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Chapter 2: Problem 30

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

Short Answer

Expert verified
The solution is \( x = 9 \).

Step by step solution

01

Isolate the variable

To isolate the variable on one side of the equation, subtract \(4x\) from both sides of the equation.Thus, the equation becomes:\[ 5x - 4x = 4x + 9 - 4x \] Which simplifies to:\[ x = 9 \]
02

Verify the solution

To check if \( x = 9 \) is the correct solution, substitute \( x \) with 9 in the original equation and see if both sides are equal. Substituting \( x = 9 \) into the original equation:\[ 5(9) = 4(9) + 9 \]This simplifies to:\[ 45 = 36 + 9 \]Since both sides of the equation are equal (\( 45 = 45 \)), \( x = 9 \) is indeed the correct solution.

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Key Concepts

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

Isolating Variables
One of the most important steps in solving linear equations is isolating the variable. This means we want to get the variable (often represented as 'x') alone on one side of the equation.

Let's look at our example equation: \(5x = 4x + 9\).
  • First, we subtract \(4x\) from both sides of the equation. This keeps the equation balanced as we're doing the same operation on both sides.

  • Subtracting \(4x\) from both sides looks like this: \(5x - 4x = 4x + 9 - 4x\).

With this step, we simplify the equation to \(x = 9\). Now, the variable \(x\) is isolated on one side of the equation. This clear and simple result tells us \(x\) without any additional operations needed.

In more complicated equations, you might need to use additional operations like addition, subtraction, multiplication, or division to isolate the variable. Remember, the goal is always to get the variable alone.
Substitution Method
The substitution method helps verify that our solution is correct. This is done by substituting our solution back into the original equation to see if both sides are equal.

In our example, we found \(x = 9\):
  • Substitute 9 for x in the original equation: \(5(9) = 4(9) + 9\).
  • Perform the multiplications: \(45 = 36 + 9\).

If both sides of the equation are equal, this confirms our solution is correct. In this case:
    \(45 = 45\).

As the equation holds true, \(x = 9\) is indeed the correct solution.

The substitution method is excellent for double-checking your work. If the sides of the equation do not match after substitution, there was likely an error in your initial isolation steps.
Verification of Solutions
Verifying solutions ensures the solution we found is valid for the original equation. This step is crucial in mathematics as it confirms the accuracy.

We鈥檝e already substituted our solution into the original equation in the previous section. However, let鈥檚 underline the importance of this step:
  • Verification builds confidence in your result.

  • It catches mistakes that might occur during isolation or simplification.

  • Ensures thorough understanding and correctness.

It鈥檚 not just a rote step; it鈥檚 a meaningful part of making sure your math is sound and reliable. Always make it a habit to verify your solution.

When you're solving more advanced problems or applying mathematical solutions in real life, verification can save you from potentially costly errors. Always complete this final step in problem-solving.

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