Problem 43 - Elementary and Intermediate Algebra

Question

Solve each formula for the specified variable. $$ T-W=m a \text { for } W $$

Step-by-Step Solution

Answer

The solution for \( W \) is \( W = T - ma \).

Step 1: Understand the Goal

We need to rearrange the equation \( T - W = ma \) so that \( W \) is expressed in terms of the other variables. In other words, we want to solve for \( W \).

Step 2: Isolate the Variable W

To solve for \( W \), we need to move \( W \) to one side of the equation by itself. We can do this by subtracting \( T \) from both sides of the equation. This removes \( T \) from the left-hand side:\[ -W = ma - T \]

Step 3: Solve the Equation for W

Next, we need to get \( W \) alone, which involves removing the negative sign. We can multiply both sides of the equation by -1:\[ W = T - ma \]

Step 4: Verify the Solution

To verify, substitute \( W = T - ma \) back into the original equation to ensure it holds true:Original: \( T - (T - ma) = ma \)Simplifying: \( ma = ma \)Both sides of the equation are balanced, confirming our solution is correct.

Key Concepts

Solving for a Variable

Solving for a variable means finding its value in terms of the other elements in the equation. In our exercise, we aimed to find the value of the variable \( W \). This requires the isolation of \( W \), so it stands alone on one side of the equation.
Understanding the equation \( T - W = ma \), our primary goal was to express \( W \) on its own:

• Our target is to rearrange so that \( W = \text{{something in terms of }} T, m, \text{ and } a \).
• Isolate \( W \) by reversing operations that affect it.

This process of solving for \( W \) required moving other terms, which we tackled by subtracting \( T \) and switching the sign to expose \( W \). Once \( W \) is isolated, we reach a clearer understanding of its relationship with the other variables involved.

Rearranging Equations

Rearranging equations is a key skill in algebra that allows one to manipulate the structure of an equation to suit solving needs. This involves moving terms around while maintaining the equation's balance. The goal is to make the desired variable the subject of the equation.
In the example \( T - W = ma \), rearranging the equation required a few steps:

• Subtract \( T \) from both sides to remove it from the left side. This left us with \( -W = ma - T \).
• Change the sign by multiplication to transition \( -W \) into \( W \), giving us \( W = T - ma \).

These steps showcase the purposeful manipulation of equation elements to focus on simplifying the expression in terms of one variable. Rearranging equations effectively is vital for problem-solving across different mathematical and scientific contexts.

Verifying Solutions

Verifying solutions is an essential step in algebra to confirm that our manipulation and calculations have led to a correct result. Verification involves substituting the solved value back into the original equation and checking if the equality holds true.
For the solved equation \( W = T - ma \), verification included:

• Substitute \( W \) with \( T - ma \) in the original formula: \( T - (T - ma) = ma \).
• Simplify the equation by performing operations: effectively showing that \( ma = ma \).

The equality on both sides confirms the solution's correctness. This checking process not only validates our answer but reinforces the understanding of variable relationships within the equation. Consistently verifying solutions ensures accuracy and builds confidence in algebraic manipulation.