Problem 76 - Elementary Algebra

Question

Translate the following sentences into linear equations and then solve. Twelve times \(x\) is equal to 36 .

Step-by-Step Solution

Answer

The solution for \(x\) is 3.

Step 1: Translate the Sentence into an Equation

To translate the sentence "Twelve times \(x\) is equal to 36" into a linear equation, identify the multiplication and equality. The phrase "twelve times \(x\)" translates to \(12x\) and "is equal to 36" translates to "= 36". Thus, the equation is \(12x = 36\).

Step 2: Solve the Equation for \(x\)

Now, solve the equation \(12x = 36\) for \(x\). To isolate \(x\), divide both sides of the equation by 12:\[x = \frac{36}{12}\]

Step 3: Simplify the Equation

Simplify the division on the right side of the equation:\[x = 3\]

Key Concepts

Solving Equations

When it comes to solving equations, the goal is to isolate the variable, often represented by letters like \( x \). This requires a series of logical operations until the unknown variable is on one side of the equation, and a known value is on the other. Let's see how this works with the example "Twelve times \( x \) equals 36."
First, translate the sentence into an equation: \( 12x = 36 \). Solving this equation involves dividing both sides by 12 to get \( x \) by itself:

• Divide both sides by 12: \( x = \frac{36}{12} \)
• Simplify the division to find \( x = 3 \)

Breaking down each operation, step by step, helps simplify the equation and ultimately solve the problem. Knowing how to manipulate both sides of an equation using operations like addition, subtraction, multiplication, and division is crucial for success in solving linear equations.

Algebra Problems

Algebra problems can initially seem complex, but they are essentially puzzles where you find the unknown value. In this exercise, "Twelve times \( x \) equals 36," we set up a linear equation that represents the problem. The equation is \( 12x = 36 \). This is a straightforward introduction to algebra and linear equations, where the relationship between numbers and operations is used to find what \( x \) represents.
Consider these elements of algebra problems:

• Numbers are constants or fixed values.
• Letters (like \( x \)) represent unknown values you need to find.
• Arithmetic operations express relationships between the numbers.
• Equations are balanced statements where both sides are equal.

Understanding these components ensures you can approach any algebraic problem logically and systematically.

Mathematical Translation

Mathematical translation is the process of converting words into mathematical expressions and equations. This skill is crucial in making math problems understandable and solvable using numerical methods. For instance, with the phrase "Twelve times \( x \) is equal to 36," you need to translate it into the algebraic form \( 12x = 36 \).
Here's how you can do this effectively:

• Identify key terms like "times," "plus," or "equals" which indicate operations.
• Convert phrases into mathematical symbols: "times" becomes multiplication (\( \times \)), and "is equal to" becomes \( = \).
• Recognize that the number after these keywords becomes part of the equation.

Effective translation from words to equations is essential for solving real-world problems using math. This ensures that the mathematical model accurately represents the given scenario, making it much simpler to find a solution.