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Chapter 4: Problem 62

Simplify each expression. $$ -\frac{2}{3}(3 w-6) $$

Short Answer

Expert verified
The simplified expression is \(-2w + 4\).

Step by step solution

01

Distribute the Negative Sign

Begin by distributing the negative sign across the expression inside the parenthesis. The expression is given as: \[-\frac{2}{3}(3w - 6)\] Apply the distributive property: \[-\frac{2}{3} \times 3w\] and \[-\frac{2}{3} \times (-6)\].
02

Simplify Each Term

Calculate each term from Step 1 separately.For the first term: \[-\frac{2}{3} \times 3w = -2w\]For the second term:\[-\frac{2}{3} \times (-6) = 4\] (This is because \(-\times - = +\)).
03

Combine the Terms

After simplifying, you will have two terms: \[-2w + 4\] This is the simplified form of the original expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra that helps simplify expressions. This property allows you to multiply a number by a sum or a difference, effectively distributing the multiplication over each term within the parentheses. This is written as: \[ a(b + c) = ab + ac \] Similarly, this applies when dealing with negative numbers. Here, it is crucial to distribute the minus sign as well, such as in expressions like \[ - rac{2}{3}(3w - 6) \].
  • Multiply \[ - rac{2}{3} \] with \[ 3w \], resulting in \[ -2w \].
  • Multiply \[ - rac{2}{3} \] with \[ -6 \], which becomes \[ 4 \] since two negatives make a positive.
By understanding the distributive property, algebraic expressions become more manageable, especially when dealing with complex or large expressions.
Negative Numbers
Handling negative numbers can be confusing, but it is essential to grasp their properties especially when simplifying expressions. Negatives can change the outcome of multiplications significantly. The rule of thumb is:
  • Multiplying two positive numbers yields a positive number: \[ 2 \times 3 = 6 \]
  • Multiplying a positive number with a negative number gives a negative result: \[ 2 \times (-3) = -6 \]
  • Multiplying two negative numbers results in a positive product: \[ (-2) \times (-3) = 6 \]
In our expression \[ -\frac{2}{3} \times (-6) \], both numbers are negative, so their product is positive, leading to \[ 4 \]. When working with negatives, always pay attention to the signs to ensure accuracy.
Combining Like Terms
Once each term is calculated using the distributive property, the next step is to combine like terms to reach the simplest form of the expression. Like terms are terms that have the same variable part, such as \[ 2x \] and \[ -3x \]. Only these can be added or subtracted from each other.
  • Identify all like terms in the expression. Terms with the same variable and exponent can be combined.
  • Add or subtract the coefficients of these like terms.
In the expression \[ -2w + 4 \], there are no like terms to combine further so this is the simplest form. By systematically finding and combining like terms, you simplify expressions and make your algebraic work more efficient.

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Most popular questions from this chapter

Use the elimination method to solve each system. If there is no solution, or infinitely many solutions, so state. $$ \left\\{\begin{array}{l} {2 x+5 y-13=0} \\ {-2 x+13=5 y} \end{array}\right. $$

Solve each system of equations by graphing. $$ \left\\{\begin{array}{l} {x=3} \\ {3 y=6-2 x} \end{array}\right. $$

Solve each system by substitution. If a system has no solution or infinitely many solutions, so state. $$ \left\\{\begin{array}{l} {x+3 y=6} \\ {x=-3 y+6} \end{array}\right. $$

Determine whether the ordered pair is a solution of the given system of equations. $$ (3,-2),\left\\{\begin{array}{l} {2 x+y=4} \\ {y=1-x} \end{array}\right. $$

Solve each system by substitution. If a system has no solution or infinitely many solutions, so state. $$ \left\\{\begin{array}{l} {a=\frac{2}{3} b} \\ {9 a+4 b=5} \end{array}\right. $$
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