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Chapter 2: Problem 29

Solve each equation, and check your solution. $$ 3 x=2 x+7 $$

Short Answer

Expert verified
x = 7

Step by step solution

01

Subtract \(2x\) from both sides

To isolate the variable on one side of the equation, subtract \(2x\) from both sides:\[ 3x - 2x = 2x + 7 - 2x \]This simplifies to:\[ x = 7 \]
02

Check the solution

To verify the solution, substitute \(x = 7\) back into the original equation:\[ 3(7) = 2(7) + 7 \]This simplifies to:\[ 21 = 14 + 7 \]Since both sides equal 21, the solution is correct.

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Key Concepts

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

Isolating Variables
When solving linear equations, the goal is often to isolate the variable on one side of the equation. This means manipulating the equation so that the variable stands alone on one side, making it easy to solve for its value.

Let's take a closer look at the exercise: \3x = 2x + 7\.

First, we aim to isolate the variable \(x\). To do this, we subtract \(2x\) from both sides of the equation:
\[ 3x - 2x = 2x + 7 - 2x\]This step simplifies the equation to:\[ x = 7\]Now, the variable \(x\) is isolated on one side and its value is found.

Remember, isolating the variable makes solving the equation straightforward. Always perform the same operation on both sides to keep the equation balanced.
Substitution Method
After finding the value of the variable, we should verify our solution by substitution. This method involves replacing the variable in the original equation with the value we found, ensuring both sides remain equal.

From the exercise, we found \(x = 7\). To verify, we substitute this value back into the original equation: \3(7) = 2(7) + 7\.

Substituting \(x = 7\) gives us:
\[ 3(7) = 2(7) + 7\]This simplifies to:\[ 21 = 14 + 7\]Since \(21 = 21\), the left-hand side equals the right-hand side, confirming our solution is correct.

Substitution is a powerful method to ensure that the solution is accurate. It double-checks the work done in isolating and solving the variable.
Equation Verification
Equation verification is the final step in solving linear equations. Once you have a solution, you should always check it to ensure its accuracy.

Using the example \(3x = 2x + 7\), we found \(x = 7\). By substituting \(7\) back into the equation, we checked if the original equation is satisfied:
\[ 3(7) = 2(7) + 7\]Simplifying the right-hand side, we get:\[ 21 = 14 + 7\]Confirming \(21 = 21\).

Verification ensures that no mistakes were made during the isolation or substitution steps. It adds confidence to your solution. Always double-check your answers to avoid errors.

Key Points to Remember:
  • Isolate the variable first.
  • Substitute the value back into the original equation.
  • Verify that both sides are equal.
By following these steps, you can solve linear equations accurately and confidently.

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