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Chapter 2: Problem 30

Solve each equation, and check your solution. $$ 5 x=4 x+9 $$

Short Answer

Expert verified
x = 9

Step by step solution

01

Combine like terms

First, transfer all the terms containing the variable to one side of the equation. Subtract \( 4x \) from both sides of the equation: \[ 5x - 4x = 4x + 9 - 4x \]This simplifies to: \[ x = 9 \]
02

Check the solution

Substitute \( x = 9 \) back into the original equation to verify the solution. The original equation is: \[ 5x = 4x + 9 \]Substitute \(x\) with 9: \[ 5(9) = 4(9) + 9 \]Simplify both sides: \[ 45 = 36 + 9 \]\[ 45 = 45 \]Since both sides are equal, \( x = 9 \) is a correct solution.

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Key Concepts

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

Combining Like Terms
Combining like terms is an essential part of solving linear equations. It means bringing all the terms with the same variable together on one side of the equation. This step makes it easier to isolate the variable and solve for it. Let's look at our example equation: \[ 5x = 4x + 9 \]To combine like terms, we need to transfer all the x terms to one side. We do this by subtracting \(4x\) from both sides: \[ 5x - 4x = 4x + 9 - 4x \]This step simplifies the equation to: \[ x = 9 \]Now, all the terms with the variable are combined on one side, and we have a simpler equation with just \(x\) on the left and a number on the right. This makes it much easier to solve.
Substitution Method
The substitution method involves replacing the variable with its value to check if it satisfies the original equation. After solving for \(x\), we need to make sure that our solution is correct. In our example, we found that \(x = 9\). To verify this, we'll substitute \(9\) back into the original equation: \[ 5x = 4x + 9 \]Replace \(x\) with \(9\): \[ 5(9) = 4(9) + 9 \]Now, we'll simplify both sides: \[ 45 = 36 + 9 \]This gives us: \[ 45 = 45 \]Since both sides of the equation are equal, this shows that the value \(x = 9\) is indeed a correct solution. The substitution method is a reliable way to confirm the validity of a solution.
Verifying Solutions
Verifying solutions ensures that our answer makes the original equation true. This step is crucial because it helps us catch any mistakes we might have made during the solving process. Here's how to verify the solution step-by-step:
  • Solve for the variable: In our case, we simplified \(5x = 4x + 9\) to find \(x = 9\).
  • Substitute the solution back into the original equation.
  • Simplify both sides of the equation.
  • Check if both sides are equal.
By following these steps, we confirmed that \(45 = 45\), validating that \(x = 9\) is the correct solution. Verification is an effective way to ensure accuracy in any algebraic problem-solving process.

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