Problem 59 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ x+12=10 $$

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Identify the Equation Structure

The equation given is in the form of a simple linear equation: \( x + 12 = 10 \). Our goal is to isolate \( x \) to find its value.

Step 2: Isolate the Variable

To solve for \( x \), we need to isolate it on one side of the equation. This can be done by subtracting 12 from both sides of the equation.

Step 3: Perform the Subtraction

Subtract 12 from both sides of the equation: \( x + 12 - 12 = 10 - 12 \). Simplifying both sides, we get \( x = -2 \).

Step 4: Verify the Solution

Check the solution by substituting \( x = -2 \) back into the original equation: \( -2 + 12 = 10 \), which simplifies to \( 10 = 10 \). Thus, the solution is verified.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that represent a straight line when plotted on a graph. They typically involve one or more variables raised to the power of one. The general structure of a simple linear equation can be described as follows:

• It ideally takes the form of \( ax + b = c \).
• The term \( x \) is the variable we want to solve.
• \( a \), \( b \), and \( c \) are coefficients and constants that are known values.

In the example equation \( x + 12 = 10 \), our coefficients and constants are easily identifiable: \( a = 1 \), \( b = 12 \), and \( c = 10 \). Linear equations form the backbone of algebra, serving as a foundation for more complex equations and systems. They are pervasive in various real-world applications, where they model relationships between quantities that have a constant rate of change.

The Art of Isolating Variables

Isolating variables is a crucial skill in solving linear equations. The goal here is to get the unknown, \( x \), by itself on one side of the equation. To do this, inverse operations are used to eliminate other components. Here's how in three intuitive steps:

• Identify the Term to Remove: Spot the terms not involving \( x \). In \( x + 12 = 10 \), our target is \( +12 \).
• Apply the Inverse Operation: Use subtraction because it's the opposite of addition. Subtract \( 12 \) from both sides: \( x + 12 - 12 = 10 - 12 \).
• Simplify the Equation: Once the inverse operation is applied, the problematic term vanishes. We are left with \( x = -2 \).

By consistently following this method, solving for any linear equation becomes an accessible task, ensuring that students grasp the process intuitively.

Equation Verification Techniques

Verifying your solution is an essential step in the problem-solving process. By ensuring your solution is correct, you increase confidence and reinforce learning. Let's look at how verification is done for \( x = -2 \) in the equation \( x + 12 = 10 \):

• Substitute Back into the Original: Replace \( x \) in the original equation with your solution. For this example, it transforms the equation to \( -2 + 12 \).
• Compute and Compare: After substitution, the expression becomes \( 10 \). Check if this equals the right-hand side of the original equation, \( 10 \). Since they match, your solution is indeed correct.
• Understand Mistakes Quickly: If a mismatch occurs, go through your steps to find a potential error quickly, enhancing accuracy.

Verification not only serves as a check for mistakes but also aids in building a strong understanding of linear equations. Ironclad verification means the solution withstands scrutiny, boosting problem-solving skills.