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Chapter 2: Problem 14

Solve each equation, and check your solution. $$ 6 q-2=3 q $$

Short Answer

Expert verified
The solution is \( q = \frac{2}{3} \). Verification confirms this.

Step by step solution

01

- Move variable terms to one side

Subtract 3q from both sides of the equation: \[ 6q - 2 - 3q = 3q - 3q \]This simplifies to: \[ 3q - 2 = 0 \]
02

- Isolate the variable term

Add 2 to both sides to isolate the variable term: \[ 3q - 2 + 2 = 0 + 2 \]This simplifies to: \[ 3q = 2 \]
03

- Solve for the variable

Divide both sides by 3 to solve for q: \[ \frac{3q}{3} = \frac{2}{3} \]This simplifies to: \[ q = \frac{2}{3} \]
04

- Check the solution

Substitute \( q = \frac{2}{3} \) back into the original equation to verify: \[ 6 \left( \frac{2}{3} \right) - 2 = 3 \left( \frac{2}{3} \right) \]Simplify each side:\[ 4 - 2 = 2 \]This checks out as both sides are equal.

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Key Concepts

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

Variable Isolation
To solve linear equations, we often need to isolate the variable on one side of the equation. In our example, the equation is initially: \[ 6q - 2 = 3q \]
By moving all the terms involving the variable to one side, we make it easier to solve for the variable. We do this by subtracting 3q from both sides: \[ 6q - 2 - 3q = 3q - 3q \]
Simplifying, we get: \[ 3q - 2 = 0 \]
Now our variable term 3q is isolated on one side. Bullet points can help you understand the steps better:
  • Identify the variable terms on both sides of the equation
  • Use addition or subtraction to eliminate variable terms on one side
  • Simplify the equation after each step

Isolating the variable is a crucial initial step because it sets the groundwork for solving the equation.
Equation Simplification
Once the variable is isolated on one side, the next step is to simplify the equation further to solve for the variable. From our isolated equation: \[ 3q - 2 = 0 \]
We need to move the constant term to the other side. We do this by adding 2 to both sides: \[ 3q - 2 + 2 = 0 + 2 \]
Which simplifies to: \[ 3q = 2 \]
Now, we solve for q by dividing both sides of the equation by 3: \[ \frac{3q}{3} = \frac{2}{3} \]
Simplifying, we get: \[ q = \frac{2}{3} \]
Simplifying equations helps to gradually break down the problem into manageable steps. Key actions to remember include:
  • Use inverse operations (like addition to counter subtraction)
  • Perform the same operation on both sides of the equation
  • Always ensure the equation is balanced

Following these steps rigorously ensures we reach the correct solution.
Solution Verification
Verifying the solution is an essential step to ensure that our answer is correct. We do this by substituting the value we obtained for the variable back into the original equation. In our example, we solved for: \( q = \frac{2}{3} \)
We substitute this value back into the original equation: \[ 6 \times \frac{2}{3} - 2 = 3 \times \frac{2}{3} \]
Simplifying both sides yields: \[ 4 - 2 = 2 \]
Since both sides are equal, our solution is verified. Important steps for solution verification include:
  • Substitute the solved value back into the original equation
  • Simplify both sides rigorously
  • Check if both sides of the equation are equal

Verification helps to ensure that no steps were missed and that the calculated solution is indeed correct.

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Most popular questions from this chapter

Write the answer to each problem in terms of the variable. A clerk has \(v\) dollars, all in \(\$ 10\) bills. What expression represents the number of \(\$ 10\) bills the clerk has?

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