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Chapter 4: Problem 55

Solve. If no equation is given, perform the indicated operation. $$ \frac{1}{5} y=10 $$

Short Answer

Expert verified
The solution is \( y = 50 \).

Step by step solution

01

Understand the Equation

The given equation is \( \frac{1}{5} y = 10 \). This implies that \( y \) is multiplied by \( \frac{1}{5} \), and the result is 10.
02

Isolate the Variable

To find the value of \( y \), we need to isolate it on one side of the equation. Since \( y \) is multiplied by \( \frac{1}{5} \), we can multiply both sides of the equation by 5 to cancel out the fraction. This gives us:\[5 \times \left( \frac{1}{5} y \right) = 10 \times 5\]
03

Simplify the Equation

Simplifying the left side, the 5 and \( \frac{1}{5} \) cancel each other out, leaving just \( y \). On the right side, multiply 10 by 5. The equation becomes:\( y = 50 \).So, the value of \( y \) is 50.

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Key Concepts

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

Equation Manipulation
When solving linear equations, you often need to manipulate the equation to simplify or rearrange it in order to find the unknown variable. This involves performing operations on both sides to maintain equality.

Here are a few common manipulations:
  • Addition/Subtraction: Used to eliminate terms on one side. For example, if an equation has a term like "+3," subtract 3 from both sides to remove it.
  • Multiplication/Division: Used to eliminate coefficients. If a variable is multiplied by 2, divide both sides by 2 to solve for the variable.
  • Combining Like Terms: Simplify the equation by merging similar terms to reduce clutter.
In the original exercise, by multiplying both sides by 5, we effectively reversed the division by 5. This is a perfect demonstration of how multiplication can be used to simplify equations.
Isolating Variables
Isolating variables is a key step in solving equations, and it essentially means getting the variable alone on one side of the equation. This allows us to see clearly what the variable equals.

To isolate a variable:
  • Perform opposite operations to cancel out other terms. For example, to eliminate addition, subtract that value from both sides.
  • If there is a fraction, multiply by the denominator to clear it.
The primary goal is to leave the variable by itself on one side, often through a series of operations to undo what was done to the variable. In our example, multiplying both sides by 5 effectively isolated the variable "y," allowing us to find its value.
Working with Fractions
Fractions in equations often seem intimidating, but they are just another type of division. Working with fractions usually involves one of two main steps: clearing the fraction or performing operations directly on it.

To clear a fraction:
  • Multiply all terms by the denominator, which can remove the fraction entirely and simplify the equation.
Alternatively, if you decide to work with the fraction, treat it as a number:
  • Perform operations directly, such as adding or subtracting similar fractions.
  • Multiply or divide fractions by reciprocals to simplify.
In our exercise, the fraction \( \frac{1}{5} \) was addressed by multiplying by its denominator, 5, to simplify the problem and solve for \( y \). This method ensures that the fraction no longer poses a hurdle in isolating the variable.

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Most popular questions from this chapter

Perform the indicated operations. $$ -20 \frac{2}{5}+\left(-30 \frac{3}{10}\right) $$

Translate each phrase into an algebraic expression. The sum of \(8 \frac{3}{4}\) and a number

Write each mixed number as an improper fraction. See Example 20. $$166 \frac{2}{3}$$

Write each mixed number as an improper fraction. See Example 20. $$114 \frac{2}{7}$$

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$4-\frac{1}{5}$$
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