Solve for n 9(2n+5)-12=51

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:

1. Distributive Property: This property states that $a(b+c)=ab+ac$. We use this to expand $9(2n+5)$ into $18n+45$.

2. 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 $18n+33$.

3. 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.

4. 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.

5. 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.