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When we talk about simplifying expressions in algebra, we mean making them shorter or easier to work with while keeping the same value. This process helps us to understand the mathematical expression better and can make future calculations easier and quicker to perform.

For instance, in the original problem, the expression is given as \[7(y+4) - 10y\].
We simplify this expression to make it more concise.

• First, we start by using the distributive property to get rid of the parentheses.
• Next, we look to combine all like terms, which will consolidate our expression.

By simplifying expressions, not only do we make it easier to interpret the mathematical sentence, but we also lay a better foundation for solving algebraic equations.

The distributive property is a fundamental principle of algebra which allows you to multiply a sum by multiplying each addend separately and then adding the products. Crucially, it helps in transforming expressions, especially when parentheses are involved.

In symbolic terms, the distributive property is written as:\[a(b + c) = ab + ac\].
It's invaluable when an expression has parentheses as it lets us eliminate them.

Looking at the example \[7(y+4)\],we apply the distributive property by multiplying 7 with both \(y\) and 4.

• This results in \(7y + 28\).
• Notice how the property helps in breaking down the expression inside the parentheses.

Using the distributive property is essential for simplifying expressions as it reveals the terms that we might be able to further simplify by combining like terms.

Like terms in algebra refer to terms that have the same variable raised to the same power. Only the coefficients (numbers in front of the variables) may be different. Identifying and combining like terms is a vital step in simplifying expressions.

For our expression, once the distributive property is applied and the parentheses are removed, we have:\[7y + 28 - 10y\].Here, \(7y\) and \(-10y\) are like terms because both involve the variable \(y\).

Combining them involves:

• Calculating \(7y - 10y\), which equals \(-3y\).
• These computations condense the expression into a simpler form.

Recognizing and combining like terms simplifies expressions and is especially useful when solving equations, as it reduces the complexity of the terms involved.