Which of the following is the equation \(3(x-y)=5\) solved for \(y ?\) A. \(y=x-\frac{5}{3}\) B. \(y=\frac{5}{3}-x\) C. \(y=15-x\) D. \(y=x-15\) E. \(y=\frac{5}{3} x\)

Algebraic manipulation is a foundational skill in solving mathematical equations, especially in subjects like algebra. It involves applying various algebraic techniques to rearrange and simplify equations. The goal of manipulation is often to solve for a particular variable, make complex equations more manageable or discern patterns.

For example, consider the equation from the ACT Math preparation exercise, 3(x-y)=5. Manipulating this equation requires several steps, such as distributing the 3, moving terms to different sides of the equality, and simplifying fractional components. Each step relies on a different manipulation technique, which, when applied correctly, leads to finding the value of y in terms of x. Through practice and understanding the rules of algebraic manipulation, students can solve equations efficiently and accurately.

To solve for a specific variable means to isolate that variable on one side of an equation. It's a critical process in algebra to find the value of the variable in question. Isolating variables requires the understanding and application of addition, subtraction, multiplication, and division, as well as handling the inverse operations.

Looking at the exercise 3(x-y)=5, we isolated y by first eliminating the term containing x before proceeding with division to completely isolate y. Mastery in isolating variables not only aids in solving textbook exercises but also in understanding more complex relationships between variables in algebraic expressions.

The distributive property is a critical algebraic principle that allows you to multiply a number by a sum or difference within parentheses. It is expressed as a(b + c) = ab + ac or a(b - c) = ab - ac. Applying this property simplifies expressions and is a necessary step in many algebraic problems.

In our exercise, we applied the distributive property to the expression 3(x-y), which resulted in 3x - 3y. Understanding how and when to use the distributive property is crucial for solving linear equations and will be a vital skill when facing algebra problems on standardized tests like the ACT.

Preparation for the ACT Math section includes getting comfortable with various algebraic concepts, including solving linear equations through algebraic manipulation, isolating variables, and using the distributive property. Practice is essential as it builds familiarity with the types of questions and the format of the test.

Working through problems like the one presented, 3(x-y)=5, helps reinforce these concepts. The key to ACT preparation is not just solving the problems but understanding the underlying principles that lead to the solution. This allows students to be prepared for similar, yet unique, problems they may encounter during the test, ensuring a more confident and successful outcome.