Problem 85 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. In each exercise, solve the given equation. $$5(2 y-3)-4 y=9$$

Step-by-Step Solution

Answer

The solution for y in the equation is 4.

Step 1: Distribute the Multiplication

To start off, distribute the \(5\) to both \(2 y\) and \(-3\) inside the bracket. This leads to a new equation: \(10 y - 15 - 4 y = 9\)

Step 2: Combine Like Terms

Combine like terms (i.e., terms with the same variable y). The equation becomes \( 6 y - 15 = 9 \)

Step 3: Add 15 to Both Sides

Next, add 15 to both sides in order to isolate y. This leads to another equation \( 6 y = 24 \)

Step 4: Divide by 6

Lastly, the equation \( 6 y = 24 \) is simplified by dividing both sides by 6. Hence, the solution to the equation is \( y = 4 \)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can represent numbers, relationships, or even real-world scenarios. For example, in the equation given in our exercise, 5(2y - 3) - 4y = 9, the expression 5(2y - 3) - 4y includes variables (y), constants (5, -3, and -4), and operations (multiplication and subtraction).

Understanding algebraic expressions is fundamental as they form the basis of algebra and help in formulating and solving equations. When faced with an algebraic expression, it is crucial to identify each part: coefficients (numerical factors of variables), terms (the components separated by + or - signs), operators (like +, -, *, or /), and constants (specific numbers). Recognizing these elements aids in simplifying and solving the expressions accurately.

Combining Like Terms

Combining like terms is a critical step in simplifying algebraic expressions. Like terms are terms that have exactly the same variables raised to the same powers. In other words, they 'look' the same. For example, in 10y - 15 - 4y from the solution's Step 2, the terms 10y and -4y are 'like terms because they both contain the variable y to the first power.

By combining these like terms, we consolidate them into a single term, which makes the equation simpler and easier to solve. In our exercise, combining 10y and -4y gives us 6y, thus reducing the equation to 6y - 15 = 9. Whenever you encounter multiple terms with the same variables, always check if you can combine them to simplify the expression.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis. Specifically, it states that a(b + c) = ab + ac. Applying the distributive property makes it possible to eliminate parentheses from an algebraic expression.

Let’s consider the Step 1 of our original exercise. When distributing the 5 across the expression (2y - 3), we multiply 5 by 2y and by -3 separately, resulting in 10y - 15. And this step is vital: without using the distributive property correctly, we cannot move forward to combine like terms or isolate the variable and solve the equation. Recognizing when and how to use the distributive property is crucial to solving linear equations efficiently and is a skill that applies to a multitude of algebraic tasks.