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Chapter 2: Problem 3

Solve each equation and check. $$ -10=x+12 $$

Short Answer

Expert verified
The solution is \(x = -22\).

Step by step solution

01

Isolate the Variable

To solve the equation \(-10 = x + 12\), we need to isolate \(x\). Start by subtracting \(12\) from both sides of the equation.
02

Simplify the Equation

Subtract \(12\) from both sides:\[-10 - 12 = x + 12 - 12\]This simplifies to:\[-22 = x\]Therefore, \(x = -22\).
03

Verify the Solution

To check if \(x = -22\) is a correct solution, substitute \(-22\) back into the original equation.\[-10 = (-22) + 12\]Calculate the right-hand side:\(-22 + 12 = -10\).Since both sides match, the solution \(x = -22\) is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Variable
When solving linear equations, the first step is often to isolate the variable. In this exercise, we want to solve for the unknown variable, which is \(x\). To "isolate" a variable means to get it alone on one side of the equation. This makes it easy to see what \(x\) equals.To begin, look at the equation: \(-10 = x + 12\). Here, \(x\) is accompanied by \(+12\). To isolate \(x\), we need to remove the \(12\) from the right side. You do this by performing the opposite operation: subtraction. So, subtract \(12\) from both sides of the equation:
  • Left side: \(-10 - 12\)
  • Right side: \(x + 12 - 12\)
By doing this, you keep the equation balanced. This results in the new equation: \(-22 = x\). Congratulations, you have isolated \(x\)!
Simplifying the Equation
After isolating the variable, the next step is to simplify the equation fully. Simplification makes the equation neat and easy to understand. In this exercise, when you subtract \(12\) from both sides, you simplify the terms. Here's how it looks using the given values:Starting with \(-10 - 12\):
  • Calculate the sum: \(-22\)
This simplifies the equation to \(x = -22\). Simplifying the equation is crucial because it shows exactly what \(x\) equals. Simplification removes any extra terms or numbers, leaving you with a clean and final solution. This streamlined approach lets you easily verify the solution in the next step.
Verifying the Solution
The final step in solving an equation is to verify the solution. This ensures your solution is correct. Verification involves substituting the solution back into the original equation to see if it holds true. For the equation \(-10 = x + 12\) and solution \(x = -22\), substitute \(-22\) for \(x\):
  • Substitute \(-22\) on the right side: \(-22 + 12\)
  • Calculate it: \(-10\)
Now, compare both sides of the original equation:
  • Left side: \(-10\)
  • Right side: \(-10\)
Since both sides are equal, the solution \(x = -22\) is verified as correct. Verification is a reliable way to build confidence in your mathematical solutions and ensures accuracy.

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