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Chapter 2: Problem 10

Solve each equation, and check your solution. $$ x-5=-7 $$

Short Answer

Expert verified
x = -2

Step by step solution

01

- Add 5 to both sides

To isolate the variable x, add 5 to both sides of the equation to eliminate the -5 on the left side:\(x - 5 + 5 = -7 + 5\)
02

- Simplify both sides

Simplify the equation to solve for x:\(x = -2\)
03

- Check the solution

Substitute \(x = -2\) back into the original equation to verify the solution:\(-2 - 5 = -7\)Both sides of the equation are equal, confirming 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 variables
When solving linear equations, one of the first steps is often to isolate the variable you're solving for. This means getting the variable by itself on one side of the equation. In the example equation given \(x - 5 = -7\), we want to isolate \(x\).

Here鈥檚 what you can do:
  • Identify what is being done to the variable. In this case, 5 is being subtracted from \(x\).
  • Perform the opposite operation to both sides of the equation to cancel it out. Since 5 is subtracted, you need to add 5 to both sides. This gives: \(x - 5 + 5 = -7 + 5\).
  • Simplify the result. On the left side, \(-5 + 5\) cancels out, leaving \(x\). On the right, \(-7 + 5\) gives \(-2\).
Now the equation simplifies to: \(x = -2\). Congratulations, you've isolated the variable!
verification of solution
After isolating the variable and finding a solution, it's crucial to verify that your solution is correct. This step ensures that there has been no mistake in your calculations. To verify the solution for the equation \(x - 5 = -7\), we substitute the value we found for \x\ back into the original equation:

  • We determined that \(x = -2\).
  • Replace \(x\) in the original equation with \(-2\). This gives: \(-2 - 5\).
  • Now simplify that expression: \(-2 - 5 = -7\).
Since both sides of the equation are equal, \(-7 = -7\), the solution \(x = -2\) is verified as correct.
basic algebra steps
Mastering basic algebra steps is essential for solving equations efficiently and accurately. Let's break down the common steps you should follow:

  • Identify the operations affecting the variable (addition, subtraction, multiplication, division).
  • Use inverse operations to isolate the variable on one side of the equation.
  • Simplify both sides as necessary. This might involve combining like terms or simplifying numbers.
  • Once the variable is isolated and you've found its value, substitute it back into the original equation.
  • Simplify the equation again to ensure both sides match. This verifies your solution.
In our example equation \(x - 5 = -7\), the steps were:
  • Add 5 to both sides to eliminate \(-5\).
  • Simplify to \(x = -2\).
  • Substitute \(-2\) back into the original equation to check: \(-2 - 5 = -7\).
Following these basic algebra steps will make solving linear equations straightforward and help build a strong mathematical foundation.

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