Problem 102 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$6(w+2)=4 w-10 ;-9$$

Step-by-Step Solution

Answer

-9 is not a solution of the equation \(6(w+2)=4 w-10\).

Step 1: Substitute -9 for w

Replace \(w\) in the equation with -9: \(6(-9+2)=4(-9)-10\)

Step 2: Simplify the equation

First calculate within the brackets: \(6*(-7)= 4*(-9)-10\), which simplifies to \(-42= -36-10\)

Step 3: Simplify further

Calculate the right side of the equation to determine its value: \(-42= -46\)

Step 4: Compare the two sides of the equation

Both sides of the equation are not equal (-42 ≠ -46). Thus, -9 is not a solution to the equation.

Key Concepts

Solution Verification

Solution verification is an essential process in algebra. It helps determine whether a proposed value is indeed a solution to an equation.
The goal is simple: check if both sides of an equation are equal when the variable is substituted with the given number. This process ensures that the proposed solution satisfies the original equation. In the exercise, we were tasked to verify if -9 is a solution for the equation \(6(w+2) = 4w - 10\). To verify a solution:

• Substitute the proposed value into the equation.
• Simplify both sides independently.
• Compare the results.

If both sides are equal, the proposed value is a solution. If they are not, as shown in this exercise, the value does not satisfy the equation.

Substitution Method

The substitution method is an effective way to solve and verify solutions in equations. It involves replacing the variable with a given number to test if the equation holds true.In our case, we replaced \(w\) with -9 in the equation \(6(w+2)= 4w -10\). Here's a step-by-step look at how the substitution method works:

• Replace the variable in the equation with the number provided or found.
• Evaluate the expression to find a numerical result for both sides of the equation.
• Substitution is complete once the variable is successfully replaced with a consistent number.

This method makes it clear whether the proposed value is a solution by showing if it results in a true equation. In the given exercise, substituting -9 resulted in unequal sides, proving it wasn't a solution.

Simplifying Equations

Simplifying equations is a crucial step to make calculations easier and find solutions quickly. Once a number has been substituted into the equation, each side of the equation needs to be simplified to reveal if they balance.For the exercise, we first substituting \(w\) with -9 in the equation gave:\[6(-9+2) = 4(-9) - 10\]This required us to:

• Simplify expressions within parentheses.
• Perform arithmetic operations according to order (parentheses, multiplication, subtraction).
• Compare the simplified forms of both sides.

By simplifying both expressions fully, we found:\(-42 eq -46\), showing the substitution didn't satisfy the equation. Understanding and practicing simplification ensures not only correctness but also builds foundational skills for tackling more complex algebraic problems.