Problem 99 - Intermediate Algebra

Question

Simplify each expression. $$-2[4(z-9)-6(3 z-7)]-7(2 z-1)$$

Step-by-Step Solution

Answer

The simplified expression is \( 14z - 5 \).

Step 1: Distribute Inside the Brackets

Start by distributing the constants inside the brackets. For the first bracket, distribute the 4 and the -6. Calculate: 1. Distribute 4: \[ 4(z-9) = 4z - 36 \] 2. Distribute -6: \[ -6(3z - 7) = -18z + 42 \] Combine these to simplify inside the brackets: \[ 4z - 36 - 18z + 42 \] This simplifies to: \[ -14z + 6 \]

Step 2: Multiply by the Outer Coefficient

Multiply the result from Step 1 by the -2 in front of the brackets: \[ -2(-14z + 6) \] Distribute -2: \[ 28z - 12 \]

Step 3: Distribute in the Second Expression

Distribute the -7 across the expression inside the brackets: \[ -7(2z - 1) \] Calculate: \[ -14z + 7 \]

Step 4: Combine Like Terms

Combine all the terms obtained from Step 2 and Step 3:1. From Step 2: \[ 28z - 12 \] 2. From Step 3: \[ -14z + 7 \] Combine these: \[ 28z - 12 - 14z + 7 \] Combine like terms: \[ (28z - 14z) + (-12 + 7) = 14z - 5 \]

Step 5: Final Step: Simplified Expression

After combining all like terms, the simplified expression is: \[ 14z - 5 \]

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, allowing us to make complex expressions more manageable by reducing them to their simplest form. The goal here is to simplify the given mathematical statement so that it is easier to understand and solve.
One of the key benefits of simplifying expressions is that it helps eliminate unnecessary terms and makes it easier to solve equations later on. It involves various steps, such as using the distributive property and combining like terms.
You begin by looking for opportunities to apply the distributive property. This means multiplying each term within a parentheses by a factor outside the parentheses. After that, you combine like terms, which are terms that have the same variables raised to the same power. This consolidation reduces the number of terms, simplifying your expression overall.

Distributive Property

The distributive property is a key algebraic rule that allows us to remove parentheses by distributing, or multiplying, an external factor into each term within the parentheses. This can be seen as taking each component individually and applying multiplication with the factor outside.

• This property is written as: \[ a(b + c) = ab + ac \]
• In the problem, we first distribute 4 across \( (z-9) \): \[ 4(z-9) = 4z - 36 \]
• Then, distribute \(-6\) across \( (3z-7) \): \[ -6(3z - 7) = -18z + 42 \]

These distributed terms can then be combined to rewrite the expression without any inner parentheses, allowing you to move forward with further simplification steps.
Often, we apply further distribution after initial distribution, as seen when \(-2(-14z + 6)\) led to \( 28z - 12 \), employing the distributive property through consecutive steps to handle expressions effectively.

Combining Like Terms

Combining like terms is another essential process in algebra, allowing us to simplify expressions by merging terms that have the same variable components. It's about seeing which terms can be "added" together to cut down on complexity, making equations cleaner and easier to solve.

• Like terms have the same variable part, for example, \( 2z \) and \( 3z \) are like terms.
• To combine them, simply add or subtract their coefficients. For instance, \( 4z - 18z \) becomes \( -14z \).

In our exercise, once distribution was completed, the expression was:\[ 28z - 12 - 14z + 7 \]By combining like terms, we simplified it to:\[ (28z - 14z) + (-12 + 7) = 14z - 5 \]
This final step reduced multiple \'like\' terms into a cleaner and more comprehensible format, stripping away any redundant parts of the expression to reach the simplified answer.