Problem 16
Question
Write the phrase as a variable expression. Let x represent the number. 15 increased by a number
Step-by-Step Solution
Verified Answer
The variable expression is \(15 + x\).
1Step 1: Understand the phrase
In the phrase, '15 increased by a number', 'increased by' means addition. The number 15 is being increased and the amount of increase is represented as 'x'.
2Step 2: Translate to mathematical expression
Having understood the meaning of the phrase, it can be written as a mathematical expression: 15 + x
Key Concepts
Algebraic ExpressionsAdditionTranslating Verbal Phrases to Expressions
Algebraic Expressions
Algebraic expressions are a fundamental part of algebra and mathematics as a whole. They involve combining numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.
Variables in these expressions serve as placeholders for unknown values, much like in a puzzle where we try to learn the value of these unknowns. For instance, in the phrase "15 increased by a number," the "number" is represented by the variable 'x'.
By using variables, algebraic expressions become powerful tools that can model real-life situations, allowing us to systematically solve problems that involve unknown quantities. In our example, the expression becomes "15 + x," representing a scenario where 15 is increased by an unknown number 'x'.
This expression can change as the value of 'x' changes, making algebraic expressions dynamic and versatile. They form the backbone of equations and inequalities, helping to solve for unknowns in various contexts.
Variables in these expressions serve as placeholders for unknown values, much like in a puzzle where we try to learn the value of these unknowns. For instance, in the phrase "15 increased by a number," the "number" is represented by the variable 'x'.
By using variables, algebraic expressions become powerful tools that can model real-life situations, allowing us to systematically solve problems that involve unknown quantities. In our example, the expression becomes "15 + x," representing a scenario where 15 is increased by an unknown number 'x'.
This expression can change as the value of 'x' changes, making algebraic expressions dynamic and versatile. They form the backbone of equations and inequalities, helping to solve for unknowns in various contexts.
Addition
Addition is one of the most basic arithmetic operations, essential for understanding how numbers interact in algebraic expressions.
When you hear terms like "increased by" in algebraic contexts, it translates into addition. For example, in our phrase "15 increased by a number," the increase operation suggests adding 15 to an unknown value, represented by the variable 'x'.
Mathematically, this is expressed as \[15 + x\].
Addition is commutative, meaning that the order in which you add numbers doesn't matter — \[x + 15\] equals \[15 + x\]. This property is useful because it allows for flexibility in how you approach adding numbers and variables.
Understanding addition as a process of combining values enhances problem-solving capabilities, making it easier to handle more complex expressions and equations.
When you hear terms like "increased by" in algebraic contexts, it translates into addition. For example, in our phrase "15 increased by a number," the increase operation suggests adding 15 to an unknown value, represented by the variable 'x'.
Mathematically, this is expressed as \[15 + x\].
Addition is commutative, meaning that the order in which you add numbers doesn't matter — \[x + 15\] equals \[15 + x\]. This property is useful because it allows for flexibility in how you approach adding numbers and variables.
Understanding addition as a process of combining values enhances problem-solving capabilities, making it easier to handle more complex expressions and equations.
Translating Verbal Phrases to Expressions
Translating verbal phrases into algebraic expressions is a critical skill in mathematics that bridges language and math.
Words often give clues about the mathematical operations needed. For example, phrases like "increased by" clearly point towards addition, whereas "decreased by" would indicate subtraction.
In the exercise provided, we see the phrase "15 increased by a number" being translated to the expression \[15 + x\].
The key to successful translation is to understand the vocabulary and the mathematical operation it represents. Terms like "sum," "total," "more than," or "plus" are all synonymous with addition. To translate effectively:
Words often give clues about the mathematical operations needed. For example, phrases like "increased by" clearly point towards addition, whereas "decreased by" would indicate subtraction.
In the exercise provided, we see the phrase "15 increased by a number" being translated to the expression \[15 + x\].
The key to successful translation is to understand the vocabulary and the mathematical operation it represents. Terms like "sum," "total," "more than," or "plus" are all synonymous with addition. To translate effectively:
- Identify the numbers and the unknown quantity in the phrase.
- Determine the operation described by keywords.
- Write the mathematical expression using the identified numbers, variables, and operations.
Other exercises in this chapter
Problem 16
Evaluate the variable expression when x = 3. $$ \frac{24}{x} \cdot 5 $$
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Check to see if \(a=5\) is or is not a solution of the equation. $$ \frac{40}{a}=8 $$
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Determine whether the table represents a function. $$ \begin{array}{|c|c|} \hline \text { input } & {\text { Output }} \\ \hline 1 & {2} \\ \hline 3 & {3} \\ \h
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An appliance store sells two stereo models. The model without a CD player is \(\$ 350 .\) The model with a CD player is \(\$ 480 .\) Your summer job allows you
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