Subtract 2(5v+6)-6(-9v+2)
The problem asks you to perform subtraction on two algebraic expressions that are both in parentheses and being multiplied by numerical coefficients. Specifically, you need to subtract the expression 2(5v+6), which represents doubling the sum of 5 times a variable v and 6, from the expression -6(-9v+2), which represents taking six times the quantity of negative 9 times the variable v and adding 2. Before you can actually perform the subtraction, you will need to apply the distributive property to remove the parentheses by multiplying the coefficients with each term inside the parentheses, and then combine like terms.
$2 \left(\right. 5 v + 6 \left.\right) - 6 \left(\right. - 9 v + 2 \left.\right)$
Answer
Solution:
Step 1: Expand the expressions
- Apply the distributive property to both terms: $2(5v + 6)$ and $-6(-9v + 2)$.
Step 1.1: Distribute the first term
- Distribute $2$ over $(5v + 6)$ to get $2 \cdot 5v + 2 \cdot 6$.
Step 1.2: Calculate $2 \cdot 5v$
- Multiply $2$ by $5v$ to obtain $10v$.
Step 1.3: Calculate $2 \cdot 6$
- Multiply $2$ by $6$ to get $12$.
Step 1.4: Distribute the second term
- Distribute $-6$ over $(-9v + 2)$ to get $-6 \cdot (-9v) - 6 \cdot 2$.
Step 1.5: Calculate $-6 \cdot (-9v)$
- Multiply $-6$ by $-9v$ to obtain $54v$.
Step 1.6: Calculate $-6 \cdot 2$
- Multiply $-6$ by $2$ to get $-12$.
Step 2: Combine like terms
- Add together the terms from the expansion to simplify the expression.
Step 2.1: Cancel out opposite terms
- Subtract $12$ from $12$ to eliminate those terms.
Step 2.1.1: Simplify to zero
- The subtraction of $12$ from $12$ results in $0$.
Step 2.1.2: Combine $10v$ and $54v$
- Add $10v$ to $54v$ to combine like terms.
Step 2.2: Sum the variable terms
- The sum of $10v$ and $54v$ is $64v$.
Knowledge Notes:
The problem-solving process involves several key knowledge points:
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand algebraic expressions by multiplying the term outside the parentheses by each term inside the parentheses.
Combining Like Terms: Terms that contain the same variable raised to the same power can be combined by adding or subtracting their coefficients. In this case, $10v$ and $54v$ are like terms because they both contain the variable $v$ to the first power.
Simplification: The process of combining like terms and performing arithmetic operations to reduce an expression to its simplest form.
Arithmetic Operations: Basic operations such as addition, subtraction, multiplication, and division are used throughout the problem-solving process. Multiplication of negative numbers, as seen with $-6 \cdot (-9v)$, results in a positive product because a negative times a negative is a positive.
Variables and Coefficients: In algebra, a variable is a symbol used to represent an unknown value, while a coefficient is a numerical factor of a term that contains a variable. In the expression $10v$, $10$ is the coefficient and $v$ is the variable.
Understanding these concepts is essential for solving algebraic expressions and equations.
