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Chapter 3: Problem 59

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

Short Answer

Expert verified
The equation in standard form is \(x - y = 0\), with \(A = 1\), \(B = -1\), \(C = 0\).

Step by step solution

01

Rewrite the Equation

The given equation is \(x = y\). First, we want to rewrite this equation into the standard form of a linear equation, which is \(Ax + By = C\). To do this, subtract \(y\) from both sides of the equation to obtain \(x - y = 0\).
02

Identify A, B, and C

Now that we have the equation \(x - y = 0\), we compare it with the standard form \(Ax + By = C\). We see that \(A = 1\), \(B = -1\), and \(C = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of a Linear Equation
Linear equations are often expressed in what's called the "standard form." This form provides a consistent structure that makes it easier to analyze and compare equations. The standard form of a linear equation is written as \(Ax + By = C\).

Here, \(A\), \(B\), and \(C\) are integers, and ideally, \(A\) should be a positive integer. To convert an equation into standard form, you may need to rearrange terms or perform operations like addition or subtraction.

Some key benefits of standard form include:
  • Clarity in comparing different linear equations.
  • Simplifies the process of finding intercepts of the equation on the coordinate plane.
  • Provides a consistent way to log or note multiple equations for mathematicians and students alike.
It's important to note that every linear equation can be expressed in this form.
Understanding Coefficients in Linear Equations
Coefficients are the numerical or constant multipliers of the variables in an equation. In the standard form \(Ax + By = C\), \(A\) and \(B\) are the coefficients of \(x\) and \(y\) respectively.

Let's break down their roles:
  • **The coefficient \(A\):** This represents how much the term \(x\) contributes to the equation. If \(A = 0\), the equation has no \(x\) term, depicting a horizontal line when graphed.
  • **The coefficient \(B\):** Similarly, this affects the contribution of \(y\) in the equation. If \(B = 0\), the equation only considers the \(x\) term, resulting in a vertical line when plotted.


Being mindful of coefficients helps to understand how different changes in their values shift or skim the graph's slope and intercepts. Each coefficient plays a crucial role in defining the precise behavior and graphical representation of the linear relationship.
Equation Transformation Techniques
Equation transformation is the process of adjusting an equation to a different form while still representing the same relationship. In this context, transforming an equation to standard form involves rearranging the terms to fit the \(Ax + By = C\) model.

Here are common techniques:
  • **Adding or subtracting terms:** This is often used to isolate variables or shift them from one side of the equation to the other, as seen in the transformation of \(x = y\) to \(x - y = 0\).
  • **Multiplying or dividing by constants:** These operations help in achieving integer coefficients if initial fractions exist.


Transforming equations can make their properties more apparent and simplify solving systems of equations or graphing. Whether you're balancing terms or shifting variables, maintaining the equation's equality is essential for a successful transformation.

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Most popular questions from this chapter

Solve each system of equations. \(9 a+7 b=-30\) \(8 b+5 c=11\) \(-3 a+10 c=73\)

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{2 x+2 y \geq 4} \\ {2 y \geq 3 x-6} \\ {4 y \leq x+8} \\\ {f(x, y)=3 y+x}\end{array} $$

Solve each system of equations. \(r-3 s+t=4\) \(3 r-6 s+9 t=5\) \(4 r-9 s+10 t=9\)

Solve each system of equations. \(2 r+s+t=14\) \(-r-3 s+2 t=-2\) \(4 r-6 s+3 t=-5\)

In order to practice at home, Tadeo purchased a basketball and a volleyball that cost a total of \(\$ 67,\) not including tax. If the price of the basketball \(b\) is \(\$ 4\) more than twice the cost of the volleyball \(v\) which system of linear equations could be used to determine the price of each ball? A. \(b+v=67\) \(b=2 v-4\) B. \(b+v=67\) \(b=2 v+4\) C. \(b+v=4\) \(b=2 v-67\) D. \(b+v=4\) \(b=2 v+67\)
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