Problem 37 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$-\frac{3}{5}=-\frac{3}{2}+s$$

Step-by-Step Solution

Answer

The solution to the equation is \(s=\frac{3}{10}\).

Step 1: Isolate \(s\)

First, we need to isolate \(s\) on one side of the equation. We can do this by adding \(\frac{3}{2}\) to both sides of the given equation \(-\frac{3}{5}=-\frac{3}{2}+s\). After doing this, our equation becomes \(-\frac{3}{5}+\frac{3}{2}=s. Now, we simplify the left-side of the equation \(s=\frac{3}{10}\).

Step 2: Check Proposed Solution

We are asked to check our proposed solution. To do this, we substitute \(s=\frac{3}{10}\) into the original equation \(-\frac{3}{5}=-\frac{3}{2}+s\). This gives us \(-\frac{3}{5}=-\frac{3}{2}+\frac{3}{10}\), which simplifies to \(-\frac{3}{5}=-\frac{3}{5}\), confirming that our solution \(s=\frac{3}{10}\) is correct.

Key Concepts

Solving Equations

Solving equations is the fundamental process in algebra that involves finding the value of an unknown variable that makes the equation true. In our example, the equation given is \(-\frac{3}{5}=-\frac{3}{2}+s\).

To solve such equations effectively, we often start by identifying what operation will help us isolate the variable. Here, the goal is to solve for \(s\), which means getting \(s\) by itself on one side of the equation. When solving this equation, we apply the addition property of equality.

This property tells us that if we add the same number or expression to both sides of an equation, the two sides remain equal. This is a crucial step in maintaining the balance of the equation.

• Isolate the variable by using the addition property.
• Perform the same operation on both sides to ensure equality is maintained.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the variable, making them easier to solve. In this particular example, we started with the equation \(-\frac{3}{5}=-\frac{3}{2}+s\).

To manipulate the equation algebraically, we added \(\frac{3}{2}\) to both sides. This step was essential in moving the term involving \(s\) to one side of the equation. Our equation transformed to \(-\frac{3}{5}+\frac{3}{2}=s\). Simplifying further, involved finding a common denominator.

This means converting fractions to have the same base in order to perform the addition accurately:

• Identify the least common denominator.
• Convert fractions accordingly, then perform the necessary operations.

By using these steps, we arrived at \(s=\frac{3}{10}\). The algebraic manipulation made the problem straightforward and solved it efficiently.

Proposed Solutions

Once we have a proposed solution to an equation, it's important to verify its correctness. This step ensures that our solution satisfies the original equation. In our example, the proposed solution was \(s=\frac{3}{10}\).

To check this, we substitute \(s=\frac{3}{10}\) back into the original equation \(-\frac{3}{5}=-\frac{3}{2}+s\):

• Insert \(\frac{3}{10}\) where \(s\) was in the equation.
• Simplify the equation to see if both sides equal each other.

After substitution, simplifying gives \(-\frac{3}{5}=-\frac{3}{5}\), confirming the solution is correct. Always check your solutions to ensure accuracy, reinforcing your understanding and confidence in algebraic problem-solving.