Algebra 2
Chapter 3
3.1 Solve Systems Using Tables & Graphs
3.1 Solving Systems Using Tables and Graphs
A solution to a system of linear equations is an _______________________________ that makes all of
the equations ___________.
To solve a system of equations by graphing:
1. ______________ each line very carefully.
2. _____________ for where the lines intersect.
3. _____________ the solution in both equations.
Example: Solve each system by graphing
1.
y 2x 1
1
y 2 x 4
2.
x 4 y 12
3x 4 y 4
1
Algebra 2
8 x 4 y 12
2 x y 3
3.
Chapter 3
3.1 Solve Systems Using Tables & Graphs
7 x y 6
7 x y 6
4.
has at least ______ solution
Independent: has _________________ solution
So what words do we use
to describe these systems?
Consistent:
Inconsistent:
has _____ solution
Dependent: has __________________ solutions.
2
Algebra 2
Chapter 3
3.2 Solving Systems Algebraically
3.2 Solving Systems Algebraically
Sometimes, it is difficult to graph equations accurately by hand.
Sometimes, the points of intersection are not integer values and are difficult to see.
Sometime, we need a different way to find the solution to a system of equations.
A: Solving Systems by Substitution
Example: Solve by substitution
Make a Plan:
This method works well when
One equation is already solved
for one variable in terms of
another like y= or x =
1.
x 2 y 5
x 2 3y
2.
2 x y 6
7 x 6 y 1
One equation has a term with
a coefficient of 1 or -1.
Make it Happen:
If necessary, solve one
equation for one of the
variables.
Get x = or y =
Substitute the expression for
the variable in the other
equation.
Solve the equation. You will
almost always have to
distribute first.
Substitute your solution into
either original equation.
Solve for the remaining
variable.
Write your answer as an
ordered pair.
CHECK YOUR ANSWER IN BOTH
EQUATIONS!
3
Algebra 2
(Solve by Substitution, Continued)
4 x 3 y 10
3.
x 2 y 10
Chapter 3
3.2 Solving Systems Algebraically
2 x y 12
3x 7 y 1
4.
Make a Plan:
This method works well when
No variable is already solved for
B: Solving by Elimination
5.
3x 2 y 14
2 x 2 y 6
No coefficient is 1 or -1
Make it Happen:
GOAL – to create additive opposites
Ex. 4x, -4x or -7y, 7y
If necessary, multiply one or both
equations by a constant to create additive
opposites.
Add the two equations. This should
eliminate one of the variables.
Solve for the variable.
Substitute into either original equations
and solve for the remaining variable.
Write your answer as an ordered pair.
CHECK YOUR ANSWER IN BOTH
EQUATIONS!
4
Algebra 2
Chapter 3
3.2 Solving Systems Algebraically
(Solve by Elimination)
6.
x 2 y 10
3x y 9
7.
2 x 7 y 4
3x 5 y 5
8.
x y 2
2 x 2 y 0
9.
4 x y 6
12 x 3 y 18
5
Algebra 2
Chapter 3
3.3 Systems of Inequalities
3.5 Systems with Three Variables
In lesson 3.1 & 3.2, we learned about systems of 2 equations with 2 variables, that may have a solution
(x, y) that makes both equations true.
In this lesson, we will learn about systems of 3 equations and 3 variables that may have a solution
(x, y, z) that makes all three equations true.
Example: Solve each system of equations. Check your answers.
Make a Plan
GOAL: Use elimination (lesson 3.2)
to change your 3 x 3 system to a 2 x
2 system.
Look for a variable that can easily
be eliminated from all three
equations.
1.
x y z 1
x y 3z 3
2 x y 2 z 0
Make it Happen
Choose a variable to eliminate.
Choose 2 equations & eliminate the
variable.
Choose a different pair of equations
& eliminate the SAME VARIABLE
again.
You now have TWO new equations
each with TWO variables. Use
substitution or elimination to solve
(Like 3.2)
Substitute your solutions into one
of the ORIGINAL 3 variable
equations and solve for the missing
term.
Write your answer as an
alphabetical ordered triplet. (x, y, z)
6
Algebra 2
2.
x y 2 z 7
3x y 2 z 7
x 3 y z 9
3.
x 2 y 3z 12
2 x y 2 z 5
2 x 2 y z 4
Chapter 3
3.3 Systems of Inequalities
7
Algebra 2
Chapter 3
3.3 Systems of Inequalities
3.3 Systems of Inequalities
A: Graphing a Linear Inequality:
(REVIEW)
Graph the Inequality like an equation
If < or > use a __________________ line.
If ≤ or ≥ use a __________________ line.
Test a point on one side of your boundary line.
If the point satisfies the inequality, shade that side
of the line. If the point does not, shade the
opposite side of the line.
In general: If line is y = mx + b
Graph: y > 3x – 4
< or ≤ means shade ______________
> or ≥ means shade _______________
B: Solving a System of Linear Inequalities
The solution to a system of equations is the _________ that makes both equations true.
The solution to a system of inequalities is the ____________ of the two regions that make each
inequality true.
Examples: What is the solution of the system of inequalities
1.
x 3
y4
2.
y2
y 3
x 4
8
Algebra 2
Chapter 3
3.3 Systems of Inequalities
(Solve the systems of inequalities)
3.
1
x 3
2
y 2x
y
4.
x 3y 3
x 2y 4
5.
y3
y x4
9
Algebra 2
Chapter 3
3.5 Systems of Three Variables
3.4 Linear Programming
A: Terminology
Graph the following system of inequalities
x 2
3 y 6
x y 10
Some real-world problems involve multiple linear relationships.
We call the inequalities the _____________________________.
The overlapping region (the solution to the system) is called the __________________________.
The corners of the region are called the __________________.
In real world problems, you will be trying to MAXIMIZE or MINIMIZE a quantity that depends
upon the constraints. Most often this is __________________ or ________________.
The equation that models what you are trying to maximize or minimize is called the
______________________________________________.
For the feasible region above, find the maximum and minimum value of the objective function:
C = 2x + y
10
Algebra 2
Chapter 3
3.5 Systems of Three Variables
B: Linear Programming Procedure
Example 1: Find the values of x and y that
maximize the objective function for the graph.
Example 2: Find the values of x and y that
minimize the objective function for the graph.
P = 4x – y
C = x + 9y
Example 3: Graph each system of constraints. Name all vertices.
Then find the values of x and y that maximize or minimize the
objective function.
Make it Happen
1. Graph the Inequalities
(the constraints)
2. Form the feasible
region.
3. Find the coordinates of
each vertex.
4. Evaluate the objective
function at each vertex.
x y 8
2 x y 10
x 0
y 0
Maximum for
N = 100x + 40y
5. State the Max/Min and
where it occurs.
11
Algebra 2
Chapter 3
3.5 Systems of Three Variables
Example 4:
x 2y 6
2x y 7
x 2
y 3
Find the Maximum and Minimum for A = x – y
12
0
