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The distributive property is an important fundamental concept in algebra. It states that multiplying a single term by each term inside a parenthesis leads to the same result as multiplying each term separately and then adding the products. This property helps to simplify expressions and solve equations efficiently.
To apply the distributive property, simply multiply each term inside the parentheses by the term outside. For example, if you have the expression \( a(b + c) \), it becomes \( ab + ac \). In the equation given in the exercise, we see the application as follows:

• For the left side of the equation: \( 4(y-3) \) distributes to \( 4y - 12 \).
• For the right side of the equation: \( 3(y-4) \) distributes to \( 3y - 12 \).

Breaking down complex expressions using the distributive property helps in achieving a clearer and simplified equation, paving the way for easier solution finding.

Another key aspect of solving linear equations involves identifying and combining like terms. Like terms are terms that contain the same variable raised to the same power.
In our given example, once the distributive property has been used, we obtained expressions that include terms like \( 4y \) and \( -y \) on the left side of the equation.
Combining these like terms together simplifies the expressions. Here's what it looks like:

• Combine \( 4y \) and \( -y \) to get \( 3y \) on the left side.

This simplification through combining like terms helps to streamline the equation, reducing potential mistakes and confusion. Recognizing and combining like terms is a key skill in algebra that you'll use often as you work with equations.

Understanding what is meant by real numbers is crucial when solving equations. Real numbers include all numbers that can be found on the number line. This includes:

• Whole numbers like 0, 1, 2, etc.
• Integers such as -3, 4, and 7.
• Fractions and decimals like 1/2 and 3.14.
• Irrational numbers which cannot be expressed as simple fractions, such as \( \sqrt{2} \) and \( \pi \).

In the given equation, the solution for \( y \) being any real number indicates that there is no single answer to the equation. Instead, any real number substituted for \( y \) will satisfy the equation. This occurrence happens because both sides of the equation turn out to be identical after simplification, showing infinite solutions. Understanding real numbers and their properties is fundamental in determining solutions for equations across various contexts.