Problem 52 - Introductory and Intermediate Algebra for College Students 4th

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 \(20-7 s=26-8 s\) is \(s = 6\).

Step 1: Re-arrange equation

Rearrange the equation such that the variable \(s\) is on one side of the equation: \(8s - 7s = 26 - 20 \)

Step 2: Simplify each side

Perform the addition or subtraction of each side to simplify the terms: \(s = 26 - 20 = 6\)

Step 3: Check solution

Substitute \(s = 6\) back into the original equation \(20-7s=26-8s\) to check if both sides are equal after substitution: \(20 - 7*6 = 26 - 8*6 \Rightarrow 20 - 42 = 26 - 48 \Rightarrow -22 = -22 \). It checks out which means the solution \(s = 6\) is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental principle in algebra that allows us to keep an equation balanced as we work to isolate the variable. In essence, when you add or subtract the same number from both sides of an equation, you maintain the equality. This property is the backbone of solving equations like the one given: 20 - 7s = 26 - 8sHere, since we are dealing with subtraction and the variable s on both sides, we can apply the addition property of equality by adding 8s to each side to move all terms involving s to one side and the constants to the other. Music to an algebraist's ears, this maneuver helps us inch closer to isolating the variable and finding the solution.

Remember, being diligent with this property helps prevent errors and ensures that every step in solving the equation is legitimate.

Algebraic Manipulation

Algebraic manipulation is the art of reshaping equations to make them simpler to solve, and it’s like a puzzle where each move sets up the next. The question at hand involves strategically transforming the equation so the variable of interest, s, is alone on one side. Let’s break down the process applied to our problem:

1. Combine like terms: 8s - 7s on one side simplifies to s.
2. Perform arithmetic operations:Subtract 20 from both sides, turning 26 - 20 into 6, which reveals the solution s = 6.

This kind of manipulation is the bread and butter of solving linear equations. By understanding each step, we chip away at the complexity until we’re left with a clear answer.

Tips to excel at algebraic manipulation include practicing the distributive property, combining like terms, and keeping track of positive and negative signs—each can turn a hairy problem into a smoothly combed solution.

Solution Checking

Finding an answer is satisfying, but verifying it’s the correct one is where you truly seal the deal. Solution checking is the safety net that catches the mistakes we might make in our calculations. To check the proposed solution of s = 6, we substitute it back into the original equation to see if both sides match:20 - 7s = 26 - 8sReplacing s with 6 gives us:20 - 7(6) = 26 - 8(6)After simplification, we get:-22 = -22Voilà! Both sides are equal, confirming our solution is as snug as a bug in a rug. Regular practice of this essential step can help squash doubts and boost your confidence in your solutions. Always remember, a solution isn’t truly a solution until it's checked and verified.