Problem 7 - Algebra for College Students

Question

Solve each equation. \(s=9+0.25 s\)

Step-by-Step Solution

Answer

\(s = 12\)

Step 1: Isolate the variable

To isolate the variable \(s\), we need to get all the \(s\) terms on one side of the equation. Subtract \(0.25s\) from both sides to move the \(0.25s\) term to the left side. This gives us: \(s - 0.25s = 9\).

Step 2: Factor out the variable

Simplify the left-hand side by combining like terms. \(s - 0.25s\) becomes \(0.75s\). The equation now reads: \(0.75s = 9\).

Step 3: Solve for the variable

Divide both sides of the equation by \(0.75\) to solve for \(s\). This gives \(s = \frac{9}{0.75}\).

Step 4: Simplify the division

Perform the division \(\frac{9}{0.75}\), which is \(12\). So, \(s = 12\).

Key Concepts

Isolate the Variable

When solving linear equations, one of the first steps is to isolate the variable you need to solve for. This means getting the unknown, or variable, by itself on one side of the equation.

• In the given problem, the variable is \(s\), and we have the equation: \(s = 9 + 0.25s\).
• The aim is to have all terms containing \(s\) on one side and all the constant terms (numbers) on the other side.

To start isolating \(s\), subtract \(0.25s\) from both sides:\[ s - 0.25s = 9\]This step moves all \(s\) terms to the left side of the equation. It’s important to remember we can subtract or add the same amount from both sides without changing the equality, keeping the equation balanced. This crucial step sets up the equation so we can eventually solve for the variable.

Combining Like Terms

After isolating the variable on one side, the next step is to simplify the equation. This often involves combining like terms. Like terms are terms that contain the same variable raised to the same power.

• The terms \(s\) and \(0.25s\) are like terms because they both contain the variable \(s\).
• By combining them, we can simplify the equation even further: \(s - 0.25s = 0.75s\).

This simplification is a key part of making the problem manageable. Simplifying helps reduce the equation to its simplest form, which makes solving it straightforward and less prone to mistakes. This idea of combining like terms is foundational in algebra, ensuring that equations can be efficiently solved.

Division in Algebra

Once you've combined like terms, the equation often needs one more step to solve for the variable, which is division.

• In our example, the simplified equation is \(0.75s = 9\).
• To find \(s\), divide both sides of the equation by \(0.75\).

This division allows us to isolate \(s\) completely:\[ s = \frac{9}{0.75}\]Dividing both sides by the coefficient of \(s\) (which is \(0.75\) in this case) is a common way to solve for variables in algebraic equations. After performing the division, you find that \(s = 12\). Division in algebra is key for solving equations because it allows us to directly solve for the variable by simplifying the coefficient to 1, effectively leaving the variable by itself.