Problem 23 - Algebra 2

Question

Write each equation in standard form. Identify A, B, and C. \(y=12 x\)

Step-by-Step Solution

Answer

Standard form is \(12x - y = 0\) with \(A = 12\), \(B = -1\), \(C = 0\).

Step 1: Understand the Problem

We need to express the equation of the line in the standard form, which is given by \(Ax + By = C\). In this form, \(A, B,\) and \(C\) are integers, and \(A\) should be non-negative.

Step 2: Rewrite in Standard Form

The given equation is \(y = 12x\). To rewrite this in standard form \(Ax + By = C\), rearrange the equation to move everything to one side: subtract \(12x\) from both sides to get \(-12x + y = 0\). Multiply the entire equation by -1 to get a positive A, resulting in \(12x - y = 0\).

Step 3: Identify A, B, and C

Now that the equation is in the standard form \(12x - y = 0\), identify \(A = 12\), \(B = -1\), and \(C = 0\). These values correspond directly to the coefficients and constant term in the equation.

Key Concepts

Coefficients in Equations

Understanding the role of coefficients in equations is foundational to mastering algebra. Coefficients are numbers that multiply variables, affecting how steeply a graph rises or falls. In the standard form of a linear equation, expressed as \(Ax + By = C\), both \(A\) and \(B\) are coefficients.

• \(A\) is the coefficient of \(x\)
• \(B\) is the coefficient of \(y\)

The coefficient \(A\) might tell you something about the direction and steepness of a line along the x-axis, while \(B\) does the same along the y-axis. For example, in the equation \(12x - y = 0\), the coefficient of \(x\) is 12, meaning every unit increase in \(y\) leads to a 12-unit increase along \(x\). Meanwhile, the coefficient of \(y\), which is -1, shows the inverse relationship.Coefficients can also impact the alignment and orientation of a line in the coordinate plane, playing a critical role in defining the line's properties.

Rearranging Linear Equations

Rearranging equations to fit a desired form is a key skill in algebra. The standard form, \(Ax + By = C\), helps in analyzing properties like slope and intercepts. Beginning with an equation such as \(y = 12x\), you rearrange terms to achieve this format.Start by moving all terms to one side of the equation. By subtracting \(12x\) from both sides, we get \(-12x + y = 0\). To satisfy the requirements of standard form, where \(A\) should be positive, multiply the equation by -1, resulting in \(12x - y = 0\).This process may seem simple, but it ensures that the equation follows a conventional form. It allows easier comparison between equations and makes properties like slopes more discernible: the slope \(m\) corresponds to \(-\frac{A}{B}\). Moreover, standard form is often required in various mathematical applications.

Identifying Constants

Constants in equations are numbers that are not multiplied by variables. In standard form, the constant is \(C\) in \(Ax + By = C\). It is the term that remains in the equation when all variable terms are set to zero.In our example equation \(12x - y = 0\), the constant term \(C\) is 0. This constant shows the y-intercept of the line—where the line crosses the y-axis when \(x = 0\). When no constant term appears on the right side, such as in this equation, we understand the line passes through the origin (0, 0).By identifying \(C\), you can quickly determine more about the equation's graph. Understanding the role of constants helps in graphing lines and solving real-world problems, as constants can represent fixed values in various scenarios.