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Chapter 2: Problem 17

Solve each equation, and check your solution. $$ 7+r=-3 $$

Short Answer

Expert verified
The solution is \( r = -10 \), and it checks out.

Step by step solution

01

Isolate the variable

To solve for the variable \( r \), we need to isolate it on one side of the equation. We can do this by subtracting 7 from both sides of the equation. \[ 7 + r - 7 = -3 - 7 \]
02

Simplify the equation

After subtracting 7 from both sides, simplify to get:\[ r = -10 \]
03

Check the solution

To check the solution, substitute \( r = -10 \) back into the original equation: \[ 7 + (-10) = -3 \] Simplify the left side: \[ -3 = -3 \] Since both sides of the equation are equal, the solution \( r = -10 \) 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 goal is to get the variable, in this case, 'r', by itself on one side of the equation. To do this, we perform operations that will remove any other numbers or variables attached to 'r'.
In the equation, \(7 + r = -3\), 'r' is on the left side with 7 added to it. To isolate 'r', you need to subtract 7 from both sides of the equation. This is because subtracting cancels out the addition, leaving 'r' alone:
\(7 + r - 7 = -3 - 7\).
Once you perform this operation, you get:\br\(r = -10\).
Now, the variable 'r' is isolated, and you can see that \(r = -10\).
Following this process is critical because it helps you clearly see what the variable equals.
Simplify the Equation
After you isolate the variable, you need to simplify the equation. Simplifying means performing the arithmetic operations to reduce the equation to a simpler state. In our example, after isolating 'r', we have:
\(r = -10\).
This step is straightforward because there are no further operations to perform. The equation is already simplified, and 'r' equals -10.
Simplifying helps ensure that the equation looks as straightforward as possible and that the solution makes sense. It鈥檚 a good practice to double-check that all calculations are correct. This way, there will be no mistakes in the final simplified result.
Check the Solution
After solving the equation, always check your solution to ensure it's correct. To do this, substitute your solution back into the original equation. For the equation \(7 + r = -3\), we found that \(r = -10\).
Substitute -10 in place of 'r':
\(7 + (-10) = -3\).
Now, simplify the left side:
\(7 - 10 = -3\).
Both sides of the equation are equal, so we confirm that the solution \(r = -10\) is correct.
Checking your solution is important to catch any mistakes you may have made during calculations. It ensures that your answer satisfies the original equation and is, therefore, the right solution.

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