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Chapter 8: Problem 17

In the following exercises, solve each equation. $$ x-15=-42 $$

Short Answer

Expert verified
x = -27.

Step by step solution

01

Identify the Equation

Recognize the given equation: Equation: x - 15 = -42.
02

Isolate the Variable

To isolate the variable term, add 15 to both sides of the equation. : x - 15 + 15 = -42 + 15. This simplifies to: x = -27.
03

Solve for the Variable

Combine the constants on the right side: x = -42 + 15 results in x = -27.
04

Verify the Solution

Substitute x = -27 back into the original equation to confirm: x - 15 = -42. -27 - 15 = -42 is true.

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Key Concepts

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

Isolating Variables
In mathematics, isolating the variable is a crucial step in solving linear equations. This means getting the variable by itself on one side of the equation.
To achieve this, we often need to undo any operations being performed on the variable.
In our example equation, \( x - 15 = -42 \), the variable \( x \) is being subtracted by 15.
To isolate \( x \), we need to get rid of the -15.
How do we do that? We add 15 to both sides of the equation.
Why both sides? Because if we add 15 to one side only, the equation won't balance anymore, and the equality will no longer hold true.
Addition Property of Equality
The Addition Property of Equality is a fundamental principle in algebra. It states that if you add the same number to both sides of an equation, the two sides remain equal.
Using this property helps preserve the balance of the equation. Let's revisit our equation: \( x - 15 = -42 \).
To apply this property, we add 15 to both sides: \( x - 15 + 15 = -42 + 15 \).
This simplifies to \( x = -27 \).
Notice how adding 15 cancels out the -15 on the left side.
This operation brings us closer to having the variable isolated.
Verification of Solutions
Verification is the final step to ensure our solution is correct.
To verify, we substitute the solved value of the variable back into the original equation.
Let's substitute \( x = -27 \) into the original equation \( x - 15 = -42 \).
Replacing \( x \) with -27: \( -27 - 15 = -42 \).
Solving this, we get \( -42 = -42 \), which is true.
This confirms that our solution is correct and that \( x = -27 \) satisfies the original equation.
Verification is a good practice because it ensures the calculated solution is indeed satisfying the given equation.

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