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Chapter 3: Problem 479

In the following exercises, translate and solve. The product of \(-6\) and \(y\) is \(-42\)

Short Answer

Expert verified
The solution is \(y = 7\).

Step by step solution

01

Understanding the Problem

We need to translate the given statement 'The product of \(-6\) and \(y\) is \(-42\)' into an algebraic equation.
02

Write the Equation

Translate the words into a mathematical equation: The product of \(-6\) and \(y\) means \(-6 \times y\).So, the equation is: \(-6y = -42\)
03

Solve for y

To solve for \(y\), divide both sides of the equation by \(-6\): \( \frac{-42}{-6} = y\)This simplifies to: \(y = 7\)
04

Verify the Solution

To ensure the solution is correct, substitute \(y = 7\) back into the original equation: \(-6 \times 7 = -42\).Since the left-hand side equals the right-hand side, \(y = 7\) is the correct solution.

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Key Concepts

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

algebraic translation
Algebraic translation is the process of converting spoken or written statements into mathematical expressions. This is an essential skill for solving word problems in algebra.
In the given exercise, the statement is 'The product of \(-6\) and \(-42 \)'. Here, 'product' means multiplication, and 'is' serves as the symbol for the equals sign \( = \).
Translating 'The product of \(-6 \) and \(-42 \)' into an algebraic equation gives us: \(-6y = -42 \).
Breaking this down,
  • 'The product of' tells us to multiply.
  • \(-6 \) and y are the factors (the numbers we multiply).
  • 'is' indicates that the result equals \(-42 \).
This translation step is crucial in understanding how to form and solve algebraic equations from word problems.
product in algebra
The term 'product' in algebra refers to the result of multiplying two or more numbers or variables. Understanding this concept helps in interpreting and forming equations.
In our exercise, the concept of product is represented in the multiplication operation between \(-6 \) and \(\text{y} \), reflected as \(-6 \) \( \times \) \( \text{y} \) or simply \(-6y \).
Here are a few points:
  • The term 'product' signals multiplication.
  • The factors in our case are \(-6 \) and \( \text{y} \).
  • This multiplication yields the result \(-42 \).
Understanding the product helps us to set up the initial equation correctly, which leads to accurate solutions.
solving for variables
Solving for variables involves finding the value of unknowns in an equation. In algebra, this usually means isolating the variable on one side of the equation.
From our equation \(-6y = -42 \), we want to isolate \( \text{y} \):
  • We do this by performing the same operation on both sides of the equation to keep it balanced.
  • Here, we divide both sides by \(-6 \) to isolate \(\text{y} \).
  • This yields \( \text{y} = \frac{-42}{-6} \), simplifying to \( \text{y} = 7 \).
It's important to verify the solution by substituting back into the original equation. Substituting \( \text{y} = 7 \), we get \(-6 \times 7 = -42 \), confirming that our solution is correct. Verifying ensures that the solution satisfies the original problem.

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