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Chapter 1: Problem 67

Solve. $$ 1.8(2-n)=9 $$

Short Answer

Expert verified
n = -3

Step by step solution

01

Distribute the 1.8

Multiply 1.8 by both terms inside the parentheses: \[1.8 \times 2 - 1.8 \times n = 9\]which simplifies to: \[3.6 - 1.8n = 9\]
02

Isolate the term with the variable

Subtract 3.6 from both sides of the equation to isolate the term with the variable:\[3.6 - 1.8n - 3.6 = 9 - 3.6\]This simplifies to: \[-1.8n = 5.4\]
03

Solve for n

Divide both sides of the equation by -1.8 to solve for n:\[\frac{-1.8n}{-1.8} = \frac{5.4}{-1.8}\]This simplifies to: \[n = -3\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
The Distributive Property is a fundamental algebraic property used to simplify expressions and solve equations. In our example, the equation was: \[1.8(2-n)=9\]. To apply the Distributive Property, we multiply 1.8 by each term within the parentheses:
  • 1.8 multiplied by 2 gives us 3.6
  • 1.8 multiplied by -n equals -1.8n

This simplifies our equation to: \[3.6 - 1.8n = 9\]. This step helps break down the equation into simpler parts, making it easier to solve.
isolating variables
The next step is to isolate the variable term, which means getting the term with the variable 'n' by itself on one side of the equation. Starting from our simplified equation: \[3.6 - 1.8n = 9\], we need to remove the constant term 3.6 from the left side.

We do this by subtracting 3.6 from both sides: \[3.6 - 1.8n - 3.6 = 9 - 3.6\].
Simplifying gives us: \[-1.8n = 5.4\].
This leaves us with an equation where the term with the variable is isolated. It's crucial to perform the same operation on both sides to maintain the equality.
division in equations
The final step in solving the equation is dealing with the isolated variable term. We now have: \[-1.8n = 5.4\].
To solve for 'n', we need to eliminate the coefficient -1.8, which is currently multiplying the variable 'n'. We do this by dividing both sides by -1.8:
  • \frac{-1.8 * n}{-1.8} = \frac{5.4}{-1.8}

Simplifying this expression, we obtain: \[ n = -3 \].
Division helps us isolate the variable completely, providing the final solution for 'n'. Always remember, every step must ensure the equation remains balanced.

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