Problem 24 - Algebra Form and Function

Question

Put the equation in standard form. $$ y=2+4(x-3) $$

Step-by-Step Solution

Answer

Answer: The standard form of the given linear equation is -4x + y = 10.

Step 1: Expand the equation

Distribute the multiplication of 4 to both the terms inside the brackets: $$ y = 2 + 4(x - 3) $$ $$ y = 2 + (4 \times x) - (4 \times 3) $$

Step 2: Simplify the equation

Perform the multiplications, then simplify the equation: $$ y = 2 + 4x - 12 $$ $$ y = 4x - 10 $$

Step 3: Rearrange the equation in standard form

Subtract 4x from both sides of the equation and add 10 to both sides of the equation to arrange it in the standard form, Ax + By = C: $$ -4x + y = 10 $$ The equation is now in standard form: $$ -4x + y = 10 $$

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and represent straight lines when plotted on a graph. They typically take the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. This form is known as the 'standard form.' A linear equation might look like \(3x + 2y = 6\). Here, \(x\) and \(y\) are variables that can change, but the equation describes a linear relationship between them.
Linear equations are important because they model many real-world relationships between two variables. Understanding them is the foundation for solving systems of equations, performing graph analyses, and even in calculus. When manipulating linear equations, you aim to arrange the terms so that everything falls into the desired format, usually for ease of interpretation or to solve for one of the variables.

Expanding Equations

Expanding equations is the process of using distributive property to simplify expressions with brackets. For example, in the equation \(y = 2 + 4(x - 3)\), the term \(4(x - 3)\) requires expansion.
To do this, multiply 4 by each term inside the bracket:

• First, multiply \(4\) by \(x\) to get \(4x\).
• Next, multiply \(4\) by \(-3\) to get \(-12\).

After expanding, the expression becomes \(y = 2 + 4x - 12\). This step is crucial because it transforms an equation into a format that's easier to manage, particularly for later steps such as combining like terms or attempting to solve the equation. It helps break down complex problems into simpler parts.

Simplifying Expressions

Simplifying expressions involves combining like terms and reducing an expression to its simplest form. For instance, consider the equation obtained after expanding: \(y = 2 + 4x - 12\). Here, combining the constant terms helps simplify it.
Start by evaluating the constants:

• Combine \(2\) and \(-12\), which simplifies to \(-10\).

The simplified form then becomes \(y = 4x - 10\). Simplifying is important because it tidies up the equation, making it much clearer and easier to work with. This process is key in solving equations, as it enables you to see the essence of the expression without unnecessary complications. It's like organizing your work to focus on what's most important.