplifies to:
x = -100
Therefore, the solution to the equation -(4/5)x = 80 is x = -100.
Analyzing the Incorrect Options
Now, let's examine the incorrect options provided in the original question and understand why they do not correctly solve the equation. This will further solidify our understanding of the correct method.
The first incorrect option is:
-(4/5)(5/4)x = 80(5/4)
This option multiplies both sides of the equation by the reciprocal of (4/5), which is (5/4). However, it fails to account for the negative sign in front of the fraction -(4/5). Multiplying by (5/4) instead of -(5/4) will not isolate x correctly. While the fractions will cancel out, the negative sign will remain, leading to an incorrect solution. This highlights the crucial importance of considering the sign when dealing with coefficients in algebraic equations. To correctly isolate x, we must multiply by the reciprocal that includes the negative sign, ensuring that the product of the coefficient and its reciprocal is positive 1.
The second incorrect option is:
-(4/5)(-5)x = 80(-5)
This option attempts to eliminate the fraction by multiplying -(4/5) by -5. While this does eliminate the denominator, it does not correctly isolate x. Multiplying -(4/5) by -5 results in 4, not 1. Therefore, this operation does not lead to the isolation of x and subsequently does not solve the equation correctly. This mistake underscores the need to multiply by the entire reciprocal, not just a part of it, to achieve the desired cancellation and isolate the variable. The correct approach involves multiplying by the reciprocal, which includes both the inverted fraction and the appropriate sign to ensure the coefficient of x becomes 1.
The Correct Approach in One Step
The correct method to solve the equation -(4/5)x = 80 in one step is to multiply both sides by the reciprocal of -(4/5), which is -(5/4). This is represented as:
-(4/5)(-(5/4))x = 80(-(5/4))
This single step effectively isolates x on the left side and allows us to calculate its value on the right side. This method demonstrates the power and efficiency of using inverse operations to solve algebraic equations. By multiplying by the reciprocal, we directly address the coefficient of x, ensuring its isolation in a single, well-executed step. This approach not only simplifies the solution process but also reduces the chances of making errors along the way. Understanding and applying this method is a cornerstone of algebraic problem-solving.
Why This Method Works
This method works because it leverages the multiplicative inverse property. The multiplicative inverse of a number is the number that, when multiplied by the original number, results in 1. In the case of fractions, the multiplicative inverse is the reciprocal. By multiplying both sides of the equation by the reciprocal of the coefficient of x, we are essentially multiplying the left side by 1, which leaves us with just x. This isolates the variable and allows us to determine its value.
The principle behind this method is rooted in the fundamental properties of equality. When we perform the same operation on both sides of an equation, we maintain the equality. Multiplying both sides by the same value ensures that the equation remains balanced, and the solution remains valid. This principle is a cornerstone of algebraic manipulation, allowing us to transform equations while preserving their integrity. The multiplicative inverse property is a powerful tool in this context, enabling us to isolate variables and solve equations efficiently.
Conclusion
In conclusion, the correct way to solve the equation -(4/5)x = 80 in one step is to multiply both sides by -(5/4). This method utilizes the concept of inverse operations and the multiplicative inverse property to efficiently isolate the variable x and arrive at the solution x = -100. Understanding this method is crucial for mastering algebraic equations and building a strong foundation in mathematics. By correctly applying the reciprocal and considering the sign, we can solve such equations accurately and efficiently. This skill is invaluable for tackling more complex problems and advancing in mathematical studies.
Remember, the key to solving algebraic equations is to isolate the variable by performing inverse operations on both sides. By mastering this technique, you will be well-equipped to tackle a wide range of mathematical challenges.
