Problem 26 - Algebra Form and Function

Question

Put the equation in standard form. $$ y-6=5(x+2) $$

Step-by-Step Solution

Answer

Answer: The standard form of the equation is \(5x - y = -16\).

Step 1: Distribute the 5 on the right side of the equation

Multiply 5 by the terms inside the parentheses \((x+2)\): $$ y - 6 = 5(x + 2) $$ $$ y - 6 = 5x + 10 $$

Step 2: Move terms with x and y to one side and the constant terms to the other side

Subtract 5x from both sides and add 6 to both sides: $$ y - 6 - 5x = 5x + 10 - 5x $$ $$ -5x + y = 16 $$

Step 3: Write the equation in the standard form

Standard form is Ax + By = C. In this case, we can write the equation as: $$ 5x - y = -16 $$ The equation in standard form is: \(5x - y = -16\).

Key Concepts

Standard Form

When working with algebraic equations, converting them to a standardized format makes them easier to compare and analyze. The standard form of a linear equation is expressed as \[Ax + By = C\] where \(A\), \(B\), and \(C\) are integers, and \(A\) should be a non-negative integer. If in your equation, \(A\) is negative, you can multiply everything by \(-1\) to make it positive.

In this form, both \(x\) and \(y\) are on the left side, while the constant term \(C\) is isolated on the right side. This arrangement is beneficial because it provides a clear format for identifying the slope and intercepts when graphed.

By converting the equation \(y - 6 = 5(x + 2)\) into standard form, you ensure that the equation is in one of the most widely accepted mathematical formats. Taking each term to its assigned place allows for consistent use in further applications like graphing or solving further algebraic problems.

Distributive Property

The distributive property is an essential principle in algebra that allows for multiplying a single term across terms within parentheses. It states: \[a(b + c) = ab + ac\] and can be applied in a variety of algebraic situations.

In the context of the equation \(y - 6 = 5(x + 2)\), the distributive property enables us to multiply the 5 by both \(x\) and \(2\) inside the parentheses. This results in \(5x + 10\), effectively removing the parentheses and simplifying expression handling.

Understanding and applying the distributive property correctly simplifies initial equation forms into more workable expressions, facilitating the transition to the standard form. Teaching the importance of distributing accurately is crucial for solving a variety of algebraic equations where expressions need to be expanded or simplified.

Equation Simplification

Simplifying equations is about making algebraic expressions as clear and manageable as possible. This involves:

• Eliminating parentheses using the distributive property.
• Combining like terms on each side of the equation.
• Rearranging terms to isolate variables or constants.

Removing clutter helps see the path towards finding solutions or rearranging into specific forms like the standard form.

In our example, after applying the distributive property, the task was to collect and move terms such as \(5x\) and \(y\) to one side. Meanwhile, constants like \(-6\) must be moved to the opposite side to maintain an organized and logical equation flow: \(-5x + y = 16\)

Precise simplification is often the cornerstone of formulating effective solutions in more complex mathematical tasks. It is here that the equation finally takes on its standard form through careful reallocation of terms.