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EXAMPLES & SOLUTIONS
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They are two or more equations that solve for the same number of unknown variables.
These system of equations are either independent or dependent.
These equations can have one solution, no solution or infinite solutions.
They both have similiar unknown variables.
Different methods can be utilized to determine if these equations are functions or not.
Two techniques are the methods of elimination or subtitution.
They are also solved by observing the coordinates on a graph.
SYSTEM OF EQUATIONS
Below are the most common forms of system of equations
Each graph displays two linear lines that are formed from two equations.
Fact: Consistent linear equations have solutions.
Intersecting Lines
They intersect at a common point
EQ. 1
EQ. 2
Perpendicular lines
They intersect at a right angle
EQ. 1
EQ. 2
They both intercept at one point
therefore,
only one solution
These lines are independent,
however, they are consistent.
This means there are two linear equations that are considered independent of each other, yet they are consistent at one point. There consistency causes one y-value to depend on one x-value for one solution.
Same linear line
EQ. 1
EQ. 2
They intersect at common points
They intersect at all points
therefore,
infinite solutions
These lines are dependent
and consistent.
This means there are two linear equations that are considered dependent on each other and consistent at all points. There consistency causes all y-values to depend on all x-values for infinite solutions.
Fact: Inconsistent linear equations have no solution.
Parellel Lines
They intersect at no common points
EQ. 1
EQ. 2
m
(x - )
x
1
They intersect at no points
therefore,
no solutions
These lines are independent
and they are inconsistent
This means two linear equations are considered independent of each other. There inconsistency causes every y-value of both linear lines not to depend on any x-values from the other equation. Because of this, they do not intercept and cannot form a solution.
SOLVING SYSTEMS OF EQUATIONS
Rule: Two equations may or may not form the solutions to the equations.
Below are examples of systems of equations
2x + 3y = 5
4x + 3 y = 9
Standard Form
y = -2x/3 + 5
y = -4x/ 3 + 3
Slope-Intercept
Below are three ways to discover if a equation has a solution.
Graphing Technique
The graphing method allows you to discover points by placing them on a graph to determine the solution(s) to a equation.
Elimination Technique
The elimination method solves a system of equations.
It utilizes the arithemic properties of equality to eliminate a common variable so that a solution to an equation can be solved.
Substitution Technique
The substitution method solves a system of equations.
It solves one variable from one equation, then substitutes the variable-value into the other equation to discover the other variable-value.
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