Problem 32
Question
To prepare for Section 9.3, review simplifying expressions \((\text { Section } 1.8)\) Simplify. [ 1.8] $$ -10(5 a+3 b-c) $$
Step-by-Step Solution
Verified Answer
-50a - 30b + 10c
1Step 1: Understand the Problem
The given expression is -10(5a + 3b - c). To simplify, distribute the -10 to each term inside the parentheses.
2Step 2: Distribute the Coefficient
Multiply each term inside the parentheses by -10: -10 * 5a + (-10) * 3b + (-10) * (-c).
3Step 3: Simplify Each Term
Perform the multiplications: -10 * 5a = -50a, -10 * 3b = -30b, -10 * (-c) = 10c.
4Step 4: Write Final Expression
Combine the simplified terms: -50a - 30b + 10c.
Key Concepts
Distributive PropertyCoefficientsCombining Like Terms
Distributive Property
The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations. It's an essential tool for breaking down complex problems into more manageable pieces.
The distributive property states that multiplying a single term by each term within a parenthesis is equivalent to multiplying that single term with the entire expression. Symbolically, we write it as:
\( a(b + c) = ab + ac \).
In our exercise, we applied the distributive property by multiplying \( -10 \) with each term inside the parentheses (5a, 3b, -c).
Here's the breakdown:
-10 * 5a,
-10 * 3b,
and -10 * (-c).
This step helps distribute the multiplication over addition or subtraction, making it easier to handle terms individually.
The distributive property states that multiplying a single term by each term within a parenthesis is equivalent to multiplying that single term with the entire expression. Symbolically, we write it as:
\( a(b + c) = ab + ac \).
In our exercise, we applied the distributive property by multiplying \( -10 \) with each term inside the parentheses (5a, 3b, -c).
Here's the breakdown:
-10 * 5a,
-10 * 3b,
and -10 * (-c).
This step helps distribute the multiplication over addition or subtraction, making it easier to handle terms individually.
Coefficients
Coefficients are the numerical factors that multiply the variables in an algebraic expression. They indicate how many times a variable is taken. Understanding coefficients is crucial for manipulating and simplifying expressions.
In the expression -10(5a + 3b - c), the number -10 is the coefficient outside the parentheses.
When we distribute -10 over each term within the parentheses, the result is new products with coefficients: \(-10 \times 5a = -50a\), \(-10 \times 3b = -30b\), and \(-10 \times -c = 10c\).
Notice that the sign of the coefficient plays a significant role. Multiplying by a negative coefficient changes the sign of the products. For example, -10 multiplied by 5a changes to -50a, and -10 multiplied by -c changes to 10c.
Recognizing and correctly dealing with coefficients ensures accurate simplification of algebraic expressions.
In the expression -10(5a + 3b - c), the number -10 is the coefficient outside the parentheses.
When we distribute -10 over each term within the parentheses, the result is new products with coefficients: \(-10 \times 5a = -50a\), \(-10 \times 3b = -30b\), and \(-10 \times -c = 10c\).
Notice that the sign of the coefficient plays a significant role. Multiplying by a negative coefficient changes the sign of the products. For example, -10 multiplied by 5a changes to -50a, and -10 multiplied by -c changes to 10c.
Recognizing and correctly dealing with coefficients ensures accurate simplification of algebraic expressions.
Combining Like Terms
Combining like terms is an essential skill in algebra to simplify expressions further. Like terms are terms that have the same variables raised to the same power.
For example, 2a and 3a are like terms, but 2a and 2b are not.
In our simplified expression -50a - 30b + 10c, we can't combine the terms any further since:
Combining like terms involves adding or subtracting the coefficients while keeping the variable part unchanged. This process helps reduce expressions to their simplest form.
Always look out for like terms when simplifying to ensure the expression is as simplified as possible.
For example, 2a and 3a are like terms, but 2a and 2b are not.
In our simplified expression -50a - 30b + 10c, we can't combine the terms any further since:
- -50a has the variable 'a'
- -30b has the variable 'b'
- 10c has the variable 'c'
Combining like terms involves adding or subtracting the coefficients while keeping the variable part unchanged. This process helps reduce expressions to their simplest form.
Always look out for like terms when simplifying to ensure the expression is as simplified as possible.
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