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Chapter 2: Problem 189

Solve the following equations with variables on both sides. $$8 x+27=11 x$$

Short Answer

Expert verified
x = 9

Step by step solution

01

- Simplify the Equation

To start, move all the variable terms to one side of the equation. Subtract \(8x\) from both sides of the equation:\[8x + 27 - 8x = 11x - 8x\]which simplifies to:\[27 = 3x\]
02

- Isolate the Variable

Next, solve for \(x\) by isolating it on one side of the equation. Divide both sides by 3:\[\frac{27}{3} = \frac{3x}{3}\]which simplifies to:\[9 = x\]

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Key Concepts

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

Variables on Both Sides
When you encounter an equation with variables on both sides, it means each side of the equal sign has a term that includes the variable you're solving for. In this exercise, both sides of the equation include the variable \(x\):
\[8x + 27 = 11x\]
To solve such equations, we need to make sure all variable terms are on one side of the equation. This simplifies our job of isolating the variable later on.

Why is this important? Because having variables on both sides can make it confusing to understand the value of the variable. By bringing all the variables to one side:
  • We can better see the relationship between different terms.
  • Simplifying steps in our solution process becomes easier.
  • Reduces chances of errors in equations involving more complex terms.
Simplify the Equation
Once we have identified variables on both sides, our next step is to simplify the equation.
Begin by moving all variable terms to one side of the equation. To do this in our exercise, we subtract \(8x\) from both sides:
\[8x + 27 - 8x = 11x - 8x\]
This gives us a simplified form:
\[27 = 3x\]
Why is moving the variable terms crucial? Let鈥檚 break it down:
  • It allows us to see the equation in a more manageable form.
  • Combines like terms, making the equation simpler to solve.
  • Helps in clearly isolating the variable later on, reducing potential mistakes.
By simplifying the equation in this way, it becomes straightforward to manage and proceed to the next step.
Isolate the Variable
The final step in solving the equation is to isolate the variable. In the simplified equation:
\[27 = 3x\]
We want to completely isolate \(x\) so that we can determine its value. To achieve this, we perform the following operation: divide both sides by 3:
\[\frac{27}{3} = \frac{3x}{3}\]
This simplifies to:
\[9 = x\]
This means the value of \(x\) is 9. Isolating the variable involves:
  • Ensuring the variable stands alone on one side of the equation.
  • Performing inverse operations like addition, subtraction, multiplication, or division.
Always double-check your steps to ensure you haven鈥檛 introduced any errors during operations. By isolating the variable, you find the answer to the equation in a clear manner.

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