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Chapter 2: Problem 46

Solve each equation. See Examples 9 and \(10 .\) \(-14 y-1.8=-24 y+3.9\)

Short Answer

Expert verified
The solution to the equation is \(y = 0.57\).

Step by step solution

01

Isolate all terms with variables on one side

Start by adding or subtracting terms with variables to get them on one side of the equation. Here, add \(24y\) to both sides to move the \(y\) terms together:\[-14y - 1.8 + 24y = -24y + 3.9 + 24y\]This simplifies to:\[10y - 1.8 = 3.9\]
02

Isolate the variable term

Next, get the \(y\) term by itself by adding \(1.8\) to both sides:\[10y - 1.8 + 1.8 = 3.9 + 1.8\]This simplifies to:\[10y = 5.7\]
03

Solve for the variable

Divide both sides by 10 to solve for \(y\):\[y = \frac{5.7}{10}\]Simplify the fraction:\[y = 0.57\]

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Key Concepts

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

Linear Equations
Linear equations might be among the first concepts you'll encounter in algebra. They are called "linear" because they graph as straight lines. Think of them as equations like "y = mx + b" where you'll often seek the value of a variable, like "y." In our given equation,
  • The goal is solving for "y." The equation stems from setting two linear expressions equal.
  • Each side of the equation can be seen as a separate line if it were graphed.
  • Unlike more complex equations, linear equations do not include powers or exponents.
To solve them, we find the value that makes the equation true. In other words, we are on a mission to determine what number "y" must be, such that the left side equals the right side when we substitute "y" in. Linear equations are foundational to understanding algebra and mastering them is a step toward tackling even more complex mathematical relationships.
Variable Isolation
Variable isolation is a technique used to solve equations by getting the variable on its own. In this problem, the variable we need to focus on is "y." The steps are like untangling a set of headphones - methodical but straightforward.
  • First, we group all the terms with "y" to one side of the equation. To do this, we add or subtract similar terms from both sides.
  • Next, any other numbers, or constants, are moved to the opposite side. This is achieved using the inverse operations such as addition and subtraction.
In our example with \(-14y - 1.8 = -24y + 3.9\), we wanted to go from "scattered" terms to concise expression:
  • We actually got the "y" terms all on one side with \(10y = 3.9 + 1.8\).
  • Ensuring no y's were left out was important to isolating our focus purely on solving.
This way of isolating makes problem-solving systematic and clearer!
Simplifying Fractions
Simplifying fractions is the process of making them as simple, or as little, as possible. When an equation involves fractions, it often means division is involved at some point.
  • For example, when solving for \(y\), we simplified the fraction \(\frac{5.7}{10}\).
  • The aim here is to reduce it to its simplest form or least complex numeric expression, which is \(y = 0.57\) in this case.
  • By doing this, the answer becomes easier to digest and verify.
Simplifying makes calculations straightforward and number handling less prone to error. It's always a handy tool in your math toolkit, ensuring the solutions are clear and comprehensible.

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Most popular questions from this chapter

Solve each inequality. Write each answer using solution set notation. $$ \frac{1}{2}(x-5)<\frac{1}{3}(2 x-1) $$

Solve each inequality. Write each answer using solution set notation. $$ 7(x-2)+x \leq-4(5-x)-12 $$

The area of the Sahara Desert is 7 times the area of the Gobi Desert. If the sum of their areas is 4,000,000 square miles, find the area of each desert.

In the 2008 Summer Olympics, Korea won more gold medals than Germany, which won more gold medals than Australia. If the numbers of gold medals won by these three countries are three consecutive integers whose sum is \(21,\) find the number of gold medals won by each. (Source: Beijing 2008 Olympics)

If both sides of the inequality \(-3 x<-30\) are divided by \(3,\) do you reverse the direction of the inequality symbol? Why or why not?
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