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Chapter 8: Problem 165

Solve the linear equation using the general strategy. $$9=3(x-3)$$

Short Answer

Expert verified
x = 6

Step by step solution

01

- Distribute the 3 on the right-hand side

Apply the distributive property to eliminate the parentheses on the right-hand side of the equation. This means multiplying 3 by each term inside the parentheses:\[ 9 = 3x - 9 \]
02

- Isolate the term with the variable

To isolate the term with the variable, add 9 to both sides of the equation to get rid of the -9:\[ 9 + 9 = 3x - 9 + 9 \]This simplifies to:\[ 18 = 3x \]
03

- Solve for the variable

Divide both sides of the equation by 3 to solve for \( x \):\[ \frac{18}{3} = \frac{3x}{3} \]Simplifying gives:\[ x = 6 \]

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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 helps simplify expressions and equations by distributing a single term over others inside parentheses.

In the given equation, 9 = 3(x - 3), applying the distributive property means multiplying 3 by both x and -3 within the parentheses.
So, 3(x - 3) becomes 3x - 9.
This transforms our original equation to 9 = 3x - 9.
  • Think of it like sharing equally: each term inside the parentheses gets multiplied by what's outside.
  • This property is very useful to eliminate parentheses and simplify expressions.
Distributing terms correctly is a critical first step in solving linear equations!
isolating variables
Isolating the variable means getting the variable term on one side and the constants on the other side of the equation.

Once we have 9 = 3x - 9, the next step is to move the constant term (-9 here) to the other side.
We do this by adding 9 to both sides of the equation, giving us 18 = 3x.
  • Ensuring you do the same operation on both sides keeps the equation balanced.
  • Aim to get the variable term alone so you can solve for it directly.
Isolating the variable simplifies the equation, making it easier to solve for the variable right away.
simplifying equations
Simplifying equations involves performing arithmetic operations to make the equation easier to solve.

With 18 = 3x, we need to isolate x fully by dividing both sides by 3 (the coefficient of x).
This step simplifies 18/3 to 6 and 3x/3 simply to x, resulting in x = 6.
  • Use division or multiplication to simplify to the easiest form.
  • Ensure you apply the same operation on both sides to maintain balance.
Simplifying forms the last step in solving the linear equation, revealing the value of the variable.
After these steps, we see that x equals 6, solving the equation completely.

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