Solve for y 1/3=-3/5y+3/4y-1/2y
The problem is asking to find the value of the variable y. It presents an equation where y is involved in a combination of fraction multiplication and addition on the right-hand side of the equation, and a fraction on the left-hand side. The goal is to isolate y and solve for it by applying the principles of algebra, which includes manipulating the equation through addition, subtraction, multiplication, division, and combination of like terms to get y by itself on one side of the equation.
$\frac{1}{3} = - \frac{3}{5} y + \frac{3}{4} y - \frac{1}{2} y$
Answer
Solution:
Step 1
Start by expressing the equation in a standard form: $-\frac{3}{5}y + \frac{3}{4}y - \frac{1}{2}y = \frac{1}{3}$.
Step 2
Begin simplification of the left-hand side of the equation.
Step 2.1
Break down the simplification process for each term involving $y$.
Step 2.1.1
Associate $y$ with $\frac{3}{5}$: $-\frac{3y}{5} + \frac{3}{4}y - \frac{1}{2}y = \frac{1}{3}$.
Step 2.1.2
Rearrange the term to place $3$ before $y$: $-\frac{3y}{5} + \frac{3}{4}y - \frac{1}{2}y = \frac{1}{3}$.
Step 2.1.3
Combine $\frac{3}{4}$ with $y$: $-\frac{3y}{5} + \frac{3y}{4} - \frac{1}{2}y = \frac{1}{3}$.
Step 2.1.4
Associate $y$ with $\frac{1}{2}$: $-\frac{3y}{5} + \frac{3y}{4} - \frac{y}{2} = \frac{1}{3}$.
Step 2.2
Identify a common denominator for the terms.
Step 2.2.1 - Step 2.2.10
Perform the necessary multiplications to achieve a common denominator of $20$.
Step 2.3
Combine all terms over the common denominator: $\frac{-3y \cdot 4 + 3y \cdot 5 - y \cdot 10}{20} = \frac{1}{3}$.
Step 2.4
Simplify each term in the numerator.
Step 2.5
Combine like terms in the numerator to get: $-\frac{7y}{20} = \frac{1}{3}$.
Step 3
Multiply both sides of the equation by $-\frac{20}{7}$ to isolate $y$.
Step 4
Simplify both sides of the equation.
Step 4.1
Focus on simplifying the left side of the equation.
Step 4.1.1
Cancel out the common factors of $20$ and $7$.
Step 4.1.1.3
Complete the multiplication to isolate $y$.
Step 4.2
Simplify the right side of the equation by multiplying $-\frac{20}{7}$ by $\frac{1}{3}$.
Step 5
Present the final result in both exact and decimal forms: $y = -\frac{20}{21}$.
Knowledge Notes:
The problem involves solving a linear equation with fractions. Here are the relevant knowledge points:
Combining Like Terms: When solving equations, terms with the same variable can be combined by adding or subtracting the coefficients.
Common Denominator: To combine fractions, a common denominator is required. This is typically the least common multiple of the individual denominators.
Simplifying Fractions: Fractions are simplified by dividing the numerator and the denominator by their greatest common divisor or by canceling out common factors.
Isolating the Variable: To solve for a variable, you need to isolate it on one side of the equation. This often involves performing the same operation on both sides of the equation to maintain equality.
Multiplication and Division of Fractions: When multiplying fractions, multiply the numerators together and the denominators together. To divide fractions, multiply by the reciprocal of the divisor.
Negative Numbers: Pay attention to the signs when dealing with negative numbers, especially when multiplying or dividing, as two negatives make a positive.
Equation Solving Process: The process of solving an equation systematically involves simplifying terms, combining like terms, finding common denominators, and isolating the variable to find its value.
Representation of Solutions: Solutions to equations can be represented in different forms, such as exact fractions or decimal approximations, depending on the context and requirements.
