Problem 52 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$20-7 s=26-8 s$$

Step-by-Step Solution

Answer

The solution for the equation is \(s = 6\)

Step 1: Rearrange the Equation

First, the objective is to isolate 's' on one side of the equation. To do this, add \(8s\) to both sides of the equation to cancel out \(-8s\) on the right: \(20 - 7s + 8s = 26 - 8s + 8s\). This simplifies to \(20 + s = 26\).

Step 2: Solve for 's'

In the second step, to isolate 's' on one side, subtract '20' from both sides of the equation \(20 + s - 20 = 26 - 20\). This simplifies to \(s = 6\).

Step 3: Validate the Solution

Finally, validate the solution by substituting \(s = 6\) into the initial equation. \(20 - 7*6 = 26 - 8*6\), which simplifies to \(-22 = -22\). Since the left side equals the right side, the solution is valid.

Step 4: Conclusion

After validation, the solution to the equation \(20 - 7s = 26 - 8s\) is \(s = 6\).

Key Concepts

Solving Linear Equations

Linear equations look much like a balancing act. Imagine a seesaw in perfect balance, where both sides are equal. The essence of solving linear equations is maintaining this balance while solving for an unknown variable. In this exercise, we tackle the equation: \(20 - 7s = 26 - 8s\). Simple guidelines can make this process fun and stress-free.

The equation has the variable 's' on both sides, and your job is to isolate 's' all by itself on one side. We start by adding \(8s\) to both sides. Why? Because this cancels out the \(-8s\) on the right side. Equations can feel like a game because whatever you do to one side, you must also do to the other. This keeps things balanced.

• After adding \(8s\), the equation becomes: \(20 + s = 26\).
• This transformation simplifies the problem, making the next steps more straightforward.

Solving equations step by step not only breaks down a potentially daunting task but also ensures clarity.

Validating Solutions

Once you've found your solution, it's vital to double-check your work through validation to make sure your solution is correct. This means substituting your solution back into the original equation to see if both sides still equal.

For our exercise, we substitute \(s = 6\) back into the original equation to validate our solution: \(20 - 7 \times 6 = 26 - 8 \times 6\).

Breaking it down:

• On the left side, \(20 - 42\) calculates to \(-22\).
• On the right side, \(26 - 48\) also results in \(-22\).

This critical step confirms that both sides indeed match and proves that the value \(s = 6\) is the correct and valid solution. Validation is like proof—it reassures you that your answer is reliable and correct.

Isolation of Variables

The isolation of variables is a core part of solving linear equations. It involves reshaping the equation until the variable of interest—'s' in our exercise—is alone on one side of the equation.

Think of it as shining a spotlight on 's' so you can see it clearly without any distractions. We achieve this by performing operations that nudge 's' into the spotlight. In this exercise:

• After removing the \(-8s\) from the right side, we focus on simplifying \(20 + s = 26\).
• Subtracting \(20\) from both sides moves everything else away from 's'.

The equation thus simplifies to \(s = 6\).

Isolating variables is like tidying up a room before finding that important item you need—it's about clarity and making everything else fall away until only 's' remains, leading you directly to the solution.