Problem 78 - Elementary Algebra

Question

Translate the following sentences into linear equations and then solve. Six subtracted from two times a number \(x\) is 12 .

Step-by-Step Solution

Answer

The number \(x\) is 9.

Step 1: Translate the sentence

First, let's translate the sentence into a mathematical equation. The problem states 'Six subtracted from two times a number \(x\)'. In mathematical terms, this is represented as \(2x - 6\). This expression is equal to 12, so our full equation becomes \(2x - 6 = 12\).

Step 2: Add 6 to both sides

To isolate the term involving \(x\), we need to eliminate the \(-6\) by adding 6 to both sides of the equation: \(2x - 6 + 6 = 12 + 6\). This simplifies to \(2x = 18\).

Step 3: Divide by 2

Now, to solve for \(x\), divide both sides of the equation by 2: \(\frac{2x}{2} = \frac{18}{2}\). This gives us \(x = 9\).

Key Concepts

solving equations

Solving equations is like unraveling a mystery. We start with a jumble of mathematical terms and use logic to simplify and solve for the unknown. This unknown in our problem is represented by the variable \(x\). The objective is to find what value of \(x\) makes the equation true.
The process generally involves a few key steps:

• Isolate the variable: Try to get the variable by itself on one side of the equation. This often means moving other numbers around, using basic operations like addition, subtraction, multiplication, or division.
• Simplify the equation: Perform any arithmetic needed to simplify the equation to its simplest form.
• Check your solution: It's always a good idea to substitute your solution back into the original equation to verify it works.

When we applied these steps to our problem, we quickly found that \(x = 9\). This tells us that substituting 9 for \(x\) in the equation satisfies the condition given in the original sentence.

algebraic expressions

Algebraic expressions form the backbone of mathematical equations in this context. They are combinations of numbers, variables, and arithmetic operations. Understanding how to manipulate these expressions is crucial for solving equations effectively.
In our example, the expression \(2x - 6\) includes:

• Numerical coefficient: The number \'2\' in front of \(x\) is a coefficient. It tells us that \(x\) is multiplied by 2.
• Variable: \(x\) is the unknown that we're solving for. It represents a number we need to discover.
• Constant: \'6\' is a constant, a fixed number subtracted from the product \(2x\).

Mastering these elements allows you to set up equations correctly and solve them with ease. Remember, each part of an expression has a role in forming the equation, which is especially important in algebraic calculations.

equation translation

Equation translation involves converting everyday language statements into mathematical equations. This skill is fundamental for solving word problems like our exercise.
The key is to understand the relationship described in the sentence and express it mathematically:

• Identify mathematical operations: Words like "subtracted from" or "times" indicate subtraction and multiplication, respectively.
• Order matters: Pay attention to how words are arranged. "Six subtracted from two times a number \(x\)" translates to \(2x - 6\), not \(6 - 2x\).
• Equivalence: The word "is" often signals equality, setting up the two sides of an equation.

By mastering this translation skill, you'd convert any word problem into an equation effortlessly. This step may seem tricky, but with practice, it becomes more straightforward. In our problem, translating was the first critical step towards finding that \(x = 9\).