Problem 41 - Prealgebra and Introductory Algebra

Question

Solve. $$\frac{3}{5} y-\frac{1}{3}(1-y)=\frac{2 y-5}{15}$$

Step-by-Step Solution

Answer

The solution to the equation is \( y = \frac{1}{2} \).

Step 1: Distributing the Fractional Term

Distribute \(-\frac{1}{3}\) to each term in the parenthesis \((1 - y)\) to get a new equation: \[\frac{3}{5}y - \frac{1}{3} + \frac{1}{3}y = \frac{2y - 5}{15}\]

Step 2: Simplifying the Fractional Terms

Simplify the fractional terms in the equation to get:\[\frac{14}{15}y - \frac{1}{3} = \frac{2y - 5}{15}\]

Step 3: Combining Like Terms

Combine the like terms on both sides of the equation to get:\[\frac{14}{15}y - \frac{2y}{15} = -\frac{1}{3} + \frac{5}{15}\]

Step 4: Solving for y

Solve the above equation for \( y \) to finally get:\[y = \frac{1}{2}\]

Key Concepts

Distributing Fractional Terms

Understanding how to distribute fractional terms is essential when solving algebraic equations. This process involves multiplying a fraction outside the parentheses by each term inside the parentheses. For example, if you have a term like \( -\frac{1}{3}(1 - y) \) you should multiply \( -\frac{1}{3} \) by 1 and \( -y \). This gives you two new terms: \( -\frac{1}{3} \) and \(+\frac{1}{3}y \). These terms replace the original expression in parentheses, transforming the equation into a more workable form.

Remember that when you distribute a negative fraction, the signs of the terms inside the parentheses will change. Positive becomes negative, and vice versa. Not only does correctly distributing fractional terms set the stage for further simplification, but it also helps in maintaining accuracy throughout the problem-solving process.

Simplifying Fractions

Simplifying fractions within algebraic equations makes them more manageable and easier to solve. Start by ensuring that all fractions have a common denominator, in this case, 15. This allows you to combine fractions effectively. Take the equation \( \frac{3}{5}y - \frac{1}{3} + \frac{1}{3}y = \frac{2y - 5}{15} \),

After the distribution step, you would convert the terms to have the same denominator, so \( \frac{3}{5}y \) becomes \( \frac{9}{15}y \) and \( \frac{1}{3} \) becomes \( \frac{5}{15} \). Notice how the equation starts to look cleaner and easier to work with. The goal is to reach a point where the coefficients of the variable are combined into a single fraction, reducing the complexity of the equation.

Combining Like Terms

Once you have simplified the fractions, it's time to combine like terms. Like terms are terms that contain the same variables raised to the same power. They can be combined by adding or subtracting the coefficients. In the equation \( \frac{14}{15}y - \frac{2y}{15} = -\frac{1}{3} + \frac{5}{15} \), you have like terms on each side of the equation involving y and constants.

Subtract \( \frac{2y}{15} \) from \( \frac{14}{15}y \) to consolidate your y terms. For constants, convert \( \frac{1}{3} \) to \( \frac{5}{15} \) so that you can combine it with the other constant term. This step makes the equation much simpler and sets you up for the final step of solving for the variable.

Algebraic Problem Solving

Algebraic problem solving often culminates in solving for the unknown variable. Having completed steps involving distribution, simplification, and combination, you reach a point where the equation takes a form where the variable is isolated and can be solved for. In the given problem, after combining like terms, you are left with an equation where y's coefficient is a single fraction, and you have a numerical value on the other side.

To isolate y, you simply divide both sides of the equation by the coefficient of y. In this case, the coefficient is \( \frac{14}{15} - \frac{2}{15} \), which simplifies to \( \frac{12}{15} \) or \( \frac{4}{5} \). Dividing both sides by \( \frac{4}{5} \) gives you the value of y. This critical thinking and methodical approach are integral to solving algebraic equations efficiently and accurately.