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The distributive property is a useful algebraic principle that helps break down expressions into simpler components. It allows for the multiplication of terms within parentheses by a factor outside. In our original exercise, you see the equation is given as:

• Left side: \(8(t+5)\)
• Right side: \(4(2t+10)\)

To apply the distributive property, multiply the factor outside the parentheses by each term inside. Here's how it unfolds:

• Left side: Distribute 8 to both \(t\) and 5: \(8 \cdot t + 8 \cdot 5 = 8t + 40\)
• Right side: Distribute 4 to both \(2t\) and 10: \(4 \cdot 2t + 4 \cdot 10 = 8t + 40\)

After distributing, both sides of the equation become \(8t + 40\). It's important to understand this step because it simplifies complex expressions, making it easier to solve equations.

Properties of equality are essential tools that allow you to maintain balanced equations while manipulating their components. These properties include addition, subtraction, multiplication, and division across both sides of an equation by the same non-zero number. They ensure that the equality holds.In our original equation: \(8t + 40 = 8t + 40\)We used the subtraction property of equality to subtract \(8t + 40\) from both sides. This step aims to simplify the equation, ultimately reducing it to:\[0 = 0\]This subtraction keeps the equation true and equal on both sides. It demonstrates the integrity of the equality throughout solving process. Always remember that using any of these properties correctly won't disrupt the balance of your equation. Understanding this concept is crucial to solving algebraic expressions.

The idea of infinite solutions in linear equations is fascinating because it suggests that an equation holds true for all variable values. After applying the distributive property and using properties of equality, we have:\[0=0\]This final form is an identity, meaning it is always true. No matter what value 't' is substituted with, both sides remain equal. Therefore, the original equation \(8(t+5)=4(2t+10)\) is valid for all real numbers. Unlike equations with a single solution, or no solution at all, identity equations symbolize infinite possibilities. When faced with a result like \(0=0\), it's a hint that any number will satisfy the original equation, offering a broad array of solutions.