Problem 8 - Algebra for College Students

Question

Solve each equation. \(s=15+0.4 s\)

Step-by-Step Solution

Answer

The solution is \(s = 25\).

Step 1: Move all terms containing 's' to one side

Start by isolating the variable terms on one side of the equation. Here, subtract \(0.4s\) from both sides of the equation to collect \(s\) terms on one side:\[ s - 0.4s = 15 \]

Step 2: Simplify the equation

Combine the 's' terms on the left side. \(s - 0.4s\) is equal to \(0.6s\):\[ 0.6s = 15 \]

Step 3: Solve for 's'

Divide both sides of the equation by \(0.6\) to solve for \(s\):\[ s = \frac{15}{0.6} \]

Step 4: Calculate the division

Perform the division on the right to find the value of \(s\):\[ s = 25 \]

Key Concepts

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. In this case, the equation we're working with is \( s = 15 + 0.4s \).
**The steps involve:**

• Rearranging the equation so that all terms involving the variable are on one side of the equation.
• Isolating the variable enables solving for its value.

Begin by moving variable terms to one side. In this instance, subtract \( 0.4s \) from both sides so that all 's' terms are together. This key step reduces the complexity of the equation, allowing us to isolate 's' on one side.
This approach facilitates focusing solely on the variable, making it easier to understand and solve the equation.

Algebraic Manipulation

Algebraic manipulation is all about rearranging and simplifying equations using different algebraic rules and properties. This process allows equations to be reshaped and transformed into more manageable forms.
In our specific example, after moving like terms together, we perform the operation:\( s - 0.4s \).
**Key Steps in Algebraic Manipulation:**

• Combine like terms: Terms that have the same variable (in this case 's') can be combined by adding or subtracting their coefficients.
• Use arithmetic operations: Subtraction was used here to consolidate terms into \( 0.6s \).

Combining like terms is crucial and makes the equation simpler and neat. It reduces potential errors and makes the solving process straightforward.

Equation Simplification

Equation simplification involves transforming the equation into its simplest form, making it much easier to solve. Once algebraic manipulation is complete, the equation \(0.6s = 15\) must be simplified further.
In this step, divide both sides of the equation by the coefficient of the variable. Here, you divide by \(0.6\) to solve for 's':

• This step isolates 's', providing its exact value once the arithmetic operation is done.
• Performing division aids in reducing the equation to its solution.

After applying this division, you arrive at \( s = 25 \), the solution to our original equation.
Proper simplification ensures that the solution is accurate and easily verifiable, enhancing understanding and accuracy in solving similar problems.