Algebra 2
Unit 3
Notes: Solving Systems
Classifying Systems of Equations:
Consistent Independent System: A system that has only one solution. The graphs of
each equation intersect at only one point.
Consistent Dependent System: A system that has more than one solution (infinitely
many solutions). The graphs of each equation appear as one graph overlaying each other.
Inconsistent System: A system that has no solution (the empty set ∅, or { }). The graphs of
the equations do not intersect.
Examples:
Consistent Independent
Consistent Dependent
Inconsistent
Solving Systems Algebraically:
Two methods used to solve systems algebraically are Substitution and Elimination.
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When using either method the goal is write and solve an equation in one variable from
the equations in the system.
Then use the value of the variable in the original equations to find the value of the
other variables.
In order to be a solution to the system, the solution must satisfy all equations in the
system.
The number of variables dictates the number of equation necessary to solve the
system, i.e. if two variables, you need two equations, if three variables, then you need
three equations and so on.
Solving Systems using the Substitution Method (Two Variables):
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Solve one equation for one variable.
Substitute value for variable into other equation, so you have an equation in one
variable. Solve for the variable.
If necessary substitute the value of this variable back into the original equation to find
the value of the original variable.
Check solution to system in each equation, must be true for both to be a solution to the
system.
Example:
3𝑎 − 2𝑏 = −3
{
3𝑎 + 𝑏 = 3
Solve 3𝑎 + 𝑏 = 3 for b, 𝑏 = −3𝑎 + 3,
replace the variable b in equation 3𝑎 − 2𝑏 = −3, with −3𝑎 + 3:
3𝑎 − 2(−3𝑎 + 3) = −3 and solve for a:
Substitution
3𝑎 + 6𝑎 − 6 = −3
Distribution
9𝑎 − 6 = −3
Simplify
9𝑎 = 3
Subtraction Property
1
𝑎=3
Division Property
1
Solve for b by replacing a in 𝑏 = −3𝑎 + 3, with 3 and solve for b:
1
𝑏 = −3(3) + 3
Substitution
𝑏 = −1 + 3
Simplify
𝑏=2
Simplify
1
The system is Consistent Independent and (3 , 2) is the solution.
Check in each equation to verify.
 A system will be Consistent Dependent if when solving the variable goes to zero and
you have a true statement.
 A system will be Inconsistent if when solving the variable goes to zero and you have a
false statement.
Example Consistent Dependent System: {
Solve 𝑥 + 3𝑦 = 2 for x, 𝑥 = −3𝑦 + 2
𝑥 + 3𝑦 = 2
4𝑥 + 12𝑦 = 8
Subtraction Property
Substitute for x in 4𝑥 + 12𝑦 = 18: 4(−3𝑦 + 2) + 12𝑦 = 18
−12𝑦 + 8 + 12𝑦 = 8
8=8
Substitution
Distribution
Simplify
Variable went to zero and a true statement remained, Infinitely Many Solutions and the
system is Consistent Dependent.
Example Inconsistent System:
{
𝑥 − 2𝑦 = 4
−4𝑥 + 8𝑦 = 17
Solve 𝑥 − 2𝑦 = 4 for x, 𝑥 = 2𝑦 + 4
Addition Property
Substitute for x in −4𝑥 + 8𝑦 = 17: −4(2𝑦 + 4) + 8𝑦 = 17
−8𝑦 − 16 + 8𝑦 = 17
−16 = 17
Substitution
Distribution
Simplify
Variable went to zero and a false statement remained, No Real Solutions and the system is
Inconsistent.
Solving Systems using the Elimination Method (Two Variables)
 To solve a system of linear equations by elimination.
 Add or subtract the equations to eliminate one of the variables.
 You may first need to multiply one or both of the equations by a constant so that one of
the variables has opposite coefficients in one equation as it has in the other.
 The goal is to produce an equation in one variable to solve for the variable.
 Substitute the value of the variable into one of the original equations and solve for the
other variable.
 If the variables go to zero when adding or subtracting the equations to produce
an equation in one variable. The system is consistent dependent if true
statement or the system is inconsistent if false statement.
Example: {
2𝑥 − 4𝑦 = −26
3𝑥 − 𝑦 = −24
Multiply the equation 3𝑥 − 𝑦 = −24 by – 4 so the coefficients of y are opposites, then add the
equations to eliminate the variable y.
