3-1 Solving Linear Systems by Graphing
Name:__________________
Objective: To solve systems of linear equations by graphing.
Algebra 2 Standard 2.0
*A system of two linear equations in two variables, _____ and _____, also called a linear
system, consists of two equations that can be written in the following form:
Ax + By = C (equation 1)
Dx + Ey = F (equation 2)
A __________________ of a system of equations is an ordered pair (x, y) that satisfies both
equations. On a graph, the solution is where the two lines ___________________.
* Graph the linear system and estimate the solution. Then check the solution algebraically.
Ex. 1: 5x – 2y = -10
2x – 4y = 12
You Try: 3x + 2y = -4
x + 3y = 1
*Number of Solutions of a Linear System
Exactly one solution
Lines intersect at ________
point.
Algebra 2 Chapter 3A Notes Page 1
Infinitely Many Solutions
No Solutions
Lines are the _____________.
Lines are ________________.
* Solve the system.
Ex. 2: 6x – 2y = 8
3x – y = 4
Ex. 3: -4x + y = 5
-4x + y = -2
You Try: 3x – 2y = 10
3x – 2y = 2
You Try:
2x + 5y = 6
4x + 10y = 12
Ex. 4: A soccer league offers two options for membership plans. Option A includes an initial
fee of $40 and costs $5 for each game played. Option B costs $10 for each game played. After
how many games will the total cost of the two options be the same?
A. 2 games
B. 4 games
Algebra 2 Chapter 3A Notes Page 2
C. 8 games
D. 12 games
3-3 Graph Systems of Linear Inequalities
Name:_______________
Objective: To graph a system of linear inequalities
Algebra 2 Standard 2.0
*A solution to a system of inequalities is an ordered pair that is a solution of each inequality in
the system. The graph of a system of inequalities is the graph of all solutions of the system.
*How to Graph a system of linear inequalities:
1. Graph each inequality in the system. Remember _________________ and
_____________________ lines.
2. Shade the part that is common to all graphs of the inequalities. This region is the
solution of the system.
Examples: Tell whether the ordered pair is a solution of the inequality.
1.  x  4 y  2 ; (0, 0)
2. 3 x  y  1 ; (1, 1)
Examples: Graph the system of inequalities.
3.
y  3x  2; y   x  4
4.
2x 
1
y  4; 4 x  y  5
2
You Try: y  3x  2; y  x  4
You Try: Write an inequality for the graph shown.
(1, 2)
(–2, –3)
Algebra 2 Chapter 3A Notes Page 3
5. y  2; y  x  1
You Try: x  2; y   x  2
*Graphing more than two inequalities.
6.
y  2 x  1; y  5 x  8;
1
5
y  x
4
2
7. Tell whether each ordered pair is a
solution of the system of inequalities.
You Try:
y  3; y  2;
x  4; x  3
8. Write a system of inequalities for
the shaded region.
a. (3, 3)
b. (0, 0)
c. (3, 3)
Algebra 2 Chapter 3A Notes Page 4
(4, 1)●
(–4, 1) ●
(0, -3) ●
3-2 Solve Linear Equations Algebraically
Name:______________
Objective: To solve linear systems using Substitution method.
Algebra 2 Standard 2.0
*Substitution Method:
1. Solve one equation for one of its variables. (Hint, look for a “singleton”)
2. Substitute the expression from Step 1 into the __________________ equation and solve
for the remaining variable.
3. Substitute the value from Step 2 into the revised singleton equation from Step 1.
Solve the linear system using substitution method.
Ex. 1: 3x + 2y = 1
-2x + y = 4
You Try:
4x + 3y = -2
x + 5y = -9
**If after Step 2, both variables have been eliminated
and what’s left is true, then there are ___________________ solutions.
and what’s left is false, then there is __________________ solution.
Ex. 2: x – y = 4
-6x + 6y = -24
You Try:
x – 2y = 4
3x – 6y = 8
Algebra 2 Chapter 3A Notes Page 5
Ex. 3: 2x + y = 4
4x + 2y = 2
You Try:
5 x  3 y  20
3
 x  y  4
5
*Elimination Method:
1. Multiply one or both equations by a ____________________ to obtain the opposite
coefficients.
2. _______________ the revised equations from Step 1. Combining like terms will
eliminate one of the variables. Solve for the remaining variable.
3. Substitute the value obtained from Step 2 into either of the original equations and solve
for the remaining variable.
Solve the linear systems using elimination method.
Ex. 1: 8x + 2y = 4
-2x + 3y = 13
You Try:
Ex. 2: 2x – 3y = 4
6x – 9y = 8
Ex. 3:
You Try:
You Try: 3x + 5y = -4
2x - 3y = 29
Ex. 4:
3x + 4y = 18
6x + 8y = 18
1
2
5
x y 
2
3
6
5
7
3
x y 
12
12
4
Algebra 2 Chapter 3A Notes Page 6
3x + 3y = -15
5x – 9y = 3
12x – 3y = -9
-4x +y = 3
3-4 Solve Systems of Linear Equations in Three Variables
Objective: To solve a system of equations in three variables.
Algebra 2 Standard 2.0
Name:__________________
*A linear system in three variables is an equation in the form ax + by + cz = d where a, b, and c
are not all zero. A solution of such a system is an ordered __________________ (x, y, z)
whose coordinates make each equation true.
The graph is a plane in three-dimensional space. See page 178 for examples.
*Elimination method for a three-variable system:
1. Rewrite the three variable system as a two variable system by using the elimination
method.
2. Solve the new linear system for both of its variables.
3. Substitute the values found in Step 2 into any of the original equations and solve for the
remaining variable.
*If you obtain a false equation (ex: 0 = 1) in any of the steps,
then the system has ___________________________.
*If you obtain an identity (a true equation like 0 = 0),
then system has _______________________________.
* Solve the system using Elimination Method.
2 x  y  6 z  4
Ex. 1: 6 x  4 y  5 z  7
4 x  2 y  5 z  9
Algebra 2 Chapter 3A Notes Page 7
x yz 2
Ex. 2: 3 x  3 y  3 z  8
2x  y  4z  7
x yz 6
Ex. 3: x  y  z  6
4 x  y  4 z  24
Ex. 4: Solve using Substitution method.
x yz 4
3 x  2 y  4 z  17
x  5y  z  8
You Try: Solve using any method.
3 x  y  2 z  10
You Try: 6 x  2 y  z  2
x  4 y  3z  7
x yz 3
You Try: x  y  z  3
2x  2 y  z  6
x yz 2
You Try: 2 x  2 y  2 z  6
5 x  y  3z  8
*Solve using Substitution method.
Ex. 4: Tom, Bob, and Joe are brothers. Their combined ages are 46. Joe was born 3 years before
Bob and 4 years after Tom. How old are they?
Algebra 2 Chapter 3A Notes Page 8