Problem 49 - Fundamentals of Mathematics

Question

Calculator Exercises. $$6.1185 x-4.0031=5.1185 x-0.0058$$

Step-by-Step Solution

Answer

The value of \( x \) is 3.9973.

Step 1: Simplify the Equation

First, we need to get all terms involving the variable on one side of the equation. We will subtract \( 5.1185x \) from both sides: \( 6.1185x - 5.1185x - 4.0031 = -0.0058 \). This simplifies to \( 1.0x - 4.0031 = -0.0058 \).

Step 2: Isolate the Variable

Next, add \( 4.0031 \) to both sides of the equation to isolate the term with the variable \( x \): \( 1.0x = 4.0031 - 0.0058 \).

Step 3: Calculate the Right Side

Now, perform the arithmetic operation on the right side of the equation: \( 4.0031 - 0.0058 = 3.9973 \). So the equation becomes \( 1.0x = 3.9973 \).

Step 4: Solve for x

Since the coefficient of \( x \) is 1.0, the solution is simply \( x = 3.9973 \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that showcase a linear relationship. They often take the form of \( ax + b = c \). In the case of our exercise, the equation is \( 6.1185x - 4.0031 = 5.1185x - 0.0058 \). These equations are described as 'linear' because the variable, \( x \), is to the first power. Their graphical representation is a straight line.

Solving linear equations involves finding the value of the unknown variable that makes the equation true. This is crucial in various applications, such as predicting trends or determining unknown quantities in real-world scenarios. The process can be made simple with consistent practice once one understands the fundamental concepts of combining like terms and maintaining the equation's balance.

If you're new to working with linear equations, remember: practice makes perfect! Keep solving different equations to familiarize yourself with the process and techniques.

Variable Isolation

Variable isolation is a crucial step in solving equations, particularly linear equations. It involves manipulating the equation so that the variable is by itself on one side. This is typically done by performing operations that "undo" any operations being performed on the variable.

In the original exercise, we had to isolate \( x \) on one side of the equation. We started with \( 6.1185x - 4.0031 = 5.1185x - 0.0058 \). By subtracting \( 5.1185x \) from both sides, we begin the process of isolation:

• Subtracting one term from both sides does not change the equality of the equation; it only simplifies it.
• We then added \( 4.0031 \) to both sides to fully isolate the term with \( x \).

As a result, our transformed equation became \( 1.0x = 3.9973 \). With the variable isolated, you can easily see and calculate its value.

This method highlights the importance of understanding how to manipulate equations while keeping them balanced. It's a skill you'll find beneficial across many math and science disciplines.

Arithmetic Operations

Arithmetic operations are fundamental tools used in equation solving. They include addition, subtraction, multiplication, and division. In the process of solving our linear equation, arithmetic operations played a key role.

In our exercise, subtraction was used when simplifying the equation to collect like terms. We subtracted \( 5.1185x \) from both sides to simplify the equation:

• This helped reduce the equation to a simpler form, where \( 1.0x - 4.0031 = -0.0058 \).

Next, addition was crucial in isolating the variable by removing \( -4.0031 \) from one side:

• We added \( 4.0031 \) to both sides, achieving balance and simplifying it to \( 1.0x = 3.9973 \).

Finally, the arithmetic operation of solving \( 4.0031 - 0.0058 \) to get \( 3.9973 \) closed the solution with ease.

Mastering these operations ensures you can navigate and solve complex problems. Practicing them routinely can boost your confidence and efficiency in various math-related fields.