Download Concepts 1.1

5/24/2017
Solving Systems by
Elimination (opposites)
Objective: solve a linear system by
elimination when already having
opposites.
Students solve a system of two linear equations in
two variables algebraically and are able to interpret
the answer graphically. Students are able to solve a
system of two linear inequalities in two variables and
to sketch the solution sets.
Warm-Up
Solve the system by substitution.
1.
x  3


2 x  4 y  6
2x  4 y  6
2 3  4 y  6
6  4 y  6
6
6
4 y  12
4
4
y 3
 3, 3
2.
y  2x


3x  4 y  10
3 x  4 y  10
3x  4  2x   10
3x  8x  10
5x  10
5 5
x2
y  2x
y  2 2 
y4
 2, 4
Summary
Write a summary about today’s lesson.
opposites
When a linear system already has ________
just add up both equations and solve for one of the
variables. Next plug that solution back into any of the
variable
equations to find the other ________.
original _________
Notes
• Solving a System by Elimination
1. Arrange the like variables in columns.
- This is already done.
2. Pick a variable, x or y, and make the two equations
opposites using multiplication.
3. Add the equations together (eliminating a variable)
and solve for the remaining variable.
4. Substitute the answer into one of the ORIGINAL
equations and solve.
5. Check your solution.
Notes
Solve the system by linear combination.
Ex.
 2x  y  5

2 x  3 y  7
4 y  12
4
2x  y  5
2x   3   5
3 3
2x  2
2 2
x 1
4
y 3
1, 3
1) Arrange
the variables.
2) Make
opposites.
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Solve the system by linear combination.
Ex.
 2x  y  5

2 x  3 y  7
1, 3
Check
2x  y  5
2 1    3   5
23  5
55
2 x  3 y  7
2  1   3 3   7
2  9  7
77
1) Arrange
the variables.
2) Make
opposites.
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Solve the system by linear combination.
Ex.
 5 x  4 y  6

3x  4 y  2
2 x  4
2
2
x  2
1) Arrange
the variables.
2) Make
opposites.
5 x  4 y  6
5 2  4 y  6
10  4 y  6
 10
 10
4y  4
4 4
y 1
 2, 1
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Solve the system by linear combination.
Ex.
 5 x  4 y  6

3x  4 y  2
 2, 1
Check
5 x  4 y  6
5 2  4  1   6
10  4  6
6  6
3x  4 y  2
3 2  4  1   2
64  2
22
1) Arrange
the variables.
2) Make
opposites.
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Now you try.
Solve the system by linear combination.
Ex.
3x  5 y  2

 3x  6 y  6
1y  4
1 1
y  4
1) Arrange
the variables.
2) Make
opposites.
3x  5 y  2
3x  5 4  2
3x  20  2
 20  20
3x  18
3
 6,  4
3
x  6
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Solve the system by linear combination.
Ex.
3x  5 y  2

 3x  6 y  6
 6,  4
Check
3x  5 y  2
3 6  5 4  2
18   20  2
2   2
3x  6 y  6
3 6  6 4  6
18  24  6
66
1) Arrange
the variables.
2) Make
opposites.
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Solve the system by linear combination.
Ex.
6 x  7 y  10

 6 x  7 y  10
00
Infinite Solutions
1) Arrange
the variables.
2) Make
opposites.
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Now you try.
Solve the system by linear combination.
Ex.
 5x  8 y  9

5 x  8 y  2
0  11
No Solution
1) Arrange
the variables.
2) Make
opposites.
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Class Work
Solve the system by linear combination.
 2x  5 y  2
 x  4 y  1
1.
2.
3.


2 x  3 y  18
 x y 2
 6,  2
3, 1
 x  2 y  5
5 x  6 y  4
4.
5.
6.


3x  2 y  11
 2x  6 y  2
 3, 1
 2, 1
6 x  2 y  2
 3x  y  13
7.
8.
9.


 6 x  2 y  4
4 x  y  16
3, 4
No Solution
10.
3x  y  7

 x  y 1
 4,  5
11.
2 x  2 y  18
12.

2 x  y  9
 6, 3
4 x  3 y  7

 4 x  2 y  2
 1, 1
3x  y  14

2 x  y  1
3,  5
 5 x  4 y  13

5 x  4 y  13
Infinite Solutions
5 x  2 y  16

  x  2 y  8
 4,  2
Today’s Homework
p414 #8-15
Rules for Homework
1. Pencil ONLY.
2. Must show all of your work.
• NO WORK = NO CREDIT
3. Must attempt EVERY problem.
4. Always check your answers.