Problem 8 - Algebra A Combined Function

Question

Solve each equation. Check each solution. $$ t-9.2=-6.8 $$

Step-by-Step Solution

Answer

The solution is \( t = 2.4 \).

Step 1: Understand the Equation

The given equation is \( t - 9.2 = -6.8 \). The goal is to find the value of \( t \) that makes this equation true.

Step 2: Isolate the Variable

To find \( t \), you need to get \( t \) alone on one side of the equation. You can do this by adding 9.2 to both sides of the equation:\[ t - 9.2 + 9.2 = -6.8 + 9.2 \] Simplifying, we have:\[ t = 2.4 \].

Step 3: Check the Solution

To ensure the solution is correct, substitute \( t = 2.4 \) back into the original equation. The original equation was \( t - 9.2 = -6.8 \). Substitute to get:\[ 2.4 - 9.2 = -6.8 \] Simplifying the left side:\[ -6.8 = -6.8 \] Since both sides are equal, the solution is verified.

Key Concepts

Checking Solutions

When solving linear equations, it is important to verify that your solution is correct. This process is known as "checking solutions." To do this, you simply substitute the value you've found back into the original equation to see if it holds true.

For instance, in our equation, we found that \( t = 2.4 \). To check this, we replaced \( t \) with \( 2.4 \) in the equation \( t - 9.2 = -6.8 \). Performing the subtraction gives us \( 2.4 - 9.2 = -6.8 \). Simplifying the left-hand side results in \( -6.8 \), which equals the right-hand side of the equation. Therefore, the solution is correct.

Checking solutions helps to catch any mistakes that might have been made during the calculation process. If the initial solution does not satisfy the original equation, you know to go back and double-check your work. Ensuring accuracy in maths is key to building confidence and understanding.

Isolation of Variables

The term "isolation of variables" means moving all terms involving the variable to one side of the equation and constants to the other. This is crucial because it allows you to solve for the unknown variable, which in our case was \( t \).

Let's walk through how we isolated \( t \) in the provided equation: \( t - 9.2 = -6.8 \). The objective is to have \( t \) by itself on one side. To achieve this, we added \( 9.2 \) to both sides of the equation. Why add \( 9.2 \)? Because doing so cancels out the \( -9.2 \) on the left side:

• Original: \( t - 9.2 = -6.8 \)
• Add \( 9.2 \) to both sides: \( t - 9.2 + 9.2 = -6.8 + 9.2 \)
• Simplified: \( t = 2.4 \)

The variable \( t \) is now isolated, equaling \( 2.4 \). This step is critical as it leads you to the solution of the equation easily and accurately. It's like peeling away layers to get to the core of the mathematical problem.

Equation Simplification

Simplification is an essential part of solving equations. It involves combining like terms and performing arithmetic to reduce the equation to its simplest form. Simplifying makes equations easier to work with and understand.

In our example, after adding \( 9.2 \) to both sides of the equation, we simplified by performing the arithmetic on the right side:

• The equation was modified to: \( t = -6.8 + 9.2 \)
• Simplifying the right side: \( t = 2.4 \)

Simplification involved straightforward addition, but it can also include distributing and combining terms where applicable. This step ensures the equation is in its most basic and easy-to-understand form. In math, an organized and tidy equation is much easier to solve efficiently. Keeping equations simple helps minimize errors and makes complex problems much more approachable. Always aim to tidy up your equations before proceeding with further calculations.