Problem 14 - Elementary Algebra

Question

Solve each of the equations. $$s=24+0.6 s$$

Step-by-Step Solution

Answer

The solution is \( s = 60 \).

Step 1: Move variable terms to one side

We start by isolating the variable term on one side of the equation. The given equation is \( s = 24 + 0.6s \). Subtract \( 0.6s \) from both sides to get the variable \( s \) on one side: \[ s - 0.6s = 24 \]

Step 2: Combine like terms

In the equation \( s - 0.6s = 24 \), we combine the like terms on the left side. \[ (1 - 0.6)s = 24 \] So we have: \[ 0.4s = 24 \]

Step 3: Solve for the variable

To solve for \( s \), divide both sides of the equation \( 0.4s = 24 \) by \( 0.4 \). This isolates \( s \): \[ s = \frac{24}{0.4} \] Calculate \( \frac{24}{0.4} \) to find the value of \( s \): \[ s = 60 \]

Step 4: Verification (Optional)

To verify our solution, substitute \( s = 60 \) back into the original equation: \[ s = 24 + 0.6s \] Substitute \( s = 60 \): \[ 60 = 24 + 0.6(60) \] Calculate \( 0.6(60) = 36 \), then \[ 60 = 24 + 36 \] Since both sides equal 60, the solution is verified.

Key Concepts

Isolation of Variables

In the realm of solving linear equations, isolating the variable is often the first step you need to master. The ultimate goal is to have the variable you're solving for appear by itself on one side of the equation. In our exercise, we have:- Initial equation: \( s = 24 + 0.6s \).To isolate the variable \( s \), we first identify any terms on the other side of the equation that also contain the variable. By subtracting \( 0.6s \) from both sides, we bring all variable terms together to one side:\[ s - 0.6s = 24 \].This critical step of moving variable terms is essential and sets the stage for solving the equation effectively. Bringing all terms involving the variable onto one side consolidates the equation, allowing for more straightforward manipulation in subsequent steps.

Verification of Solutions

Verification is an optional but valuable step in solving equations. This procedure ensures the solution you derived is accurate and satisfies the original equation. After solving for \( s \) in our exercise, we got a solution of \( s = 60 \).To verify:- Substitute \( s = 60 \) back into the original equation: \( s = 24 + 0.6s \).- Replace \( s \) with 60: \( 60 = 24 + 0.6(60) \).- Perform the operations: \( 0.6 \times 60 = 36 \), leading to \( 60 = 24 + 36 \).When both sides of the equation are equal, the solution is confirmed as correct. Verification is a small detour that can prevent mistakes and boost confidence in your solution. This additional check is particularly beneficial for complex or longer equations, where errors can sneak in unnoticed.

Combining Like Terms

Combining like terms is a vital skill that simplifies and reduces expressions to their simplest form, making the equation easier to solve. Like terms are terms that contain the same variable to the same power. In the exercise, the equation:\[ s - 0.6s = 24 \]requires combining the variable terms on the left side:- \( s \) is equivalent to \( 1s \), thus \( 1s - 0.6s \) gives us \((1 - 0.6)s = 24 \).Simplifying this further by performing the arithmetic on the coefficients,- You are left with \( 0.4s = 24 \).By combining like terms, terms are simplified, revealing a cleaner path to solving for the variable. Practicing this technique strengthens algebraic manipulation skills, crucial in handling more complicated equations effortlessly.