Problem 47 - Fundamentals of Mathematics

Question

Calculator Exercises. $$8.078-9.112=2.106 y-1.106 y$$

Step-by-Step Solution

Answer

y = -1.034

Step 1: Simplify the Left-Hand Side

Start by evaluating the left-hand side of the equation, which is a subtraction: \(8.078 - 9.112\). To do this, subtract the two numbers:\[8.078 - 9.112 = -1.034\]

Step 2: Simplify the Right-Hand Side

Next, simplify the right-hand side of the equation. We have \(2.106y - 1.106y\). This involves combining the like terms:\[2.106y - 1.106y = (2.106 - 1.106)y = 1.0y\]

Step 3: Set the Equation

Now, set the simplified left-hand side equal to the simplified right-hand side of the equation:\[-1.034 = 1.0y\]

Step 4: Solve for y

To solve for \(y\), divide both sides of the equation by the coefficient of \(y\) (which is 1):\[y = \frac{-1.034}{1.0} = -1.034\]

Step 5: Conclude the Solution

After performing these calculations, the value of \(y\) is found. The answer is:\[y = -1.034\]

Key Concepts

Simplification

Simplification is the process of making an equation easier to manage. It usually involves reducing complex expressions into simpler, equivalent forms. This is often done by performing basic arithmetic operations. In our exercise, we started with the expression on the left-hand side. We had to subtract the numbers 8.078 and 9.112. Performing the subtraction gives us \(-1.034\). Simplification ensures that the equation is easier to understand and solve.

• Identify operations: First, recognize what operations are needed, such as subtraction in our case.
• Perform the arithmetic: Calculate the result, which helps in reducing complexity.
• Aim for clarity: A simplified expression is easier to work with in the next mathematical steps.

This approach streamlines the process of solving equations by focusing only on necessary calculations.

Solving for variable

Solving for a variable means finding the value of the variable that makes the equation true. In algebra, this is a common task where we manipulate the equation to isolate the variable on one side.

• Isolate the variable: In our example, we set \(-1.034\) equal to \(1.0y\). The goal was to get \(y\) alone on one side.
• Use inverse operations: Here, we divided both sides by 1.0, which gave us \(y = -1.034\).
• Check: Ensure that every operation helps maintain equality, keeping the balance of the equation.

Understanding how to solve for a variable is fundamental since it allows us to determine unknown values in a structured way.

Like Terms

Like terms are terms in an algebraic expression with the same variable raised to the same power. Combining like terms is crucial for simplification, as it reduces the amount of work needed to solve an equation.

• Identify like terms: In our equation, \(2.106y\) and \(1.106y\) are like terms because they have the same variable \(y\).
• Combine them: We subtract these coefficients, resulting in \((2.106 - 1.106)y = 1.0y\).
• Streamline expressions: By merging like terms, the equation style becomes more concise, aiding further calculations.

Combining like terms simplifies the expression, making it much easier to solve subsequent steps of the equation.