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ALGEBRA 2: CHAPTER 3
3.1 Solve Linear Systems by Graphing
Goal  Solve systems of linear equations.
Your Notes
VOCABULARY
System of two linear equations
Two equations, with the variables x and y that can be written as:
Ax + By = C
Dx + Ey = F
Equation 1
Equation 2
Solution of a system
An ordered pair (x, y) that satisfies each equation
Consistent
A system that has at least one solution
Inconsistent
A system that has no solution
Independent
A consistent system that has exactly one solution
Dependent
A consistent system that has infinitely many solutions
Example 1
Solve a system graphically
Graph the system and estimate the solution. Then check the solution algebraically.
4x + 2y = 4 Equation 1
2x  3y = 10 Equation 2
Checkpoint Graph the linear system and estimate the solution. Then check the solution algebraically.
1.
4x + y = 2
6x  3y = 12
NUMBER OF SOLUTIONS OF A LINEAR SYSTEM
Exactly one solution
Infinitely many solutions
No Solutions
Lines intersect at one
point consistent and
independent Infinitely
many solutions it y
Lines coincide; consistent
and dependent
Lines are _parallel_;
_inconsistent_
Example 2
Solve a system with many solutions
Solve the system. Then classify the system as consistent and independent, consistent and dependent, or
inconsistent.
2x + y = 4 Equation 1
4x  2y = 8 Equation 2
Example 3
Solve a system with no solution
Solve the system. Then classify the system as consistent and independent, consistent and dependent, or
inconsistent.
2x + 4y = 8
2x + 4y = 4
Equation 1
Equation 2
Checkpoint Solve the system. Then classify the system as consistent and independent, consistent and
dependent, or inconsistent.
2. 3x  2y = 6
5x + 4y = 8
3. x 2y = 5
2x 4y = 10
4. 6x 3y = 12
6x 3y = 6
5. x + y = 2
4x 3y = 1
Example 4
Writing and using a linear system
Ice Cream Shop At an ice cream shop, one customer pays $9 for 2 sundaes and 2 milkshakes. A second
customer pays $13 for 2 sundaes and 4 milkshakes. How much do each sundae and milkshake cost?
Checkpoint Complete the following exercise.
6. In Example 4, how much do each sundae and milkshake cost if the first customer pays $7 and the second
customer pays $10?
3.2 Solve Linear Systems Algebraically
Goal  Solve linear systems algebraically.
Your Notes
VOCABULARY
Substitution method
Substitute an expression into one of the equations to solve for the variable
Elimination method
Eliminate one of the variables by adding equations
THE SUBSTITUTION METHOD
Step 1 Solve one of the equations for one of its variables.
Step 2 Substitute the expression from _Step 1_ into the other equation and solve for the other variable.
Step 3 Substitute the value from _Step 2_ into the revised equation from Step 1 and solve.
Example 1
Use the substitution method
Solve the system using the substitution method.
Equation 1
x + 2y = 2
3x + 4y = 6
Equation 2
THE ELIMINATION METHOD
Step 1 Multiply one or both of the equations by a _constant_ to obtain coefficients that differ only in _sign_ for
one of its variables.
Step 2 Add the revised equations from _Step 1 _.Combining like terms will _eliminate one of the variables.
Solve for the remaining variable.
Step 3 Substitute the value obtained in _Step 2 _into either of the original equations and solve for the other
variable.
Example 2
Use the elimination method
Solve the system using the elimination method.
2x + 5y = 14
Equation 1
Equation 2
4x + 2y = 4
Checkpoint Complete the following exercises.
1.
Solve the linear system using the substitution method.
2x + y = 2
5x + 3y = 8
2.
Solve the linear system using the elimination method.
3x + 8y = 5
2x + 2y = 18
Example 3
Use the elimination method
Solve the linear system using the elimination method.
Equation 1
3x  4y = 37
Equation 2
5x + 3y = 14
Example 4
Solve linear systems with many or no solutions
Solve the linear system.
a. x 3y = 7
2x  6y = 12
b. 2x  6y = 12
5x + 15y = 30
Checkpoint Solve the linear system.
3.
2x  y = 6
8x  4y = 13
4. 2x + 3y = 7
5x + 7y = 15
3.3 Graph Systems of Linear Inequalities
Goal  Graph systems of linear inequalities
Your Notes
VOCABULARY
System of linear inequalities
A system of two or more linear inequalities in two variables
Solution of a system of linear inequalities
An ordered pair that is a solution of each inequality in the system
Graph of a system of linear inequalities
The graph of all solutions of the system
GRAPHING A SYSTEM OF LINEAR INEQUALITIES
To graph a system of linear inequalities, follow these steps:
Step 1 Graph each inequality in the system. You may want to use colored pencils to distinguish the different
_half-planes_ .
Step 2 Identify the region that is _common_ to all the graphs of the inequalities. This region is the graph of the
system. If you used colored pencils, the graph of the system is the region that has been shaded with
_every_color.
Example 1
Graph a system of two inequalities
Graph the system.
y  2x + 1
Inequality 1
y  x  3
Inequality 2
Example 2
Graph a system with no solution
Graph the system.
3
y   x+ 1
4
3x + 4y  16
Inequality 1
Inequality 2
Checkpoint Graph the system of inequalities.
1. y  2x  3
1
y x+2
2
2. x + 3y  6
1
y
x+1
3
Example 3
Graph a system with an absolute value inequality
Graph the system.
y4
Inequality 1
y  |x  2 | + 1
Inequality 2
Checkpoint Graph the system.
3. y  x 1
y
1
x2
2
x  2
3.4 Solve Systems of Linear Equations in Three Variables
Goal  Solve systems of equations in three variables.
VOCABULARY
Linear equation in three variables
An equation of the form ax+ by+ cz.= d, where a, b, and c are not all zero
System of three linear equations
A system made up of three linear equations in three variables
Solution of a system of three linear equations
The solution is the values of the three variables that make each equation true.
Ordered triple
A coordinate in three variables (x, y, z)
THE ELIMINATION METHOD FOR A THREE-VARIABLE SYSTEM
Rewrite the linear system in three variables as a linear system in two
variables by using the elimination method.
Step 2 Solve the new linear system for both of its variables.
Step 1
Step 3 Substitute the values found in step 2 into one of the original equations and
solve for the remaining variable.
If you obtain a _false_ equation, such as 0 = 1, in any of the steps, then the system has no _solution._
If you do not obtain a false equation, but obtain an _identity_ such as 0 = 0, then the system has _infinitely
many solutions_.
Example 1
Use the elimination method
Solve the system.
3x  2y + 4z = 20
x + 5y + 12z = 73
x + 3y  2z = 1
Equation 1
Equation 2
Equation 3
Checkpoint Solve the linear system.
1.
4x  3y + 5z = 19
3x + y  8z = 21
2x + y + 3z = 13