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Chapter 5: Problem 31

Solve. $$ \frac{6}{y}=\frac{18}{15} $$

Short Answer

Expert verified
y = 5

Step by step solution

01

Identify the Equation

The given equation is \[ \frac{6}{y} = \frac{18}{15} \]
02

Simplify the Right-Hand Side

Simplify the fraction on the right-hand side of the equation. \[ \frac{18}{15} = \frac{18 \/ 3}{15 \/ 3} = \frac{6}{5} \]So, the equation becomes \[ \frac{6}{y} = \frac{6}{5} \]
03

Cross Multiply the Fractions

Cross multiply to solve for y. \[ 6 \/ y = 6 \/ 5 \]Cross multiplying gives \[ 6 \times 5 = 6 \times y \]
04

Solve for y

Divide both sides by 6 to isolate y. \[ 30 = 6y \]\[ y = 30 \/ 6 \]\[ y = 5 \]

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Key Concepts

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

Cross Multiplication
Simplification of fractions means reducing a fraction to its simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.
To simplify a fraction:/
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
/br In our exercise, the given fraction was \[ \frac{18}{15} \] ./
Here are the steps to simplify it:
  • Find the GCD of 18 and 15. Both 18 and 15 are divisible by 3.
  • Divide the numerator and the denominator by 3:
  • \[ \frac{18 \div 3}{15 \div 3} = \frac{6}{5} \] As a result, we have simplified the fraction to \[ \frac{6}{5} \]
    Now the equation becomes easier to handle: \[ \frac{6}{y} = \frac{6}{5} \]
Isolating Variables
Isolating the variable means getting the unknown variable by itself on one side of the equation. This is usually done through a series of algebraic manipulations.
In our exercise:
We started from the equation
\[ \frac{6}{y} = \frac{6}{5} \]The next step was to cross-multiply to get rid of the fractions:
\`6 \times 5 = 6 \times y\` \/Here /*We then performed algebraic operations to isolate 'y':\br>
  • \ \[ 30 = 6y \] #\br>
  • Divide both sides by 6: \ \[ y = \frac{30}{6} \]
  • Simplify the fraction: y = 5 \] \br>By isolating the variable, we've found that the solution to the equation is \[y = 5\In algebra, isolating the variables is crucial as it leads directly to finding the solution for the unknown value.

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Most popular questions from this chapter

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