Problem 58 - Algebra 2

Question

Write each equation in standard form. Identify A, B, and C. \(y=7 x+4\)

Step-by-Step Solution

Answer

A = 7, B = -1, C = -4.

Step 1: Understand Standard Form Equation

The standard form of a linear equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be a positive integer.

Step 2: Rearrange the Given Equation

Starting with the equation \(y = 7x + 4\), subtract \(7x\) from both sides to rearrange it into standard form: \(-7x + y = 4\).

Step 3: Adjust Coefficients

Standard form typically requires that the coefficient of \(x\) (\(A\)) be positive. Multiply the entire equation by \(-1\) to make \(A\) positive: \(7x - y = -4\).

Step 4: Identify A, B, and C

Compare the final form \(7x - y = -4\) to \(Ax + By = C\). Here, \(A=7\), \(B=-1\), and \(C=-4\).

Key Concepts

Linear Equations

When talking about linear equations, we refer to equations that express a straight line when plotted on a coordinate plane. These equations are commonly presented in several forms, the most prominent being the slope-intercept form and the standard form.
The equation given, \(y = 7x + 4\), is in slope-intercept form, which is generally written as \(y = mx + b\). Here, \(m\) is the slope of the line, and \(b\) is the y-intercept—the point where the line crosses the y-axis.

• The slope \(m = 7\) indicates how steep the line is: a rise of 7 units for every 1 unit moved to the right.
• The y-intercept \(b = 4\) tells us that the line crosses the y-axis at the point (0, 4).

Understanding these basic components can help in converting the equation from slope-intercept form to the standard form.

Integer Coefficients

In mathematics, integer coefficients are crucial when working with equations in standard form. These coefficients are whole numbers and can be positive or negative. The reason we prefer integers in this context is to maintain simplicity and precision in mathematical calculations.
When writing an equation in standard form, which looks like \(Ax + By = C\), coefficients \(A\), \(B\), and the constant \(C\) should ideally be integers.
In the exercise, our aim is to have \(A\) positive, which might require multiplying the entire equation by \(-1\) if \(A\) is initially negative. This shift is crucial for achieving the standard representation and aligns with the general practice in algebra to keep equations neat and standardized.

Rearranging Equations

Rearranging an equation involves modifying its structure without changing its values, to reach a desired form. It can help in simplifying equations or transforming them from one form to another, such as from slope-intercept to standard form.
In our case, starting with \(y = 7x + 4\), the goal is to move all terms to one side to form \(Ax + By = C\). By subtracting \(7x\) from both sides of the equation, we get \(-7x + y = 4\).
It's essential to follow these rearranging steps carefully, ensuring that the value of the equation remains unchanged. Checking your work by plugging in simple values for \(x\) ensures that the rearranged equation is equivalent to the original.

Identifying Coefficients

Identifying coefficients involves recognizing the numerical values that multiply the variables in an equation. In the standard form equation \(Ax + By = C\), the coefficients \(A\) and \(B\) correspond to \(x\) and \(y\) respectively, while \(C\) is the constant.
In our example, after converting \(-7x + y = 4\) to the desired \(7x - y = -4\), the coefficients can be identified as follows:

• \(A = 7\): the coefficient of \(x\).
• \(B = -1\): the coefficient of \(y\).
• \(C = -4\): the constant on the right side of the equation.

Recognizing and correctly identifying these coefficients is essential for solving and understanding equations, especially as equations grow more complex in more advanced mathematics.