Problem 49 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. A quantity multiplied by seven plus twice itself is ninety.

Step-by-Step Solution

Answer

Answer: The unknown quantity is 10.

Step 1: Identify the unknown quantity

In the problem, the unknown quantity is referred to as "a quantity." We can represent this unknown quantity as a variable, such as x.

Step 2: Translate the phrase into an equation

The problem states that the quantity multiplied by seven plus twice itself is ninety. This can be translated to the following equation: 7x + 2x = 90.

Step 3: Solve the equation for x

To solve for x, we combine the terms with x on the left side of the equation: 9x = 90. Next, divide both sides of the equation by 9: x = 10.

Step 4: Verify the solution

To check if x = 10 is correct, plug the value of x back into the original equation: 7(10) + 2(10) = 90 70 + 20 = 90 90 = 90 Since the equation holds true, x = 10 is the correct solution, and the phrase has been successfully translated into a mathematical equation and solved.

Key Concepts

Translation of Phrases Into Mathematical Expressions

Understanding how to translate verbal phrases into algebraic expressions is key in solving real-world math problems. When given a sentence like "a quantity multiplied by seven plus twice itself is ninety," it might seem complex at first. But break it down step-by-step:

• Identify keywords: "a quantity" hints at a variable, commonly represented as \(x\) in algebra.
• Next, consider the math operations: "multiplied by seven" becomes \(7x\), and "twice itself" translates to \(2x\).
• The word "plus" indicates addition, and "is" tells us where the equation equals or is set to a specific number, here ninety.

So, the phrase "a quantity multiplied by seven plus twice itself is ninety" translates to the equation \(7x + 2x = 90\). It's all about recognizing the math symbols hidden in the words.

Solving Equations

Once you've turned your verbal math problem into an equation, the next task is to solve it. For our example, the equation is \(7x + 2x = 90\). Solving this requires a few straightforward steps:

• Simplify first: Combine like terms on one side of the equation. Here it is \(7x + 2x\) which simplifies to \(9x\).
• Then you have \(9x = 90\).
• To isolate the variable \(x\), divide both sides by the coefficient of \(x\), which is 9 in this case.
• This results in \(x = 10\), giving you the solution to the equation.

The goal here is to simplify and isolate the variable, solving for what "a quantity" actually is.

Variable Representation

Using variables, like \(x\), is an essential skill in algebra. They help represent the unknowns and make expressions easier to work with. Here's how to effectively use them:

• Choose a variable to stand for the unknown quantity in your problem.
• In the phrase given, "a quantity" is unknown, so we represent it as \(x\).
• This variable acts as a placeholder, allowing you to construct the equation based on the phrase.

Effective variable representation is crucial because it transforms a word problem into something solvable. By representing the unknown with \(x\), you're better able to manipulate and solve the equation, leading to a clearer understanding of the problem at hand.