Solve for n 9(2n+5)-12=51
The problem provided is a linear equation in one variable, asking to determine the value of the variable 'n' such that the equation holds true. The equation involves operations like multiplication, addition, and subtraction, and one would typically need to simplify and solve it by isolating the variable on one side of the equation.
$9 \left(\right. 2 n + 5 \left.\right) - 12 = 51$
Answer
Solution:
Step 1: Expand and simplify the expression $9(2n+5)-12$.
Step 1.1: Distribute $9$ over $(2n+5)$.
Step 1.1.1: Apply the distributive property: $9(2n) + 9 \cdot 5 - 12 = 51$.
Step 1.1.2: Perform the multiplication: $18n + 9 \cdot 5 - 12 = 51$.
Step 1.1.3: Complete the multiplication: $18n + 45 - 12 = 51$.
Step 1.2: Combine like terms by subtracting $12$ from $45$: $18n + 33 = 51$.
Step 2: Isolate the variable $n$ by moving constants to the other side.
Step 2.1: Subtract $33$ from both sides: $18n = 51 - 33$.
Step 2.2: Calculate the difference: $18n = 18$.
Step 3: Solve for $n$ by dividing both sides by $18$.
Step 3.1: Divide each term by $18$: $\frac{18n}{18} = \frac{18}{18}$.
Step 3.2: Simplify the equation.
Step 3.2.1: Reduce the left side by canceling out $18$.
Step 3.2.1.1: Cancel the common factors: $\frac{\cancel{18}n}{\cancel{18}} = \frac{18}{18}$.
Step 3.2.1.2: Simplify to find $n$: $n = \frac{18}{18}$.
Step 3.3: Simplify the right side by dividing $18$ by $18$: $n = 1$.
Knowledge Notes:
To solve the equation $9(2n+5)-12=51$, we follow a systematic approach:
Distributive Property: This property states that $a(b+c) = ab + ac$. We use this to expand $9(2n+5)$ into $18n + 45$.
Combining Like Terms: This involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. For example, $18n + 45 - 12$ simplifies to $18n + 33$.
Isolating the Variable: To solve for $n$, we need to get $n$ by itself on one side of the equation. This is done by performing the same operation on both sides of the equation to maintain equality.
Inverse Operations: To isolate $n$, we use the inverse of multiplication, which is division. By dividing both sides of $18n = 18$ by $18$, we are left with $n$ on one side and the solution on the other.
Simplification: After performing operations, we often need to simplify the expression to get the final answer. In this case, dividing $18$ by $18$ gives us $n = 1$.
By following these steps and using these principles, we can solve linear equations and find the value of the variable.
