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The distributive property is a fundamental concept in algebra that allows us to break down and simplify expressions. This property lets us distribute a single term over terms inside parentheses. For instance, if you have an expression like \( a(b + c) \), you apply the distributive property by multiplying \( a \) with both \( b \) and \( c \) separately, resulting in \( ab + ac \). In our given problem \( 27.5(5.17 - 1.44x) = 73.4 \), we apply the distributive property to spread \( 27.5 \) across the terms inside the parentheses.
You would calculate:

• \( 27.5 \times 5.17 \)
• \( 27.5 \times -1.44x \)

After these calculations, we get \( 142.075 \) and \(-39.6x \) respectively, which transforms our equation to \( 142.075 - 39.6x = 73.4 \). Applying this property is crucial for opening up the expression and making it manageable for further operations.

Isolating the variable is a significant step in solving equations. The goal is to get the variable by itself on one side of the equation. This step usually involves several operations such as addition, subtraction, multiplication, or division.
In our equation \( 142.075 - 39.6x = 73.4 \), we want all terms including \( x \) to be on one side. To achieve isolation, we first eliminate the constant term \( 142.075 \) by subtracting it from both sides of the equation:

• This gives us \( -39.6x = 73.4 - 142.075 \).
• Which simplifies to \( -39.6x = -68.675 \).

Achieving isolation ensures that \( x \) is the focus, thereby making it easier to solve for its value in the final step.

Arithmetic operations are the backbone of algebra and involve the basic operations of addition, subtraction, multiplication, and division. These operations are used to simplify and solve equations.
For instance, after isolating the \( x \) term as \( -39.6x = -68.675 \), our task is to solve for \( x \). We do this by implementing division. We divide both sides of the equation by \(-39.6\) to solve for \( x \):
\[ x = \frac{-68.675}{-39.6} \]
Calculating the above expression yields \( x \approx 1.734 \).
Using arithmetic operations correctly is essential for determining the exact value of the variable. It simplifies complex expressions and is a key skill in algebra.