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Chapter 2: Problem 56

Solve. $$ 12 x+13=25 $$

Short Answer

Expert verified
x = 1

Step by step solution

01

Subtract 13 from Both Sides

Start by isolating the term with the variable on one side. Subtract 13 from both sides of the equation to eliminate the constant term on the left side:\[12x + 13 - 13 = 25 - 13\]This simplifies to:\[12x = 12\]
02

Divide Both Sides by 12

To solve for \(x\), divide both sides of the equation by 12:\[\frac{12x}{12} = \frac{12}{12}\]This will give:\[x = 1\]

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Key Concepts

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

Isolation of Variables
In solving linear equations, one of the primary steps is isolating the variable. This means rearranging the equation so that the variable you're solving for is by itself on one side of the equality. This is crucial because it allows you to see exactly what the variable equals. To isolate the variable, follow these steps:
  • Identify the term that contains the variable. In our equation, this is 12x.
  • Use inverse operations to move other terms to the opposite side. Here, we subtract 13 from both sides to eliminate the constant from the left, simplifying the equation to 12x = 12.
Remember, isolation of variables sets the stage for understanding the relationship between the variable and the numbers involved.
Equation Simplification
To simplify an equation means to reduce it to its most basic form without changing the equality. Proper simplification ensures that the equation is easier to solve and understand. During equation simplification, two key types of simplifications occur:
  • Combining Like Terms: This step involves combining terms on each side of the equation, though our exercise doesn鈥檛 require this as we start with only two terms on each side.
  • Removing Constants: We achieve this by subtracting or adding values to both sides, such as when we subtracted 13 from each side to simplify 12x + 13 = 25 to 12x = 12.
Simplification not only makes the equation easier to handle but ensures the integrity of mathematical operations when solving for the unknown.
Arithmetic Operations
Arithmetic operations are the basic building blocks of solving equations. They include addition, subtraction, multiplication, and division. Each of these operations has its role when solving for a variable, but they must be performed correctly to maintain the balance of the equation. In our exercise:
  • We used subtraction to remove the constant: Subtracting 13 from both sides helps us focus on the term containing the variable, 12x.
  • We used division to solve for the variable: By dividing both sides of the simplified equation 12x = 12 by 12, we obtained the value of x, which is 1.
Performing arithmetic operations properly is crucial to solving any equation efficiently and correctly. They are tools that help transform the equation into a more direct representation of the unknown variable.

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