Problem 53 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$-4(5 x-y)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-20x + 4y\).

Step 1: Understand the Distributive Property

The distributive property states that for any numbers or expressions \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). In this problem, we'll apply the property to the expression \(-4(5x - y)\). Notice that \(-4\) will be distributed to both \(5x\) and \(-y\).

Step 2: Apply the Distributive Property to the First Term

Distribute \(-4\) to \(5x\) by multiplying them together. \[-4 \cdot 5x = -20x\]

Step 3: Apply the Distributive Property to the Second Term

Distribute \(-4\) to \(-y\) by multiplying them together. \[-4 \cdot (-y) = 4y\]

Step 4: Combine the Results

Combine the results of both distributions to write the expression as a whole. Thus, the expression \(-4(5x - y)\) simplifies to:\[-20x + 4y\]

Key Concepts

Algebra

Algebra is like the language of math. It's all about finding the unknown or putting real-life problems into mathematical equations. In our exercise, we deal with an algebraic expression:

• This expression is \(-4(5x - y)\).
• Here, you can see the presence of variables ("\(x\)" and "\(y\)") which are symbols that stand for numbers we don't know yet.
• The numerical values, such as "5," "4," are coefficients. They are the constants you multiply with the variable unless they become more complex expressions later.

By mastering algebra, you unlock the door to understanding complex problems and solutions. You'll often see it applied when balancing equations or dealing with expressions, just like in this problem.

Simplifying Expressions

Simplifying expressions makes math easier to understand and solve. It involves combining like terms and applying mathematical laws, such as the distributive property, to make expressions less complex.

In the expression \(-4(5x - y)\), simplicity involves turning it into a clearer form. We do this using the distributive property:

• First, multiply each term inside the parentheses by \(-4\):
  • \(-4 \cdot 5x = -20x\)
  • \(-4 \cdot (-y) = 4y\)

Now, you've simplified it to \(-20x + 4y\):

• There are no more parentheses, and it's easier to read and work with.

Essentially, simplifying is about taking complex expressions and making them more understandable.

Mathematical Operations

Mathematical operations provide the backbone for solving math problems. These operations include addition, subtraction, multiplication, and division. In this exercise, the focus is largely on multiplication and addition:

• Multiplication: We start by multiplying \(-4\) by each term inside the parentheses. This knocks the problem down a notch to easier pieces.
• Add or Subtract: If we have a minus before the term, such as "\(-y\)," we treat it as addition of the opposite (e.g., \(-4 \cdot (-y)\) turns into \(+4y\) because two negative signs make a positive sign in math).

Combining these operations allows us to break down the complexity of expressions into manageable steps.

Just remember:

• Use multiplication to distribute through the expression and simplify
• Add or subtract terms as needed to compile the final simplified expression.

Understanding these operations thoroughly leads to successfully tackling similar math problems.