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Chapter 2: Problem 42

Solve each equation, and check your solution. $$ 9.1 x+5=8.1 x $$

Short Answer

Expert verified
-5

Step by step solution

01

- Isolate the variable terms

First, subtract the term involving \( x \) on the right side of the equation from both sides. This results in: \(9.1x - 8.1x + 5 = 8.1x - 8.1x\) simplifying to \( 1x + 5 = 0 \).
02

- Solve for x

Next, subtract 5 from both sides to isolate the variable \( x \): \( 1x = -5 \).
03

- Verify the solution

Substitute \( x = -5 \) back into the original equation to check: \(9.1(-5) + 5 = 8.1(-5)\). Evaluating both sides, you get: \(-45.5 + 5 = -40.5 \) and \(-40.5 = -40.5 \), so the solution \( -5 \) is correct.

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Key Concepts

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

Isolate the Variable
When solving linear equations, the first step is to isolate the variable on one side of the equation. This means you want to get the term with the variable by itself. For example, starting with the equation 9.1x + 5 = 8.1x , you need to move all terms involving x to one side. This helps in clearly seeing what value x should have. In the given exercise, subtracting 8.1x from both sides will leave x isolated on one side.
Subtracting Terms
Subtracting terms involves removing the same term from both sides of the equation to maintain balance. Let's look at our example again: 9.1x + 5 = 8.1x . By subtracting 8.1x from both sides, you get 9.1x - 8.1x + 5 = 8.1x - 8.1x , which simplifies to 1x + 5 = 0 . This action of subtracting is crucial because it simplifies the equation and brings it closer to a form where you can easily solve for the variable.
Verify the Solution
After isolating the variable and finding its value, it's important to verify the solution to make sure it is correct. Substitute the value of x back into the original equation and see if both sides are equal. In our example, we solve for x and get x = -5 . To verify, substitute -5 back into the original equation: 9.1(-5) + 5 should equal 8.1(-5) . Calculating this gives -45.5 + 5 = -40.5 , which matches -40.5 = -40.5 . Since both sides are equal, -5 is indeed the correct solution.
Evaluate Both Sides
It is essential to evaluate both sides of the equation after substituting the solution back in. This step confirms that the solution makes the original equation true. If both sides of the equation equal the same value after substitution, the solution is correct. Always perform this verification step to ensure accuracy and build confidence in your solution. For example, returning to our exercise, substituting x = -5 back into 9.1x + 5 = 8.1x yields -40.5 on both sides, confirming that x = -5 is the correct solution.

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