Problem 96 - Algebra A Combined Function

Question

Solve. 0.06 y+2.63=2.5562

Step-by-Step Solution

Answer

The solution is \(y = -1.23\).

Step 1: Isolate the variable term

The given equation is \(0.06y + 2.63 = 2.5562\). To isolate the term containing \(y\), subtract 2.63 from both sides of the equation: \(0.06y + 2.63 - 2.63 = 2.5562 - 2.63\). This simplifies to \(0.06y = -0.0738\).

Step 2: Solve for the variable

To solve for \(y\), divide both sides of the equation by 0.06: \(y = \frac{-0.0738}{0.06}\). Calculating this gives \(y = -1.23\).

Key Concepts

Understanding How to Solve Equations

Solving equations is a foundation of algebra that involves finding the value of a variable that makes the equation true. In the given equation, our goal is to determine what value of \( y \) satisfies the condition \( 0.06y + 2.63 = 2.5562 \). This involves several key steps:

• Identify the variable: Look for the term in the equation that includes the variable, in this case, \( 0.06y \).
• Isolate the variable: Perform operations to get the variable term by itself on one side of the equation. This often involves adding, subtracting, multiplying, or dividing terms on both sides.
• Simplify both sides: After moving terms, simplify each side of the equation to make calculations easier.
• Solve for the variable: Use arithmetic to solve for the value of the isolated variable.

In this specific equation, the steps involve subtraction and division, two common techniques in solving linear equations.

The Importance of Step-by-Step Solutions

A step-by-step solution breaks down the process of solving equations into manageable stages. This approach is not just about finding the right answer but also understanding why each step is necessary. Here's what a step-by-step solution looks like for our problem:

• Step 1: Subtracting 2.63 from both sides cancels out the constant term on the left, isolating the variable term \( 0.06y \). This is a crucial step because it simplifies the equation, making it easier to work with.
• Step 2: Dividing each side by 0.06 directly leads to the solution for \( y \), providing the exact value that balances the equation. This division is critical as it clears the coefficient of \( y \), leaving \( y \) by itself.

By following these steps meticulously, you ensure that each move is logical and mathematically sound, which helps reinforce algebraic reasoning skills.

Key Algebra Concepts in Solving Equations

Algebra is a central branch of mathematics that deals with symbols and rules for manipulating those symbols. Understanding a few core concepts can greatly help when solving equations, like the one provided:

• Linear Equations: These are equations where the variable is raised to the power of one. In our problem, \( 0.06y + 2.63 = 2.5562 \) is a linear equation, which means the graph of the equation would form a straight line if plotted.
• Inverse Operations: Solving equations often requires the use of inverse operations. For example, subtraction is the inverse of addition, and division is the inverse of multiplication. These operations undo each other and are used to isolate variables.
• Balancing Equations: Whatever you do to one side of the equation, you must do to the other. This principle ensures that the equation remains true at each step of the solution.

Familiarity with these algebraic concepts ensures a better grasp of solving equations consistently and accurately.