Problem 65 - Prealgebra

Question

Solve each equation. Check your solution. $$192=4 t$$

Step-by-Step Solution

Answer

The solution is \(t = 48\).

Step 1: Understand the Equation

The given equation is \(192 = 4t\). This is a linear equation where the goal is to solve for \(t\), which is the unknown variable.

Step 2: Isolate the Variable

To find \(t\), divide both sides of the equation by 4, which is the coefficient of \(t\): \[\frac{192}{4} = \frac{4t}{4}\]This simplifies to:\[t = 48\].

Step 3: Verify the Solution

Now substitute \(t = 48\) back into the original equation to verify the solution:\[192 = 4 \times 48\]Perform the multiplication: \[4 \times 48 = 192\]Since the left side equals the right side of the original equation, the solution \(t = 48\) is correct.

Key Concepts

Solving Equations

When we first encounter an equation like \( 192 = 4t \), we are dealing with a linear equation. The ultimate aim is to solve for the unknown variable, which in this case is \( t \). Solving equations involves finding the value of the variable that makes the equation true. This process often requires a few basic algebraic steps that help simplify the equation.

• Start by identifying what needs to be found; here, it is the value of \( t \).
• Look at the operations surrounding the variable and think about how to undo them.
• In our example, \( t \) is multiplied by 4, so we aim to cancel out this multiplication.

By carefully performing these steps, we transform the problem from an equation to an expression where the value of the unknown is apparent.

Variable Isolation

Variable isolation is a critical step in solving equations. This involves manipulating the equation to have the variable alone on one side. For the equation \( 192 = 4t \), we need to "isolate" \( t \). To do this, we think in terms of inverse operations.

**Applying Inverse Operations**
In our example, the variable \( t \) is multiplied by 4. The inverse operation of multiplication is division. To isolate \( t \), divide both sides of the equation by 4:
\[\frac{192}{4} = \frac{4t}{4}\]
This simplifies to:
\[t = 48\]
By dividing both sides by 4, \( t \) is now by itself, perfectly isolated.

• *Always perform the same operation on both sides to maintain balance.*
• *Check if any further simplification is possible.*

Equation Verification

Once you've found a solution, it's always wise to verify it, ensuring no mistakes were made during calculations. Verification involves substituting your solution back into the original equation to check if it holds true.

For our exercise, substitute \( t = 48 \) back into the original equation \( 192 = 4t \):
\[192 = 4 \times 48\]
Perform the multiplication:
\[4 \times 48 = 192\]
Since both sides of the equation match, our solution \( t = 48 \) is verified as correct.

• *Substitution helps confirm the solution is accurate.*
• *Equations should balance to prove correctness.*

Verification is an excellent habit that provides assurance that the calculations were performed correctly and the solution is reliable.