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Chapter 2: Problem 48

Solve each equation, and check your solution. $$ 7.7 r-6=6.7 r-6 $$

Short Answer

Expert verified
The solution is r = 0 and it checks out.

Step by step solution

01

Move Variables to One Side

Subtract the term with the variable on the right side from both sides of the equation. This will help isolate the variable on one side. 7.7r - 6 - 6.7r = 6.7r - 6 - 6.7r
02

Simplify Both Sides

Combine like terms to simplify both sides of the equation. The equation simplifies to: (7.7 - 6.7)r - 6 = -6 This results in: r - 6 = -6
03

Isolate the Variable

Add 6 to both sides of the equation to isolate the variable r. r - 6 + 6 = -6 + 6 This simplifies to: r = 0
04

Check the Solution

Substitute r = 0 back into the original equation to ensure the solution is correct. 7.7(0) - 6 = 6.7(0) - 6 Simplifies to: -6 = -6 Since both sides are equal, the solution is verified.

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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, one of the primary goals is to isolate the variable. This means you want to get the variable alone on one side of the equation. In the given problem, the variable is \( r \). To do this, you typically use inverse operations. For example, if the equation has addition, you would use subtraction. Similarly, if there's multiplication, you would use division. Isolating the variable helps to simplify the equation and makes it easier to find the solution.
Combine Like Terms
Combining like terms is an essential step when solving linear equations. Like terms are terms that contain the same variable raised to the same power. In our exercise, combining like terms helps to simplify the equation.
For instance, in the step 7.7r - 6 - 6.7r = 6.7r - 6 - 6.7r, we subtracted 6.7r from both sides. This left us with (7.7r - 6.7r) - 6 = -6 on the left side of the equation. Combining like terms 7.7r and -6.7r results in 1r or just r.
Simplify Both Sides
After combining like terms, it鈥檚 necessary to simplify both sides of the equation. This step makes the equation more manageable and brings it closer to the solution. In the exercise, the equation r - 6 = -6 is much simpler than the original. By keeping the equation simplified, you minimize the risk of making errors and it becomes clearer what the next steps should be. Simplifying both sides is a routine part of solving equations and is crucial for clarity.
Check the Solution
Checking the solution is the final, but equally important step in solving linear equations. This step ensures that your solution is correct. You substitute the value you found back into the original equation to see if both sides are equal. For the exercise's final step, when we substitute r = 0 back into 7.7(0) - 6 = 6.7(0) - 6, it simplifies to -6 = -6, confirming our solution is correct. This verification step helps to catch any mistakes made during the solving process and assures the accuracy of your answer.

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