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Chapter 5: Problem 23

Put the equation in standard form. $$ x=3 y-2 $$

Short Answer

Expert verified
Question: Put the given equation in standard form: $$2y - x = 2$$ Answer: The equation in standard form is: $$x - 3y = -2$$

Step by step solution

01

Rewrite the equation with both x and y on the same side

We should move the term containing y to the left side of the equation. $$ x - 3y = -2 $$
02

Check if the coefficients are integers and the coefficient of x is non-negative

The coefficients of x and y are already integers (1 and -3, respectively), and the coefficient of x is non-negative (1). So the equation is now in standard form.
03

Final Answer:

The equation in standard form is: $$ x - 3y = -2 $$

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Key Concepts

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

Linear Equations
Linear equations are mathematical expressions that represent a straight line when graphed. In algebra, these equations generally are in the form of \( ax + by = c \). Here, \( a \), \( b \), and \( c \) are constants, while \( x \) and \( y \) are variables. Linear equations are foundational in algebra and problem-solving because they form the basis of much of algebraic reasoning.

These equations are called 'linear' since their graph forms a straight line. For example, in our original exercise, converting the given equation to standard form helps in understanding the linear relationship between \( x \) and \( y \).
  • The slope of the line represents how steep the line is. It is derived from the coefficients \( a \) and \( b \) in the equation \( ax + by = c \).
  • The y-intercept is found where the line crosses the y-axis, which is when \( x = 0 \).
Equation Transformation
Equation transformation refers to the process of manipulating an equation to convert it into a specific desired form, such as standard form. This is an essential technique in algebra because it simplifies solving equations and understanding the relationships between variables. In the original exercise, the transformation involves rearranging terms so that both variables, \( x \) and \( y \), are on the same side of the equation.

To transform an equation, you need to:
  • Use operations such as addition, subtraction, multiplication, or division to both sides of the equation to maintain equality.
  • Ensure the coefficients are organized to match the desired form, such as having integers as coefficients.
  • Balance the equation by treating all changes equally on both sides.
These transformations allow equations to be handled in a uniform manner, facilitating easier problem-solving.
Integer Coefficients
Integer coefficients in an equation are simply whole number values assigned to variables, making equations simpler and often easier to interpret and solve. A crucial condition for standard form is that the coefficients for \( x \) and \( y \) should be integers.

Integer coefficients are favorable because:
  • They make arithmetic operations like addition and subtraction straightforward.
  • Integer coefficients simplify the process of factoring and finding greatest common divisors.
  • Working with integer coefficients avoids rounding errors that might arise from working with floating-point numbers.
In our solved example, the coefficients of \( x \) and \( y \) were confirmed to be integers, meeting the standard form's requirement. The coefficient of \( x \) is also checked to ensure it is non-negative, adhering to the conventional representation of standard form.

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Most popular questions from this chapter

Put the equation \(y=3 x t+2 x t^{2}+5\) in the form \(y=b+m x .\) What are the values of \(b\) and \(m ?\) [Note: Your answers could include \(t\).]

Are the lines parallel? $$ y=5+6(x+2) y=5+6(3 x-1) $$

\(f(t)=2 t+7\). Does the equation have no solution, one solution, or an infinite number of solutions? $$ f(t)+1=f(t+1) $$

For the system $$ \left\\{\begin{array}{l} 2 x+3 y=5 \\ 4 x+6 y=n \end{array}\right. $$ what must be true about \(n\) in order for there to be many solutions?

(a) Find the equation of the line with intercepts (i) (2,0) and (0,5) (ii) Double those in part (i) (b) Are the two lines in part (a) parallel? Justify your answer. (c) In words, generalize your conclusion to part (b). (There are many ways to do this; pick one. No justification is necessary.)
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