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Chapter 8: Problem 127

Solve the equation for the variable. $$6 x+27=9 x$$

Short Answer

Expert verified
x = 9

Step by step solution

01

Move all terms involving 'x' to one side

Subtract \(6x\) from both sides of the equation to start isolating the variable \(x\) on one side: \[6x + 27 - 6x = 9x - 6x\] Simplifying this gives: \[27 = 3x\]
02

Solve for 'x'

Now, isolate \(x\) by dividing both sides of the equation by 3: \[\frac{27}{3} = \frac{3x}{3}\] Simplifying this yields: \[9 = x\]

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Key Concepts

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

headline of the respective core concept
One-variable equations are equations that involve only one variable. This means that there is only one unknown quantity we need to solve for. These equations can take many forms, but they all have the same goal: to find the value of the variable that makes the equation true. For example, in the equation \(6x + 27 = 9x\), our variable is \(x\).

Understanding one-variable equations is key to solving them. Here are some general steps:
  • Identify the variable you need to isolate.
  • Combine like terms that involve the variable on one side of the equation.
  • Perform operations to isolate the variable.
This method can be applied to solve simpler to more complex one-variable equations.

In our example, we need to solve for \(x\) in \(6x + 27 = 9x\). Let's see how we can achieve that.
headline of the respective core concept
Isolating the variable is the process of manipulating an equation to get the unknown variable by itself on one side of the equation. In our example, the equation \[6x + 27 = 9x\] has the variable \(x\) on both sides.

Here's how to isolate it:
  • Subtract \(6x\) from both sides to move all variable terms to one side: \[6x + 27 - 6x = 9x - 6x\]
  • Simplify to remove similar terms: \[27 = 3x\]

Now, \(x\) is on one side of the equation. The next step is to completely isolate \(x\) by eliminating the coefficient next to it. This process helps in clearly seeing the value of the variable.
headline of the respective core concept
Equation simplification is essential to make equations easier to solve. When we simplify equations, we aim to break them down into more manageable steps.

In our example \[27 = 3x\], we need to isolate \(x\).

Follow these steps to simplify and solve for \(x\):
  • Divide both sides of the equation by the coefficient of \(x\), which is 3:
    \[\frac{27}{3} = \frac{3x}{3}\]
  • Simplify the division: \[9 = x\]

This tells us that \(x = 9\).

Always remember to check your solution by plugging the value back into the original equation to ensure it is correct. In our example, substituting \(x = 9\) into \(6x + 27 = 9x\) confirms that both sides of the equation are equal, verifying our solution.

Simplifying equations step by step makes it easier to solve for the unknown variable without mistakes.

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