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Chapter 8: Problem 332

In the following exercises, solve each equation. $$ x+3.8=8.2 $$

Short Answer

Expert verified
x = 4.4

Step by step solution

01

Understand the Objective

The goal is to isolate the variable (x) on one side of the equation to find its value.
02

Identify the Operation to Undo

Since 3.8 is added to x, we need to subtract 3.8 from both sides of the equation to isolate x.
03

Perform the Operation

Subtract 3.8 from both sides of the equation: \[ x + 3.8 - 3.8 = 8.2 - 3.8 \]
04

Simplify Both Sides

Simplify the equation by performing the subtraction:\[ x = 4.4 \]
05

Verify the Solution

Plug the value back into the original equation to ensure it holds true:\[ 4.4 + 3.8 = 8.2 \] Since both sides of the equation are equal when x = 4.4, 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 the variable
In solving linear equations, the first step is to isolate the variable. This means getting the variable (usually represented by 'x') by itself on one side of the equation. In our given exercise, the equation is:\[ x + 3.8 = 8.2 \] We need to move all the other numbers to the opposite side of the equation. This way, we can find the value of 'x'.
inverse operations
To isolate the variable, we use inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, as well as multiplication and division. Taking our exercise again, we see that 3.8 is added to 'x'.To undo this addition, we perform the inverse operation, which is subtraction. Therefore, we subtract 3.8 from both sides of the equation:\[ x + 3.8 - 3.8 = 8.2 - 3.8 \] Subtracting the same number from both sides maintains the balance of the equation. This simplifies to:\[ x = 4.4 \]
verification of solution
After isolating the variable and solving the equation, it is crucial to verify the solution to ensure it is correct. Verification involves substituting the found value of the variable back into the original equation to check if both sides are equal.In our case, we substitute 4.4 for 'x' in the original equation:\[ 4.4 + 3.8 = 8.2 \] Simplifying the left side, we get:\[ 8.2 = 8.2 \] Since both sides of the equation are equal, our solution \( x = 4.4 \) is verified as correct.Always remember to verify your solution to avoid any potential mistakes.

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