Problem 91
Question
Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between 6 times a number and \(-5\) times the number.
Step-by-Step Solution
Verified Answer
The required algebraic expression is \(11x\).
1Step 1: Translation Step
Translate the English phrase into an algebraic expression. '6 times a number' can be represented as \(6x\). 'Negative 5 times the number' can be represented as \(-5x\). The 'difference between' these two expressions signifies subtraction, so the full expression becomes \(6x - (-5x)\).
2Step 2: Simplification Step
Simplify the expression by performing the subtraction. Remember that subtracting a negative number is the same as adding a positive number, so \(6x - (-5x)\) simplifies to \(6x + 5x\). This in turn simplifies further to \(11x\), as both terms could be combined because they are alike, which are terms that contain the similar variable, in this case 'x'.
Key Concepts
SimplificationTranslation of PhrasesVariables in Algebra
Simplification
Simplification in algebra refers to reducing expressions to their simplest form. It's like cleaning up a mess to see the clear picture underneath.
For instance, consider the expression from our exercise: \(6x - (-5x)\). At first glance, it looks a bit complex with parentheses and negative signs. However, there's a simple rule in algebra that helps: "subtracting a negative number is the same as adding a positive number."
This means we can change \(6x - (-5x)\) to \(6x + 5x\). Once we have this new setup, the next step is to combine like terms. This is where both parts of the expression have the same variable, 'x' in this case. Combining yields \(11x\).
Always aim to simplify wherever possible to make equations easier to handle!
For instance, consider the expression from our exercise: \(6x - (-5x)\). At first glance, it looks a bit complex with parentheses and negative signs. However, there's a simple rule in algebra that helps: "subtracting a negative number is the same as adding a positive number."
This means we can change \(6x - (-5x)\) to \(6x + 5x\). Once we have this new setup, the next step is to combine like terms. This is where both parts of the expression have the same variable, 'x' in this case. Combining yields \(11x\).
Always aim to simplify wherever possible to make equations easier to handle!
Translation of Phrases
Translation of phrases in mathematics is about converting words into numbers and symbols. This skill is crucial because it allows you to write equations that you can then solve practically.
In this exercise, we start with the phrase: 'the difference between 6 times a number and -5 times a number.' Breaking this down, '6 times a number' translates to \(6x\), and '-5 times a number' translates to \(-5x\).
To find the "difference," you need to subtract the second expression from the first. In a mathematical sense, this translates to \(6x - (-5x)\).
Understanding how to translate phrases is key because it's the first step in tackling word problems and applying mathematical logic to real-life situations.
In this exercise, we start with the phrase: 'the difference between 6 times a number and -5 times a number.' Breaking this down, '6 times a number' translates to \(6x\), and '-5 times a number' translates to \(-5x\).
To find the "difference," you need to subtract the second expression from the first. In a mathematical sense, this translates to \(6x - (-5x)\).
Understanding how to translate phrases is key because it's the first step in tackling word problems and applying mathematical logic to real-life situations.
Variables in Algebra
Variables are symbols, usually letters like 'x', that stand in for unknown numbers in mathematical equations. They are a fundamental part of algebra.
In our exercise, the variable 'x' represents an unknown number we are working with. Think of 'x' as a placeholder that could be any number we are trying to find or calculate with.
Variables help create general formulas that are applicable in many situations. For example, when you see an expression like \(6x\), it means '6 times whatever x stands for'. As a result, algebra becomes highly flexible and powerful in problem-solving.
Grasping the concept of variables allows you to transition easily from concrete problems to abstract thinking, which is a crucial skill in higher-level mathematics.
In our exercise, the variable 'x' represents an unknown number we are working with. Think of 'x' as a placeholder that could be any number we are trying to find or calculate with.
Variables help create general formulas that are applicable in many situations. For example, when you see an expression like \(6x\), it means '6 times whatever x stands for'. As a result, algebra becomes highly flexible and powerful in problem-solving.
Grasping the concept of variables allows you to transition easily from concrete problems to abstract thinking, which is a crucial skill in higher-level mathematics.
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