Solve the equation for the given variable. Do NOT convert the answer to a decimal.
[tex]$\frac{y}{5}+\frac{y}{6}=\frac{3}{5}$[/tex]
Answer (1)
Find a common denominator and rewrite the equation: 30 6 y + 30 5 y = 5 3 .
Combine the fractions: 30 11 y = 5 3 .
Isolate y by multiplying both sides by 11 30 : y = 5 3 × 11 30 .
Simplify the expression: y = 11 18 .
Explanation
Problem Analysis We are given the equation 5 y + 6 y = 5 3 and we need to solve for y .
Finding Common Denominator To solve for y , we first need to find a common denominator for the fractions on the left side of the equation. The least common multiple of 5 and 6 is 30.
Rewriting Fractions Now, we rewrite the fractions on the left side with the common denominator of 30: 30 6 y + 30 5 y = 5 3
Combining Fractions Combine the fractions on the left side: 30 6 y + 5 y = 30 11 y = 5 3
Isolating y Now we have the equation 30 11 y = 5 3 . To isolate y , we multiply both sides of the equation by 11 30 : y = 5 3 × 11 30
Simplifying Simplify the expression to find the value of y : y = 5 × 11 3 × 30 = 55 90
Reducing Fraction Reduce the fraction to its simplest form by dividing both numerator and denominator by their greatest common divisor, which is 5: y = 55 ÷ 5 90 ÷ 5 = 11 18
Final Answer Therefore, the solution for y is 11 18 .
Examples
Imagine you're baking a cake and need to measure ingredients. If 5 1 of a cup of flour and 6 1 of a cup of sugar make up 5 3 of the dry ingredients, solving this equation helps you determine the exact amount of the mixture needed. This type of problem is useful in various real-life scenarios, such as adjusting recipes, managing finances, or calculating proportions in construction projects. Understanding how to solve such equations allows for precise measurements and better decision-making in practical situations.
