Problem 34 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$2 s-13=28 s$$

Step-by-Step Solution

Answer

The solution to the equation \(2s - 13 = 28s\) is \(s = -1/2\).

Step 1: Rearrange the equation

First step is to isolate the term with the variable on one side of the equation. Move \(2s\) from the left side of equation to the right to collect all terms involving \(s\) at one side. This is done by subtracting \(2s\) from both sides of the equation, leading to the equation \(-13 = 28s - 2s\).

Step 2: Simplify the right side of the equation

On the right side, subtract \(2s\) from \(28s\) to simplify the equation, leading to the equation \(-13 = 26s\).

Step 3: Solve for the variable \(s\)

Now, to solve for \(s\), divide both sides of the equation by the coefficient of \(s\) which is 26, thus, \(s = -13/26\). After simplifying the fraction, the solution is \(s = -1/2\).

Step 4: Checking the solution

Substitute the obtained \(s\) value into the original equation to check if the left side equals the right side. So, \(2*(-1/2) - 13 = 28*(-1/2)\). Simplifying both sides reveal that they are indeed equal: \(-1 - 13 = -14\) and \(-14 = -14\). It confirms that \(s = -1/2\) is the solution to this equation.

Key Concepts

Variable Isolation

When solving linear equations, the initial step is often to isolate the variable, like finding a single piece of a puzzle. Here, you're trying to get the variable onto one side of the equation to make it easier to solve. In our given equation, we have two expressions with the term with the variable, \(2s\) and \(28s\). Moving terms involves applying the opposite operation to eliminate unwanted terms.

• If a term is added, you subtract it from both sides.
• If it's subtracted, you add it back.

By subtracting \(2s\) from both sides, we effectively moved \(2s\) over to the right side along with \(28s\). This leaves us with \(-13\) on the left, a crucial move that simplifies the equation to a single variable term and number on either side, setting the stage for further simplification.

Simplification of Equations

Simplifying an equation might sound complex, but it's quite straightforward. It's essentially about combining like terms to reduce the equation to its simplest form. Once we have isolated the variable to one side, the next step in the process is to simplify. Simplification often involves basic arithmetic like addition, subtraction, multiplication, or division.In the equation we rearranged earlier, \(-13 = 28s - 2s\), we simplify by subtracting \(2s\) from \(28s\), leaving us with \(-13 = 26s\). This step is vital as it reduces clutter and transforms the equation into something we can easily manage. With fewer terms, it's simpler to identify operations necessary to solve for the desired variable.

Checking Solutions

After reaching a solution, checking your work ensures that the variable value satisfies the original equation. This step is a must to confirm that no mistakes were made during variable isolation or simplification. To check your solution, substitute the found value back into the original equation.For our equation, the solution yielded \(s = -1/2\). By replacing \(s\) with \(-1/2\) in the original equation \(2s - 13 = 28s\), we perform the operations:

• On the left side: \(2 \times (-1/2) - 13 = -1 - 13 = -14\)
• On the right side: \(28 \times (-1/2) = -14\)

Comparing both sides, \(-14 = -14\), confirms that our solution is indeed correct. This safeguard step assures us the solution is accurate and that the variable value is valid for the equation.