Question

$$(-6 x-z)-\\{3 y+[7 x-(3 z+8 y+x)]\\}$$

Short answer

The simplified expression is \( -12x + 2z + 5y \).

Step-by-step solution

Expand the Inner Brackets

Start by simplifying the expression inside the innermost brackets. Apply the distributive property of multiplication over subtraction to the term \(7 x - (3 z + 8 y + x)\).

Simplify the Innermost Expression

Simplify the inner brackets by distributing the negative sign through the parentheses: \(7 x - 3 z - 8 y - x\). Combine like terms, if any, within the innermost brackets.

Distribute the Outer Brackets

After simplifying the terms inside the innermost brackets, distribute the negative sign outside the curly brackets across the terms within the curly brackets: \(-3 y - (7 x - 3 z - 8 y - x)\).

Simplify the Entire Expression

Combine all like terms to simplify the entire expression. This includes \( -6 x - z - 3 y - 7 x + 3 z + 8 y + x\).

Key concepts

Understanding the Distributive Property

The distributive property is a cornerstone in simplifying algebraic expressions. It refers to the distribution of a value over a sum or difference within parentheses. For example, when we see an expression like \(a(b + c)\), we apply the distributive property by multiplying \(a\) with both \(b\) and \(c\), ending up with \(ab + ac\).

But what happens when subtraction or negative signs are involved? Let's look at our original problem when we encounter \(7x - (3z + 8y + x)\). The negative sign before the parentheses acts like a -1 multiplier, meaning we need to distribute it across each term inside. So, \(-1\) times \(3z\), \(8y\), and \(x\) individually lead to \(7x - 3z - 8y - x\), which demonstrates the application of the distributive property under subtraction.

Combining Like Terms

After the distributive property has been applied, combining like terms is the next step towards simplifying an algebraic expression. Like terms are terms that have the same variables raised to the same power; they can be added or subtracted from one another.

In our problem, once the expression is expanded, we identify like terms and add or subtract them. It's similar to organizing fruits; apples with apples and bananas with bananas, never mixing apples and bananas together. So in the case of \( -6x - z - 3y - 7x + 3z + 8y + x\), we group the terms with \(x's\), \(y's\), and \(z's\) separately, resulting in a more simplified expression after performing the arithmetic for each group.

The Art of Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying an algebraic expression into a more manageable or preferred form. This involves using a combination of the distributive property, combining like terms, and other algebraic properties such as the associative and commutative properties.

In our exercise, we use algebraic manipulation to simplify the complex expression \( (-6 x-z) -\{3 y+[7 x-(3 z+8 y+x)]\} \). The entire process involves distributing negatives across brackets, rearranging terms, and combining like terms. This manipulation leads to the understanding that algebra is more than just numbers and letters; it's a language of its own, with a set of rules that allow for systematic simplification, making complex-looking expressions into something much simpler and comprehensible.