-4 (3𝑥 − 𝑦 = −24) = −𝟏𝟐𝒙 + 𝟒𝒚 = 𝟗𝟔
Equivalent equation
−12𝑥 + 4𝑦 = 96
+ 2𝑥 − 4𝑦 = −26
Add equations
−10𝑥 = 70
Simplify
𝑥 = −7
Division Property
2(−7) − 4𝑦 = −26, substitute the value of x into one equation and solve for y.
−14 − 4𝑦 = −26
−4𝑦 = −12
𝑦=3
Simplify
Addition Property
Division Property
The solution is (-7,3), the system is Consistent Independent.
Consistent Dependent System:
Example: {
4𝑥 − 𝑦 = 8
𝑦
2𝑥 − 2 = 4
𝑦
Multiply the equation 2𝑥 − 2 = 4 by – 2 to and add the equivalent equation to the other
equation.
𝑦
−2 (2𝑥 − 2 = 4) = −𝟒𝒙 + 𝒚 = −𝟖
(+)
4𝑥 − 𝑦 = 8
−4𝑥 + 𝑦 = −8
𝟎=𝟎
Equivalent equation
Add equations
True Statement
Consistent Dependent System, Infinitely Many Solutions.
Inconsistent System:
2𝑥 − 3𝑦 = 4
Multiply 2𝑥 − 3𝑦 = 4 by 2 and 3𝑥 + 2𝑦 = 6 by 3 and add the
−3𝑥 + 2𝑦 = 6
equivalent equations to eliminate the y variable.
Example: {
2(2𝑥 − 3𝑦 = 4)
=
3(−3𝑥 + 2𝑦 = 6)
(+)
6𝑥 − 6𝑦 = 8
−6𝑥 + 6𝑦 = 18
0 = 26
Add equivalent equations
False statement.
Inconsistent system, No Real Solutions.
Solving Systems of Equations by Graphing:
You can solve a system of equations by graphing the equations on the same coordinate plane.
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Write equations in graphing form and graph
If the graphs intersect the point of intersection is the solution and the system is
consistent independent.
If the graphs do not intersect, i.e. parallel lines, the solution is no real solution, and the
system is inconsistent.
If the graph is the same for both, meaning it appears as one graph, the solution is
infinitely many solutions, and the system is consistent dependent.
The following chart summarizes the possibilities for graphs of two linear equations in two
variables.
Graphs of
Equations
Lines intersect
Lines coincide
(same line)
Lines are parallel
Slopes of Lines
Classification of Systems
Number of Solutions
Different slopes
Same slope, same
y - intercept
Same slope, different
y - intercepts
Consistent Independent
Consistent Dependent
One
Infinitely many
Inconsistent
No Real Solutions
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On the TI graphing calculator the following key strokes will identify the
intersection of a graph in the viewing window:
 2nd Trace, #5 Intersection, enter,enter,enter.
Solving Systems of Inequalities by Graphing
To solve a system of Inequalities:
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Graph the inequalities in the same coordinate plane.
Shade each inequality solution set.
The overlapping of the solution sets is the solution to the system.
If the solution sets do not overlap, then no real solutions.
The graph identifies the solution, choose a test point to test in each inequality.
Linear Programming:
Linear programming is use in business applications to determine data that would produce
maximum profit or minimize cost for a business.
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Inequalities are used as constraints to define a feasible region.
If the inequalities form a polygonal region that contains the feasible region, then
it is considered to be bounded and the minimum and maximum points occur at
the vertices.
If the inequalities don’t form a polygonal region that bounds the feasible region,
then it is considered to be unbounded and will only have a minimum or a
maximum value at one of the intersections of the inequalities.
 Write and graph system inequalities
 Find the coordinates of the vertices of the feasible region
 Write an expression(objective function) to be maximized or minimized.
 Substitute the coordinates of the vertices in the expression.
 Select the greatest or least result to maximize or minimize the value.
Solving Systems of Equations in Three Variables:
Use the methods used for solving systems of linear equations in two variables to solve systems
of equations in three variables.
1. Graphing
2. Substitution
3. Elimination
A system of three equations in three variables can have one unique solution, infinitely many
solutions, or no real solutions.
A solution is an ordered triple.
Solving Quadratic Systems:
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Like linear systems, systems of quadratic equations can be solved by substitution and
elimination.
If the graphs are a conic section and a line, the system will have 0, 1,or 2 solutions.
If the graphs are two conic sections, the system will have 0, 1, 2, 3, or 4 solutions.
Solving Quadratic Inequalities:
Systems of Quadratic Inequalities can be solved by graphing.