Problem 48 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. A number multiplied by itself added to five is thirty-one.

Step-by-Step Solution

Answer

" The phrase can be translated into a mathematical equation as follows: Let x represent the unknown number. The phrase "multiplied by itself" means x times x, which can be written as x^2. "Added to five" means adding 5 to this expression, resulting in x^2 + 5. Finally, "is thirty-one" means the entire expression is equal to 31. Therefore, the equation translated from the phrase is x^2 + 5 = 31.

Step 1: Identify the variable

Let's denote the unknown number as "x."

Step 2: Translate 'multiplied by itself'

The phrase "multiplied by itself" means we need to multiply x by x. In mathematical terms, this is expressed as "x * x" which can also be written as x^2.

Step 3: Translate 'added to five'

Now we have to add 5 to the expression x^2. We do this by writing x^2 + 5.

Step 4: Translate 'is thirty-one'

The last part of the phrase states "is thirty-one," which means that the entire expression we've created so far is equal to 31. So, we can set our expression equal to 31 like this: x^2 + 5 = 31.

Step 5: Final equation

The final mathematical equation representing the given phrase is: x^2 + 5 = 31.

Key Concepts

Equations

An equation is like a balance scale. It's a mathematical sentence that shows that two expressions are equal. In the context of our problem, we had to translate a verbal sentence into a mathematical equation. This equation helps us solve for the unknown parts within it. For example, in the problem given, the sentence "A number multiplied by itself added to five is thirty-one" becomes the equation \( x^2 + 5 = 31 \).

• The left side of the equation describes an expression, which is "a number multiplied by itself added to five." This translates to \( x^2 + 5 \).
• The right side of the equation states the result of this expression, which is 31.

In summary, equations are essential tools used for showing relationships between different mathematical expressions, allowing us to find the values of unknown numbers.

Variable

A variable acts as a placeholder or a symbol for an unknown number in a mathematical expression or equation. It's often represented by letters such as \( x \), \( y \), or \( z \). In our exercise, the variable \( x \) stands for a number we do not initially know.

• Variables allow us to write expressions and equations in a general form, applicable to many situations or different numbers.
• They are particularly useful when solving equations, as they represent the unknown parts we aim to discover.

For the problem at hand, the unknown number was represented by \( x \). This enabled us to write the equation \( x^2 + 5 = 31 \) and ultimately solve for \( x \) to find the exact value of this number.

Translate Phrases

Translating phrases into mathematical expressions or equations is a critical skill in mathematics. It involves converting words and real-world scenarios into the language of math. This allows us to analyze and solve problems more efficiently. Let's break down the translation process from our exercise.

• "A number multiplied by itself" turns into \( x \times x \) or simply \( x^2 \).
• "Added to five" means we increase \( x^2 \) by 5, making it \( x^2 + 5 \).
• "Is thirty-one" sets the entire expression equal to 31, resulting in \( x^2 + 5 = 31 \).

With practice, translating phrases becomes straightforward, turning descriptions into solvable math problems, and eventually allowing you to find unknown quantities.

Unknown Number

The unknown number is the value we are trying to find when solving an equation. In the exercise, the unknown number was related to the sentence: "A number multiplied by itself added to five is thirty-one." Initially, we didn't know the value of this number, so we introduced the variable \( x \).

• Having an unknown means we need to set up an equation to find it, through the process called solving the equation.
• In our final equation, \( x^2 + 5 = 31 \), solving it was key to identifying the unknown number.

The unknown number became clear once we rearranged and calculated the equation to find \( x \), revealing the solution.