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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 \( r = -10 \).

Step by step solution

01

Isolate the variable

To solve the equation \text{ } \(7 + r = -3\), isolate the variable \(r\) by subtracting \(7\) from both sides of the equation: \text{ } \( 7 + r - 7 = -3 - 7\).
02

Simplify the equation

Now simplify the equation by performing the subtraction: \text{ } \( r = -10\).
03

Check the solution

To verify the solution, substitute \(r = -10\) back into the original equation: \text{ } \( 7 + (-10) = -3\). Simplify: \text{ } \( -3 = -3\). \text{ } 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.

Isolating Variables
When solving linear equations, our first goal is to isolate the variable we are solving for. In the given example, the variable is \( r \). To isolate it:
- Look at the equation: \( 7 + r = -3 \).
- To isolate \( r \), we need to remove the \( 7 \) from the left side of the equation. We do this by performing the opposite operation of what is given. Since \( 7 \) is added to \( r \), we subtract \( 7 \) from both sides.
- This gives us: \( 7 + r - 7 = -3 - 7 \), which simplifies to: \( r = -10 \).
By isolating the variable, we transform the equation into a simpler form. This basic technique is crucial for solving any kind of linear equation.
Checking Solutions
Once we have isolated the variable and obtained a solution, it's important to check if our solution is correct. We do this by substituting our solution back into the original equation.
For our example, the solution we found is \( r = -10 \). Let's substitute it back into the original equation:
- The original equation: \( 7 + r = -3 \).
- Substitute \( r = -10 \): \( 7 + (-10) = -3 \).
- Simplify the left side: \( -3 = -3 \).
Since both sides of the equation are equal, our solution \( r = -10 \) is verified as correct.
This step ensures that no mistakes were made during the isolation of the variable and the simplification process.
Simplifying Equations
Simplifying equations is a key step in solving them effectively. Here鈥檚 how it works:
- After isolating the variable, we often need to perform basic arithmetic operations to simplify the equation further.
- In our example, once we subtract \( 7 \) from both sides, we get \( r = -10 \).
- This is already simplified, but sometimes equations might need further steps like combining like terms or distributing multiplication operations.
Simplification makes complex equations more manageable and helps in understanding the relationships between the variables involved. Always ensure each step follows logically to maintain the equation's balance.

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