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Chapter 2: Problem 146

Translate to an equation and then solve. \(u\) divided by 7 is equal to -49.

Short Answer

Expert verified
u = -343

Step by step solution

01

- Translate the sentence to an equation

The problem says that the variable \(u\) divided by 7 is equal to -49. So, write this as an equation: \( \frac{u}{7} = -49 \).
02

- Multiply both sides by 7

To solve for \(u\), multiply both sides of the equation \( \frac{u}{7} = -49 \) by 7: \( u = -49 \times 7 \).
03

- Calculate the value of \(u\)

Now, perform the multiplication: \( -49 \times 7 = -343 \). So, \( u = -343 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translating Word Problems
Understanding how to translate word problems into algebraic equations is crucial. It involves interpreting the text and recognizing the mathematical operations implied.
In the given exercise, 'u divided by 7 is equal to -49', the phrase 'u divided by 7' translates to the fraction \( \frac{u}{7} \), and 'is equal to' translates to the equal sign \( = \). Therefore, we write the equation as \( \frac{u}{7} = -49 \).
Recognize key terms:
  • 'divided by' indicates a division operation.
  • 'is equal to' shows equivalence.
By identifying these key phrases, you can set up the algebraic equation correctly. This skill is fundamental since word problems often describe real-world scenarios requiring mathematical expressions.
Multiplication of Integers
In algebra, solving equations often involves operations like multiplication. When we need to isolate a variable, we might multiply both sides of an equation by the same number. This ensures that the equation remains balanced.
For instance, in our problem, we have \( \frac{u}{7} = -49 \). To find \( u \), we multiply both sides by 7.
Multiplication of integers follows these simple rules:
  • When multiplying a positive number by a negative number, the result is negative.
  • When multiplying two negative numbers, the result is positive.
In the exercise, since \( -49 \) is negative and \( 7 \) is positive, their product is negative:
\[ -49 \times 7 = -343 \]
Hence, \( u = -343 \). Remember, understanding these rules helps you simplify and solve equations accurately.
Algebraic Equations
Algebraic equations are mathematical statements that show the equality between two expressions. They often include variables, which are symbols representing unknown values.
The goal is to manipulate the equation to solve for the unknown variable. This involves performing operations like addition, subtraction, multiplication, and division on both sides of the equation.
In our example, the equation \( \frac{u}{7} = -49 \) involves the variable \( u \). To solve for \( u \), we used multiplication as follows:
  • First, identify the operation needed to isolate \( u \). Here, we need to get rid of the division by 7, so we multiply both sides by 7.
  • Next, calculate the result of the multiplication to find the value of \( u \): \( u = -343 \).
By following these steps, you can solve most basic algebraic equations, ensuring you understand each operation's effect on the equation.

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