Problem 19 - Fundamentals of Mathematics

Question

Verify that each given value is a solution to the given equation. $$8 b+6=6-5 b, b=0$$

Step-by-Step Solution

Answer

Yes, \(b=0\) is a solution to the equation.

Step 1: Substitute the given value into the equation

Replace the variable \(b\) with the given value, \(b=0\), in the equation \(8b + 6 = 6 - 5b\). This gives us \(8(0) + 6 = 6 - 5(0)\).

Step 2: Simplify each side of the equation

Calculate each side of the equation separately. On the left side: \(8(0) + 6 = 0 + 6 = 6\). On the right side: \(6 - 5(0) = 6 - 0 = 6\).

Step 3: Verify equality of both sides

Compare the results from the left side and the right side of the equation: \(6 = 6\). Since both sides are equal, the equation holds true.

Key Concepts

Substitution Method

The substitution method is a technique used in algebra to replace a variable with a given value in an equation. This method is particularly useful when verifying solutions to equations or when solving systems of equations. The primary steps in using this method include:

• Identify the variable in the equation that needs to be replaced.
• Replace that variable with the provided numerical value. This is done to see if the substitution results in a valid equation.

Let's put this into practice using the equation from the exercise: Given the equation\(8b + 6 = 6 - 5b\) and the value \(b = 0\): start by substituting \(b\) with \(0\). This substitution changes the equation to \(8(0) + 6 = 6 - 5(0)\). By completing this substitution, we prepare the equation for the next critical step, which is simplifying both sides.

Simplifying Equations

Simplifying equations is a crucial skill in algebra that involves reducing expressions to their most basic form. This often includes performing arithmetic operations and combining like terms. Simplification helps in assessing whether both sides of the equation are equal, a necessary check to verify solutions.In our exercise, after the substitution step, we have the equation: \(8(0) + 6 = 6 - 5(0)\).Here's how you simplify both sides:

• Left side: Calculate \(8(0) + 6\). This is simplified to \(0 + 6\), which equals \(6\).
• Right side: Calculate \(6 - 5(0)\). This simplifies to \(6 - 0\), which also equals \(6\).

By simplifying both sides, the expressions become easier to directly compare. Each side simplifying to \(6\) indicates a preliminary confirmation of a solution, but it is essential to verify this equality accurately.

Verifying Solutions

Verifying solutions is the final step in confirming that a given value satisfies an equation. In algebra, it is critical to ensure that both sides of the equation are indeed equal after substituting and simplifying.For the exercise, once the equation is simplified on both sides, we compare them to check for equality.Steps involved in verifying:

• Compare the computed left side of the equation to the right side's result after simplification.
• If they are equal, as in our exercise where \(6 = 6\), this confirms that the substitution was correct, and \(b = 0\) is indeed a solution of the equation.
• If they are not equal, the value does not solve the equation, indicating an error in substitution, simplification, or given values.

Therefore, verification is the ultimate step to cross-verify the integrity of the solution process, ensuring that all calculations hold up and the initial solution is correct.