Problem 61
Question
Use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. The difference between \(\frac{2}{5}\) of a number and 8 is \(\frac{7}{5}\) of that number. Find the number.
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x = -8\).
1Step 1: Setting up the equation
Based on the problem statement, translate the words into an equation. 'The difference' means subtraction, 'of' means multiplication, and 'that number' refers to \(x\). So, this gives us: \(\frac{2}{5}x - 8 = \frac{7}{5}x\).
2Step 2: Rearrange the equation and simplify
To isolate \(x\) on one side of the equation, subtract \(\frac{2}{5}x\) from both sides. This gives \(-8 = \frac{7}{5}x - \frac{2}{5}x\), which simplifies to \(-8 = \frac{5}{5}x\), or \(-8 = x\).
3Step 3: Solve the equation
The result \(-8 = x\) says that the number we are looking for is \(-8\)
Key Concepts
Translating Words into an EquationAlgebraic OperationsSolving Linear Equations
Translating Words into an Equation
Understanding algebra begins with learning how to convert verbal statements into mathematical ones, which is the foundation of word problems. In the given exercise, the challenge is to 'use the given information to write an equation'.
Let's break down what this really means. Begin by identifying action words or phrases like 'the difference between' and 'of a number', which signal algebraic operations. 'The difference between' signals subtraction, and 'of' usually signals multiplication when related to a fraction of a number. As a hint, consider that when a word problem refers to 'a number', we often assign a variable like 'x' to represent it.
By interpreting these linguistic cues, we can construct an algebraic equation, which in this case is \(\frac{2}{5}x - 8 = \frac{7}{5}x\). It's essentially a translation task: turning English into the language of algebra.
Let's break down what this really means. Begin by identifying action words or phrases like 'the difference between' and 'of a number', which signal algebraic operations. 'The difference between' signals subtraction, and 'of' usually signals multiplication when related to a fraction of a number. As a hint, consider that when a word problem refers to 'a number', we often assign a variable like 'x' to represent it.
By interpreting these linguistic cues, we can construct an algebraic equation, which in this case is \(\frac{2}{5}x - 8 = \frac{7}{5}x\). It's essentially a translation task: turning English into the language of algebra.
Algebraic Operations
Algebraic operations include addition, subtraction, multiplication, and division, which can be presented in various forms within word problems. In our example, multiplication and subtraction are the key operations involved. Multiplication is indicated by the use of 'of', implying a part of a whole, particularly when we talk about fractions of a number, like \(\frac{2}{5}\) or \(\frac{7}{5}\) of 'x'.
Subtraction is clearly marked by the phrase 'the difference between'. When you see this, it's your clue to set up a subtraction equation. You should always pay close attention to these operations as they will guide how you structure your equation and manipulate algebraic expressions to solve for the variable.
Subtraction is clearly marked by the phrase 'the difference between'. When you see this, it's your clue to set up a subtraction equation. You should always pay close attention to these operations as they will guide how you structure your equation and manipulate algebraic expressions to solve for the variable.
Solving Linear Equations
Once you translate the word problem into an algebraic equation, solving it is the next step. A linear equation, like the one developed from the exercise, has one variable and involves no powers other than 1. The ultimate goal is to isolate the variable on one side of the equation to find its value.
To solve the provided equation, we perform basic algebraic operations to get the variable 'x' by itself. In this case, we subtract \(\frac{2}{5}x\) on both sides to eliminate the variable from one side and simplify the right side of the equation. The process of rearranging the equation is pivotal and requires careful attention to algebraic rules, such as maintaining equality by performing the same operation on both sides. Eventually, we reach \(x = -8\), indicating that the solution for 'x' is \-8\. This step is where the bulk of algebraic understanding is applied, ensuring we can find the correct value for 'x'.
To solve the provided equation, we perform basic algebraic operations to get the variable 'x' by itself. In this case, we subtract \(\frac{2}{5}x\) on both sides to eliminate the variable from one side and simplify the right side of the equation. The process of rearranging the equation is pivotal and requires careful attention to algebraic rules, such as maintaining equality by performing the same operation on both sides. Eventually, we reach \(x = -8\), indicating that the solution for 'x' is \-8\. This step is where the bulk of algebraic understanding is applied, ensuring we can find the correct value for 'x'.
Other exercises in this chapter
Problem 61
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