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Chapter 2: Problem 7

Solve each equation, and check your solution. $$ x-12=19 $$

Short Answer

Expert verified
x = 31

Step by step solution

01

Isolate the variable

To isolate the variable, add 12 to both sides of the equation.
02

Perform the addition

Adding 12 to both sides yields: \[ x - 12 + 12 = 19 + 12 \] Simplifying this: \[ x = 31 \]
03

Check the solution

Substitute the solution back into the original equation to verify: \[ 31 - 12 = 19 \] This simplifies to: \[ 19 = 19 \] which is a true statement.

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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 goal is to isolate the variable on one side of the equation. This makes it easier to see what the variable equals.
In the given exercise, we start with: \[ x - 12 = 19 \]To isolate \( x \), we need to remove the \( -12 \) from the left side. To do this, we perform the opposite operation, which is addition. By adding 12 to both sides, we ensure that the equation remains balanced.
Addition Property of Equality
The Addition Property of Equality states that if you add the same amount to both sides of an equation, the two sides remain equal. This property is useful for maintaining balance while solving equations.
Using the addition property on our equation: \[ x - 12 + 12 = 19 + 12 \] Simplifying both sides, we get: \[ x = 31 \]
This is a direct application of the addition property and shows how it helps to isolate the variable.
Checking Solutions
After finding the solution, it is essential to check if it is correct by plugging it back into the original equation.
Let's take our solution \( x = 31 \) and substitute it back into the original equation: \[ 31 - 12 = 19 \]Simplify the left side: \[ 19 = 19 \]
The left side equals the right side, which means our solution is correct. This step confirms that we have accurately solved the equation. Always check your answers to ensure they satisfy the original equation.

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