Problem 41 - 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. $$r+3.7=8$$

Step-by-Step Solution

Answer

The solution to the equation \(r+3.7=8\) is \(r = 4.3\).

Step 1: Recognize the Equation

Notice that the existing equation is \(r+3.7=8\). The aim is to solve for 'r'.

Step 2: Use the Addition Property of Equality

The addition property of equality states that adding the same number to both sides of an equation does not change the equality. Here we would subtract 3.7 from both sides to isolate 'r'. This gives: \(r = 8 - 3.7\)

Step 3: Solve for 'r'

Subtracting 3.7 from 8, we get \(r = 4.3\)

Step 4: Check the Solution

Substitute 'r' in the original equation with 4.3. This should satisfy the equation: \(4.3 + 3.7 = 8\). Thus, the solution is valid

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a fundamental concept in algebra which states that if you perform the same operation to both sides of an equation, the equation remains balanced. In simple terms, it's like keeping a scale balanced by adding or subtracting the same weight from both sides.

For instance, consider the equation presented in the exercise:

• Equation: \( r + 3.7 = 8 \)
• Operation: Subtract 3.7 from both sides.
• New Equation: \( r = 8 - 3.7 \)

By subtracting 3.7 from each side, the original equation maintains its balance, letting us safely solve for \( r \). This method ensures that we don't mistakenly alter the equation's truth, and instead keep it correct and equal.

Isolating Variables

In mathematics, isolating the variable means getting the variable by itself on one side of the equation. This is a crucial step when solving equations because it allows us to find the value of the unknown.

Take the equation from the problem, for example: \( r + 3.7 = 8 \).

• Goal: Get \( r \) alone on one side.
• Action: Subtract 3.7 from both sides.
• Resulting Equation: \( r = 8 - 3.7 \)
• Solution: \( r = 4.3 \)

By performing this operation, we successfully isolated \( r \) on its own, freeing it from any other numbers or terms. Now we have \( r = 4.3 \), which tells us the exact value of \( r \) in the context of the original equation.

Checking Solutions

Checking solutions is an important part of solving equations. This ensures that our solution not only satisfies the mathematical operations we've performed but also fits the original equation precisely.

Here's how you verify it:

• Take your solution: \( r = 4.3 \)
• Substitute \( r \) back into the original equation: \( 4.3 + 3.7 \)
• Perform the addition: \( 4.3 + 3.7 = 8 \)

If the equation holds true after substituting back the value, then you've correctly solved it. In our exercise, substituting \( r = 4.3 \) indeed satisfies \( r + 3.7 = 8 \), confirming that our work is accurate. Always conduct this small check as it strengthens the credibility and correctness of your solution.