Algebra 4.1 Solving Systems Of Linear Equations By Graphing

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System of linear equations

Back 1

consists of two or more linear equations with the same variables.

Examples:

2x + 3y = 4
3x - 3y = -5

Front 2

system of linear equations is often called?

Back 2

linear system

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A solution of a system of linear equations...

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is and ordered pair that makes both equations true at the same time.

Front 4

The set of all ordered pairs that are solutions of a system is its....

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solution set

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What is one way to find the solution set of a system of two linear equations?

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Graph both equations on the same axes.

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The coordinates of any point at which the lines intersect...

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gives a solution of the system

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Because two different straight lines can intersect at no more than one point...

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there can never be more than one solution for such a system.

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How do you solve a linear system by graphing?

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1. Graph each equation of the system on the same coordinate axes.

2. Find the coordinates of the point of intersection of the graph if possible. This is the solution of the system.

3. Check the solution in both the original equations.

4. Then, write the solution set.

Front 9

What are the pro's & con's of the graphing method for solving linear systems?

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Pro- shows geometrically how solutions are found and is useful when approximate answers will do.

Con - It may not be possible to determine from the graphs the exact coordinates of the point that represents the solution, particularly if those points are not integers.

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The graphs of parallel lines have ...

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- no points in common
- no solutions
- the solution set is 0 with a
slash through it.

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Graphs of linear systems that are the same line...

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- every point on the line is the solution of the system
- the solution set contains an infinite number of ordered pairs, each of which satifies both equations of the system.

- Write the solution set
{(x.y)| 2x = 5y = 1

read" the set of ordered pairs (x,y) such that 2x + 5y = 1.

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When a system has an infinite number of solutions...

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either equation of the system could be used to write the solution set. Preference is to use the equation in standard form with coefficients that are integers having no common factors except 1.

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Consistent System

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The graphs intersect at exactly one point, which gives the single ordered pair solution of the system. The equations are independent.

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Inconsistent System

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The graphs of parallel lines, so there is no solution and the solution set is 0 with a slash throught it. the equations are dependent.

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The graphs are the same line

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There is an infinite number of solutions, and the solution set is written in set builder notation { ( x, y ) | _________}, where one of the equations is written after the | symbol. The system is consistent and the equations are dependent.

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Dependent Equations

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Equations of a system that have the same graph (because they are different forms of the same equation).

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Independent Equations

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Equations of a system that have different different graphs