Problem 23 - Algebra 1

Question

Solve the equation. $$t-2=6$$

Step-by-Step Solution

Answer

The solution to the equation \( t - 2 = 6 \) is \( t = 8 \).

Step 1: Understand the Problem

We are given the equation \( t - 2 = 6 \). We need to find the value of \( t \) that makes this equation true.

Step 2: Rearrange the equation

To find \( t \), we need to isolate it on one side of the equation. To achieve this, we need to cancel out '-2' from the left-hand side. We do this by doing the opposite operation, which is adding '2' to both sides of the equation. This gives us: \( t - 2 + 2 = 6 + 2 \)

Step 3: Simplify

Now we simplify both sides of the equation. On the left-hand side, '-2 + 2' cancels out, leaving us with \( t \). On the right-hand side, we add the numbers together to get '8'. This gives us the equation \( t = 8 \).

Key Concepts

Isolate Variable

Isolating the variable is a fundamental strategy used when solving linear equations. The goal is to rearrange the equation to make the variable we're solving for, in this case, t, stand alone on one side of the equality sign. To do this, we perform operations that undo whatever is being done to the variable.

For example, in the equation \(t - 2 = 6\), the variable \(t\) is being subtracted by 2. To isolate \(t\), we need to cancel this subtraction by adding 2 to both sides of the equation. It's important to maintain the balance of the equation by performing the same operation on both sides. This action gives us \(t - 2 + 2 = 6 + 2\), simplifying to \(t = 8\), effectively isolating \(t\).

Algebraic Operations

Mastering algebraic operations is key to solving linear equations effectively. These operations include addition, subtraction, multiplication, and division. When an equation involves a variable, these operations help us manipulate the equation to isolate that variable.

In our example, we used addition to neutralize a subtraction on the variable. Had the equation been an addition, we would subtract; if it was a multiplication, we would divide; and for a division, we would multiply. This keeps the equation balanced and the integrity of the original equality intact. Through careful application of algebraic operations, we systematically simplify equations to reach the solution.

Simplification of Equations

Simplification of equations is the process of reducing them to their simplest form, making it easier to solve or understand them. This involves combining like terms, reducing fractions, and removing any unnecessary terms or factors.

For instance, in the step where we added 2 to both sides (\(t - 2 + 2 = 6 + 2\)), we simplified the equation by combining like terms. The \( -2 + 2 \) on the left side of the equation cancel each other out, which simplifies to just \( t \), and \(6 + 2\) simplifies to \(8\) on the right side. The equation \(t = 8\) is the simplest form, making it clear that the value of \(t\) is 8. Simplification leads to clearer solutions and is a crucial step in solving equations.