3.2 solving systems algebraically 2. solving systems by elimination
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3.2 Solving Systems Algebraically
2. Solving Systems by Elimination
2) Solving Systems by Elimination
• By adding and subtracting linear systems, you can “eliminate” a variable and solve for an unknown
2) Solving Systems by Elimination
Example 1:
Solve the system by elimination.
4x + 3y = 4
2x – y = 7{
2) Solving Systems by Elimination
Example 1:
Solve the system by elimination.
4x + 3y = 4
2x – y = 7{Elimination – add or subtract the equations of a linear system until you “eliminate” a variable
2) Solving Systems by Elimination
Example 1:
Solve the system by elimination.
4x + 3y = 4 1
2x – y = 7 2
Step 1: Number the equations.
{
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4 1
2x – y = 7 2
Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations.
{
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4 1
2x – y = 7 2 multiply by 2
Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations.
{
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4 1
2x – y = 7 2 multiply by 2
4x – 2y = 14
Step 2: Multiply one or both equations to make the coefficient in front of x OR y the same in both equations.
{2
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4 1
2x – y = 7 2 multiply by 2
4x – 2y = 14
Step 3: Equation subtract
{2
21
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4
4x – 2y = 14
Step 3: Equation subtract 21
1
2-
Use subtraction to eliminate x
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4
4x – 2y = 14
5y = -10
y = -2
Step 3: Equation subtract 21
1
2-
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4
4x – 2y = 14
5y = -10
y = -2
Step 4: Use substitution to solve for the remaining unknown.
1
2-
2) Solving Systems by EliminationExample 1:
Solve the system by elimination.
4x + 3y = 4 Sub y = -2 into
4x + 3(-2) = 4
4x – 6 = 4
4x = 10
x = 2.5
Step 4: Use substitution to solve for the remaining unknown.
11
2) Solving Systems by Elimination
Example 1:
Solve the system by elimination.
Therefore, the solution is (2.5, -2).
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
x + 6y = 2
5x + 4y = 36{
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
x + 6y = 2 multiply by 5
5x + 4y = 36{1
2
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
x + 6y = 2 multiply by 5
5x + 4y = 36 becomes…
5x + 30y = 10
{1
2 1
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
x + 6y = 2 multiply by 5
5x + 4y = 36 becomes…
5x + 30y = 10
subtract
{1
2 1
1 2
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
5x + 30y = 10
5x + 4y = 36
1
2-
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
5x + 30y = 10
5x + 4y = 36
26y = -26
y = -1
1
2-
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
5x + 30y = 10
5x + 4y = 36
26y = -26
y = -1 Sub y = -1 in either equation.
1
2-
2) Solving Systems by Elimination
Example 2:
Solve the system by elimination.
5x + 4y = 36
5x + 4(-1) = 36
5x – 4 = 36
5x = 40
x = 8
2
2) Solving Systems by Elimination
Example 2:Solve the system by elimination.
x + 6y = 25x + 4y = 36
Therefore, the solution to the system is (8, -1).
{
2) Solving Systems by Elimination
Example 2:Solve the system by elimination.
x + 6y = 25x + 4y = 36
Check: 8 + 6(-1) = 2 5(8) + 4(-1) = 36
2 = 2 36 = 36
{
2) Solving Systems by Elimination
Example 3:
Solve each system by elimination.
a) -3x + 5y = 7 b) 2x – 3y = 18
6x – 10y = -14 -2x + 3y = -6{ {
2) Solving Systems by Elimination
Example 3:
Solve each system by elimination.
a) -3x + 5y = 7 b) 2x – 3y = 18
6x – 10y = -14 -2x + 3y = -6{ {
Multiply by 2
2) Solving Systems by Elimination
Example 3:Solve each system by elimination.
a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6
0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12
{ {+ +
2) Solving Systems by Elimination
Example 3:Solve each system by elimination.
a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6
0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12
{ {
Always true.
The equations represent the same line.
The system is dependent.
There is an infinite number of solutions.
+ +
2) Solving Systems by Elimination
Example 3:Solve each system by elimination.
a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6
0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12
{ {
Never true.
The equations represent parallel lines.
The system is inconsistent.
There is no solution.
+ +
Always true.
The equations represent the same line.
The system is dependent.
There is an infinite number of solutions.
2) Solving Systems by Elimination
Example 3:Solve each system by elimination.
a) -6x + 10y = 14 b) 2x – 3y = 18 6x – 10y = -14 -2x + 3y = -6
0x + 0y = 0 0x + 0y = 12 0 = 0 0 = 12
{ {
Never true.
The equations represent parallel lines.
The system is inconsistent.
There is no solution.
+ +
Always true.
The equations represent the same line.
The system is dependent.
There is an infinite number of solutions.
Homework p.128 #18-21, 33, 36, 46, 55, 56, 57, 62
Homework
p.128 #18-21, 33, 36, 46, 55, 56, 57, 62
Tomorrow: In-class assignment…come prepared!