Problem 47
Question
For the following exercises, simplify the expression. $$ 6+12 b-3 \cdot 6 b $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(6 - 6b\).
1Step 1: Apply the Distributive Property
Start by simplifying the expression using multiplication. Multiply \(-3\) by \(6b\): \(-3\cdot 6b = -18b\). This changes the expression to: \[ 6 + 12b - 18b \]
2Step 2: Combine Like Terms
Identify like terms in the expression. The terms \(12b\) and \(-18b\) are like terms. Combine them: \[ 12b - 18b = -6b \] Now the expression simplifies to: \[ 6 - 6b \]
Key Concepts
Distributive PropertyCombining Like TermsExpression Simplification
Distributive Property
The Distributive Property is a foundational algebra concept that helps us simplify expressions by distributing a number across terms inside parentheses. This property states that for any three numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true.
For example, in the problem given, we see part of this in action with the expression \(-3 \cdot 6b\). Here, \(-3\) is distributed to \(6b\). The property allows us to multiply \(-3\) by \(6b\), giving us \(-18b\).
This step is crucial because it transforms the original expression into one that we can more easily combine and simplify. Always remember, if you're dealing with a situation where you need to distribute across parentheses, apply this rule to each term within the parentheses.
For example, in the problem given, we see part of this in action with the expression \(-3 \cdot 6b\). Here, \(-3\) is distributed to \(6b\). The property allows us to multiply \(-3\) by \(6b\), giving us \(-18b\).
This step is crucial because it transforms the original expression into one that we can more easily combine and simplify. Always remember, if you're dealing with a situation where you need to distribute across parentheses, apply this rule to each term within the parentheses.
Combining Like Terms
Combining Like Terms is another important concept in algebra. Like terms are terms that contain the same variable raised to the same power. We can combine them by adding or subtracting their coefficients.
In our problem, after using the distributive property, we had the expression \(6 + 12b - 18b\). Here, \(12b\) and \(-18b\) are like terms because they both contain the variable \(b\).
We handle these terms by performing the operation on their coefficients: \(12\) and \(-18\). When combined, they result in \(-6b\).
Combining like terms simplifies expressions, making them easier to understand and work with. This step reduces the number of terms, leading us closer to the simplest form of the expression.
In our problem, after using the distributive property, we had the expression \(6 + 12b - 18b\). Here, \(12b\) and \(-18b\) are like terms because they both contain the variable \(b\).
We handle these terms by performing the operation on their coefficients: \(12\) and \(-18\). When combined, they result in \(-6b\).
Combining like terms simplifies expressions, making them easier to understand and work with. This step reduces the number of terms, leading us closer to the simplest form of the expression.
Expression Simplification
Expression Simplification involves using various algebra rules to rewrite the expression in its simplest form. The goal is to make the expression easier to work with and understand.
When we simplify \(6 + 12b - 18b\), we first apply the distributive property and combine like terms. This process transformed the expression into \(6 - 6b\).
The simplified expression, \(6 - 6b\), is straightforward and concise. It represents the same idea as the original, just without unnecessary complexity.
Expression simplification is crucial in algebra as it helps in solving equations, graphing linear equations, and throughout calculus and beyond. It aids in recognizing patterns and relationships between variables.
When we simplify \(6 + 12b - 18b\), we first apply the distributive property and combine like terms. This process transformed the expression into \(6 - 6b\).
The simplified expression, \(6 - 6b\), is straightforward and concise. It represents the same idea as the original, just without unnecessary complexity.
Expression simplification is crucial in algebra as it helps in solving equations, graphing linear equations, and throughout calculus and beyond. It aids in recognizing patterns and relationships between variables.
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Problem 47
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