Concepts 1.1
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Transcript Concepts 1.1
Algebra 1
4/10/2016
Solving Systems by
Elimination (multiply)
Objective: solve a linear system by
elimination when first having to
make 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.
Yesterday’s Homework
1. Any questions?
2. Please pass your homework to the front.
• Make sure the correct heading is on your paper.
• Is your NAME on your paper?
• Make sure the homework is 100% complete.
• Incomplete work will NOT be accepted.
Introduction
elimination first
To solve a linear system using __________
multiply one or both of the equations to make
opposites Now add up your two _________
equations and solve
_________.
for the remaining variable. Plug that solution back into
variable
any equation to find the other ________.
Warm-Up
Solve the system by linear combination.
4x 2 y 2
1.
4 x 3 y 13
5 y 15
2x 5 y 2
2.
3x 5 y 2
1x 4
4x 2 y 2
5 4x 2 3 2
y3
5
4x 6 2
6 6
4 x 4
4
1, 3
1 1
2x 5 y 2
2 4 5 y 2
x 4
8 5 y 2
8
8
5 y 10
4
x 1
4, 2
5 5
y 2
Notes
• Solving a System by Linear Combination
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.
Rules for Special Cases
When solving a system of two linear
equations in two variables:
1. If an identity is obtained, such as 0 = 0, then the
system has an infinite number of solutions are
dependent and, since a solution exists, the system is
consistent.
2. If a contradiction is obtained, such as 0 = 7, then
the system has no solution. The system is
inconsistent.
Notes
Solve the system by linear combination.
Ex.
2 2 2
1) Arrange
the variables.
2 x 3 y 12
4 x 5 y 2
2) Make
opposites.
4x 6 y 24
4 x 5 y 2
11 y 22
11
2 x 3 y 12
2 x 3 2 12
2x 6 12
6 6
2x 6
11
y2
2
3, 2
2
x3
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.
1) Arrange
the variables.
2 x 4 y 18
3x y 3
2) Make
opposites.
4 4 4
2 x 4 y 18
12x 4 y 12
10x 30
10 10
x3
2 x 4 y 18
2 3 4 y 18
6 4 y 18
6
6
4 y 24
3, 6
4
4
y 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. 1 1 1
1) Arrange
the variables.
5 x 7 y 9
5 x 3 y 1
5x 7 y 9
5x 3 y 1
4 y 8
4
4
y 2
2) Make
opposites.
5x 7 y 9
5x 7 2 9
5x 14 9
14 14
5x 5
5
1, 2
5
x 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.
3 3 3
1) Arrange
the variables.
2x 5 y 2
3x 2 y 14
2 2
2) Make
opposites.
2
6x 15y 6
6x 4 y 28
11 y 22
11
11
y2
2x 5 y 2
2x 5 2 2
2x 10 2
10 10
2x 12
2
6, 2
2
x6
3) Add and
solve for the
variable.
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Now you try.
1) Arrange
the variables.
Solve the system by linear combination.
Ex.
4 4 4
2) Make
opposites.
3x 4 y 6
4 x 5 y 7
3 3 3
12x 16 y 24
12x 15y 21
1y 3
1 1
y 3
3) Add and
solve for the
variable.
3x 4 y 6
3x 4 3 6
3x 12 6
12 12
3x 6
2, 3
3
3
x 2
4) Substitute
into ANY
original
equation.
5) Check your
answer.
Notes
Now you try.
Solve the system by linear combination.
Ex.
2 2 2
9 x 3 y 12
7 x 2 y 5
3 3 3
18x 6 y 24
21x 6 y 15
39x 39
39
39
x 1
9 x 3 y 12
9 1 3 y 12
9 3 y 12
9
9
3 y 3
1, 1
3 3
y 1
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.
6 x 3 y 12
2 x 4 y 10
1.
2.
2 x 4 y 2
4 x 6 y 8
3, 2
1, 2
4.
7.
10.
4 x 6 y 6
2 x 3 y 15
3, 3
2x 4 y 2
3x 3 y 3
3, 2
4 x 2 y 12
3x 3 y 6
4, 2
5.
8.
11.
3.
6 x 2 y 16
3x 8 y 10
2, 2
2 x 2 y 10
6.
4 x y 17
4, 1
8 x 5 y 18
4 x 4 y 0
6, 6
4 x 3 y 7
3x 4 y 0
4, 3
3x 2 y 1
4 x 3 y 10
9.
2 x 4 y 8
12.
3x 3 y 3
2, 3
1, 2
4 x 3 y 7
3x 4 y 0
4, 3
Ticket Out the Door
Complete the Ticket Out the Door without talking!!!!!
Talking = time after the bell!
Put your NAME on the paper.
When finished, turn your paper face DOWN.
Solve the system.
2 x 3 y 12
4 x 5 y 2
3, 2
Today’s Homework
NO Homework!!!!!
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.