## What is Simultaneous Equations Bias?

Single-equation model in which a dependent (y) variable is a function of independent (x) variables, other y variables are among the independent variables in each SEM equation. The y variables in the system are jointly (or simultaneously) determined by the equations in the system. To find a solution, the solver needs to use the provided equations to find the exact value of each variable. Generally, the solver uses either a , the  method, thesubstitution method, or the elimination method. Some textbooks refer to the elimination method as the addition method, since it involves adding equations (or constant multiples of the said equations) to one another, as detailed later in this article.

This is a set of , also known as a :

Solving this involves subtracting x + y = 6 from 2x + y = 8 (using the elimination method) to remove the y-variable, then simplifying the resulting equation to find the value of x, then substituting the x-value into either equation to find y.

The solution of this system is:

Which, can also be written as an  (2, 4), representing on a graph the coordinates of the point of intersection of the two lines represented by the equations

## Finding solutions of simultaneous equations:

If the number of equations is the same as the number of variables, then probably (but not necessarily) the system is exactly solvable in the sense that the set of its solutions is finite; for a  in this case there is exactly one solution, for other systems to have several solutions is also typical.

Sometimes a system has no solution; this is typical for the case where the system has more equations than variables. If these rules about connection between number of solutions and numbers of equations and variables do not hold, then such situation is often referred to as dependence between equations or between their left parts.

For instance, this occurs in linear systems if one equation is a simple multiple of the other (representing the same line, e.g. 2x + y = 3 and 4x + 2y = 6) or if the ratio of like variables in two linear equations is the same (representing parallel lines, e.g. 2x + y = 3 and 6x + 3y = 7 where the ratio of comparable letters is 3).

The equation x2 + y2 = 0 can be thought of as the equation of a circle whose radius has shrunk to zero, and so it represents a single point: (x = 0, y = 0), unlike a normal circle containing an infinity of points. This and similar examples show the reason why the last two types described above need the qualification "usually".

An example of a system of equations of the first type described above with an infinite number of solutions is given by x = |x|, y = |y| (where the notation |•| denotes the  function), whose solutions form a quadrant of the . Another example is x = |y|, y = |x|, whose solution represents a . Another example is (x+1)(x+y)=0, (y+1)(x+y)=0, whose solution represents a line and a point.

The two example  intersect twice. Therefore, there are two solutions.

Systems of simultaneous equations can be hard to solve unless a systematic approach is used. A common technique is the substitution method: Find an equation that can be written with a single variable as the subject, in which the left-hand side variable does not occur in the right-hand side expression. Next,  that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved (whether by further application of the substitution method or by other methods), substitute the solutions found for the variables in the above right-hand side expression.

In this set of equations

x is made the subject of the second equation:

then, this result is substituted into the first equation:

After simplification, this yields the solutions

And by substituting this in x = −2y the corresponding x values are obtained. The two solutions of the system of equations are then:

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The passage primarily discusses the concept of simultaneous equations, methods to solve them, and provides examples of systems with different numbers of solutions. It also briefly touches on the conditions under which a system of equations can have no solution, a unique solution, or infinitely many solutions.
multiple_questions_with_answers

the passage contains multiple mathematical questions and provides answers to them, such as solving a system of linear equations and finding the solutions to various systems with different numbers of solutions.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
## What is Simultaneous Equations Bias?

Single-equation model in which a dependent (y) variable is a function of independent (x) variables, other y variables are among the independent variables in each SEM equation. The y variables in the system are jointly (or simultaneously) determined by the equations in the system. To find a solution, the solver needs to use the provided equations to find the exact value of each variable. Generally, the solver uses either a , the  method, thesubstitution method, or the elimination method. Some textbooks refer to the elimination method as the addition method, since it involves adding equations (or constant multiples of the said equations) to one another, as detailed later in this article.

This is a set of , also known as a :

Solving this involves subtracting x + y = 6 from 2x + y = 8 (using the elimination method) to remove the y-variable, then simplifying the resulting equation to find the value of x, then substituting the x-value into either equation to find y.

The solution of this system is:

Which, can also be written as an  (2, 4), representing on a graph the coordinates of the point of intersection of the two lines represented by the equations

## Finding solutions of simultaneous equations:

If the number of equations is the same as the number of variables, then probably (but not necessarily) the system is exactly solvable in the sense that the set of its solutions is finite; for a  in this case there is exactly one solution, for other systems to have several solutions is also typical.

Sometimes a system has no solution; this is typical for the case where the system has more equations than variables. If these rules about connection between number of solutions and numbers of equations and variables do not hold, then such situation is often referred to as dependence between equations or between their left parts.

For instance, this occurs in linear systems if one equation is a simple multiple of the other (representing the same line, e.g. 2x + y = 3 and 4x + 2y = 6) or if the ratio of like variables in two linear equations is the same (representing parallel lines, e.g. 2x + y = 3 and 6x + 3y = 7 where the ratio of comparable letters is 3).

The equation x2 + y2 = 0 can be thought of as the equation of a circle whose radius has shrunk to zero, and so it represents a single point: (x = 0, y = 0), unlike a normal circle containing an infinity of points. This and similar examples show the reason why the last two types described above need the qualification "usually".

An example of a system of equations of the first type described above with an infinite number of solutions is given by x = |x|, y = |y| (where the notation |•| denotes the  function), whose solutions form a quadrant of the . Another example is x = |y|, y = |x|, whose solution represents a . Another example is (x+1)(x+y)=0, (y+1)(x+y)=0, whose solution represents a line and a point.

The two example  intersect twice. Therefore, there are two solutions.

Systems of simultaneous equations can be hard to solve unless a systematic approach is used. A common technique is the substitution method: Find an equation that can be written with a single variable as the subject, in which the left-hand side variable does not occur in the right-hand side expression. Next,  that expression where that variable appears in the other equations, thereby obtaining a smaller system with fewer variables. After that smaller system has been solved (whether by further application of the substitution method or by other methods), substitute the solutions found for the variables in the above right-hand side expression.

In this set of equations

x is made the subject of the second equation:

then, this result is substituted into the first equation:

After simplification, this yields the solutions

And by substituting this in x = −2y the corresponding x values are obtained. The two solutions of the system of equations are then:

Transtutors.com can help you with all problems related to your homework. Our experienced and highly qualified team will give you quality assignment help for Economics related to Simultaneous equations bias at reasonable prices.

## Related Topics

All Economics Topics

## Get Simultaneous equations bias Assignment Help with these Q&A

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The passage primarily discusses the concept of simultaneous equations, methods to solve them, and provides examples of systems with different numbers of solutions. It also briefly touches on the conditions under which a system of equations can have no solution, a unique solution, or infinitely many solutions.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

The passage contains multiple mathematical questions and provides answers to them, such as solving a system of linear equations and finding the solutions to various systems with different numbers of solutions.