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Chapter 2: Problem 25

Simplify each expression as completely as possible. $$5(x-3 y)-(x-3 y)$$

Short Answer

Expert verified
The simplified expression is \[ 4x - 12y \].

Step by step solution

01

Distribute the terms

Begin by distributing the constants in the expression. The expression is: \[ 5(x - 3y) - (x - 3y) \] Distribute the 5 to both terms inside the parentheses: \[ 5 \times x - 5 \times 3y = 5x - 15y \] The expression now becomes: \[ 5x - 15y - (x - 3y) \]
02

Distribute the negative sign

Next, distribute the negative sign (or think of it as multiplying by -1) to both terms inside the second set of parentheses: \[ -(x - 3y) = -x + 3y \] So the expression now is: \[ 5x - 15y - x + 3y \]
03

Combine like terms

Combine the like terms by adding or subtracting the coefficients of each term: \[ (5x - x) + (-15y + 3y) \] Simplify the expression: \[ 4x - 12y \]

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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 basic yet powerful algebraic tool. It allows you to multiply each term inside a parenthesis by the factor outside it. In the example, the expression \[5(x - 3y) - (x - 3y)\] involves distributing the 5 to both terms inside the first parenthesis: \[5 \times x - 5 \times 3y = 5x - 15y\]. This step ensures that everything inside the parenthesis is properly accounted for by the outer factor.
Negative Sign Distribution
Distributing a negative sign is like multiplying by -1. This step changes the signs of the terms inside the parenthesis. For instance, in the expression \[-(x - 3y)\], you would distribute the \[-1\] to each term, resulting in \[-x + 3y\]. This is crucial for ensuring that each term's sign is accurately represented.
Combining Like Terms
Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable. In our example, once you distribute the terms, the expression becomes \[5x - 15y - x + 3y\]. The next step is to combine the \[x\] terms (\[5x - x\]). Likewise, combine the \[y\] terms (\[-15y + 3y\]). This results in: \[(5x - x) + (-15y + 3y) = 4x - 12y\]. Combining like terms significantly simplifies your expression.
Algebraic Simplification
Algebraic simplification is the final goal. It involves applying all the previous steps to make the expression as simple as possible. Each simplification step reduces complexity, making the expression easier to work with. After distributing the constants and the negative sign, and combining like terms, we reduced \[5(x-3y)-(x-3y)\] to \[4x - 12y\]. Simplifying algebraic expressions prepares you for more advanced mathematical concepts and problem-solving.

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