Problem 32 - 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. $$-2.7+w=-5.3$$

Step-by-Step Solution

Answer

The solution to the equation is \( w = -2.6 \).

Step 1 - Apply Addition Property of Equality

To solve for 'w' isolate it by adding 2.7 to both sides using the addition property of equality. This gives: \( -2.7 + 2.7 + w = -5.3 + 2.7 \). After simplifying, we get \( w = -5.3 + 2.7 \)=

Step 2 - Simplify the Right Side of the Equation

Perform arithmetic operation on the right side, \( w = -5.3 + 2.7 = -2.6 \)

Step 3 - Check the Proposed Solution

Checking the calculated value for 'w' by substituting it back into the original equation. This gives us: \( -2.7 + (-2.6) = -5.3 \), which simplifies to \( -5.3 = -5.3 \). Seeing that both sides of the equation are equal, the solution is indeed correct.

Key Concepts

Solving Equations

One fundamental skill in algebra is solving equations, which is the process of finding the values of variables that make the equation true. To solve any equation, it's essential to understand what an equation represents: a statement that two expressions are equal. Solving equations often involves reversing operations to isolate the variable. The goal is to have the variable on one side of the equation and its corresponding value on the other.

When we solve equations, we commonly use properties like the addition property of equality, which allows us to add the same number to both sides of the equation without changing its equality. This approach helps in maintaining balance and leads us to the solution. Always remember to check your solutions by substituting the variable back into the original equation; this verifies your work.

Isolation of Variables

Isolation of variables is a technique used to solve equations, where we aim to have the variable alone on one side of the equation. The process typically involves undoing arithmetic operations surrounding the variable. For example, if the variable has a number added to it, we need to subtract that number from both sides of the equation. Conversely, if it's subtracted, we would add that number.

In the given equation \(-2.7 + w = -5.3\), to isolate \(w\), we added 2.7 to both sides because adding cancels out the \(-2.7\) on the left, leaving us with \(w = -5.3 + 2.7\). This technique is applicable for any operations performed on the variable; if it involves multiplication or division, you multiply or divide both sides accordingly.

• Step-by-step isolation
  
• Balance both sides
  
• Re-check to ensure correctness
  

Mastering isolation techniques allows anyone to effectively solve equations.

Arithmetic Operations

Understanding arithmetic operations is crucial when working through algebraic equations. These operations include addition, subtraction, multiplication, and division. Each operation has a role in simplifying equations, especially when using the addition property of equality.

Let's consider the solving process for \(-2.7 + w = -5.3\). We added 2.7 to both sides to isolate \(w\), which resulted in \(w = -5.3 + 2.7\). This requires a solid understanding of negative numbers and addition.The operation leads us to calculate \(-5.3 + 2.7 = -2.6\). Ensuring the correct result here is essential for arriving at the right solution for the variable.

• Careful handling of negative numbers
  
• Using consistent and correct arithmetic rules
  
• Simplification leads to solution confirmation
  

Consequently, being comfortable with arithmetic operations simplifies the process of solving equations.