Problem 11 - Precalculus

Question

Solve each equation. $$t-\\{4-[t-(4+t)]\\}=6$$

Step-by-Step Solution

Answer

The solution is \(t = 14\).

Step 1: Simplify the Inner Bracket

The given equation is \(t-\{4-[t-(4+t)]\}=6\). Begin by simplifying the innermost bracket: \([t-(4+t)]\). This simplifies to \(t-4-t\), which further simplifies to \(-4\).

Step 2: Replace the Simplified Result

Substitute the result from Step 1 back into the equation: \(t-\{4-[-4]\}=6\). Simplifying inside the braces gives \(4+4=8\).

Step 3: Solve the Remaining Expression

Substitute the value from Step 2 to simplify the equation further: \(t-8=6\).

Step 4: Isolate the Variable

Add 8 to both sides of the equation to isolate \(t\): \(t = 6 + 8\).

Step 5: Solve for t

Calculate the right-hand side: \(t = 14\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that allows us to make complex equations easier to solve. Think of it as cleaning up a messy room; we aim to make it as simple as possible.
In the given problem, we start with a complex expression inside brackets: \[t-\{4-[t-(4+t)]\}=6\] It's important to focus on one part at a time. Start with the innermost brackets. Here, we have \[[t-(4+t)]\] First, distribute the subtraction and simplify to \[t-4-t\], which ultimately simplifies to \[-4\].
This process reduces the complexity and prepares the equation for further solving steps. Each step of simplification brings us closer to the solution by reducing clutter and making the equation more manageable. With practice, simplifying becomes a quick and intuitive process that eases the path to solving any algebraic equation.
By breaking down the original expression into smaller parts, we can tackle complex algebraic equations efficiently.

Isolation of Variables

Once the equation is simplified, the next goal is to isolate the variable we need to solve for, in this case, it's the variable \(t\). Isolation of variables is crucial because it means you've arranged the equation to show the variable equals something on its own.
After rewriting our equation as \[t-8=6\], our task is to 'move' the constant in the equation so that \(t\) can stand alone. This involves doing the opposite operation of what was done to \(t\). Here, \(-8\) is substracted from \(t\), so we need to add \(8\) to both sides:

• Add 8 to both sides: \[t = 6 + 8\]

This step effectively 'moves' the \(-8\) away from \(t\) and isolates it.
Isolating variables is akin to finding the needle in the haystack in problem-solving.It ensures that we are solving correctly by putting the variable in its simplest form, ready to find the precise answer.

Step-by-step Problem Solving

This approach refers to tackling a problem by breaking it into smaller achievable steps. It's like following a recipe; each step means you're closer to a delicious dish, or in this case, the solution!
In our example, we begin by simplifying the equation in manageable chunks:

• Simplify the innermost expression \([t-(4+t)]\) into \(-4\).
• Substitute the simplified expression back into the entire equation, reducing it to \[t-8=6\].
• Isolate \(t\) by adding 8 to both sides of the equation.
• Finally, solve for \(t\) by calculating the right-hand side, leading to \(t = 14\).

Each of these steps is a mini-goal; achieving each one ensures that we're on track and consistently moving towards the final solution.
Breaking the problem down in this way prevents feeling overwhelmed by complexity and helps maintain focus. It provides a clear and structured pathway to solve any algebra problem efficiently and accurately.