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The distributive property is a powerful and essential tool in algebra that lets us multiply a single term by each term inside a set of parentheses. But it's not just limited to numbers; it's crucial for handling negative signs as well. In our expression \[(21y - 4.6) - (36y - 8.2),\]we need to apply this property to simplify it correctly. The negative sign before the parentheses affects everything inside the \((36y - 8.2)\) chunk. This is like saying we have a hidden "\(-1\)" that needs multiplying with each term inside the parentheses.
This affects the given terms in two ways:

• \(-1 \times 36y \rightarrow -36y\)
• \(-1 \times -8.2 \rightarrow +8.2\)

Therefore, distributing the negative sign changes our expression to:\[21y - 4.6 - 36y + 8.2.\]
Remember, being meticulous with signs, especially negatives, is key. This helps avoid easy mistakes in simplifying expressions.

Combining like terms is about simplifying expressions by bringing together terms that have the same variables raised to the same power. Think of it like grouping similar items together in algebra to tidy things up.
In the expression \[21y - 4.6 - 36y + 8.2,\]you'll notice there are two "like terms" involving "\(y\):" these are \(21y\)and \(-36y\).
Like terms have identical variables. So you can combine their coefficients, just as you would combine numbers of apples or cars:

• The calculation for these terms is \(21y - 36y = -15y\).

The constants \(-4.6\) and \(+8.2\)are also like terms. Combining these gives \(-4.6 + 8.2 = 3.6\).
After combining like terms, the expression simplifies to \(-15y + 3.6\).
Combining like terms refines the expression into a simpler form, making problems easier to solve or understand.

Recognizing the difference between constants and coefficients is a fundamental step in algebra. Each term in an algebraic expression consists of numbers and sometimes variables.
In \(-15y + 3.6,\)\(-15\)is the coefficient. Coefficients are numbers that multiply a variable, showing how many of that variable there are. Here, \(-15\)tells us how many \(y's\) are being added or subtracted.
On the other hand, \(3.6\)is a constant. Constants are fixed numbers without variables; they remain the same regardless of any changes to the variables.
In algebraic terms:

• **Coefficients**: Multiply variables (e.g., \(-15\)in \(-15y\)).
• **Constants**: Standalone numbers (e.g., \(3.6\)).

By identifying constants and coefficients, you understand the role each number plays in an expression, aiding in simplification and evaluation.