Download Joke of the Day Systems of Linear Equations in Two

Joke of the Day
Systems of Linear Equations in Two
Variables – Elimination Method
Objective: To solve systems of
equations using the elimination
method.
Are the given ordered pairs solutions of
the given systems?
Definitions
System – 2 or more equations
together
Solution of system – any ordered
pair that makes all equations true
Note: If the lines are graphed, the
point of intersection is the solution
of the system.
We must substitute the given ordered pair (x,y)
into the equations and verify that it “works” in both
equations. SOLUTION
a)
2x + y = -6 and x + 3y = 2; (-4,2)
2(-4)+2=-6
-4+3(2)=2
-8+2=-6
-4 + (6) = 2
-6 = -6
2=2
a)
2x + y = -6 and x + 3y = 2; (-4,2)
b)
9x – y = -4 and 4x + 3y = 11; (-1,5)
SOLUTION (continued)
9x – y = -4 and 4x + 3y = 11; (-1,5)
9(-1) – 5 = -4
4(-1) + 3(5) = 11
-9 – 5 = -4
-4 + 15 = 11
-14 = -4
11=11
Even though the ordered pair worked in
one equation, it did not work in the
other. Therefore, it is NOT a solution to
the system.
b)
1
Methods for Solving Systems
STEPS for ELIMINATION
Graphing
Substitution
3. Elimination – to use math
operations to eliminate a variable
1.
2.
In this lesson, we will concentrate on
the ELIMINATION method.
Solve using elimination.
1) Look for a variable that has OPPOSITE
coefficients. If found, ADD the two
equations. Solve, substitute, solve.
2) If no opposite coefficients, look for
variables with the SAME coefficients. If
found, SUBTRACT one equation from the
other. Then, solve, substitute, solve.
Note: Possible solutions are
One point
Infinitely many solutions
No solution
SOLUTION
C) 3x – y = -7 D) 2x - y = -2
2x + y = -3
-2x - y = 2
Add 5x = -10
-2y = 0
x = -2
y=0
Sub this value into either point to find the
other value.
2(-2) + y = -3
2x – 0 = -2
-4 + y = -3
2x = -2
y=1
x = -1
Solution: (-2,1)
(-1,0)
C) 3x – y = -7
2x + y = -3
D)
2x - y = -2
-2x - y = 2
Solve using elimination.
E) x - y = 4
2x + y = 8
F) 2x + y = 7
3x + y = 12
Solve using elimination.
E) x - y = 4
F) 2x + y = 7
2x + y = 8
3x + y = 12
Add: 3x = 12 Subtract: -x = -5
x=4
x=5
Sub this value into either point to find the other value.
4–y=4
-y = 0
y=0
Solution: (4,0)
2(5) + y = 7
10 + y = 7
y = -3
(5,-3)
2
Using Multiplication with
Elimination
What if neither variable can be eliminated by
simply adding or subtracting?
We
may have to MULTIPLY
before adding or subtracting!
SOLUTION – Multiply top equation
by -2 and then add.
Solve by elimination.
G) x + 3y = 8
2x – 5y = -17
Solve by elimination.
H) 7x + 8y = -2
3x – 2y = 10
Write both equations in standard form.
Look for the easiest way to get one
variable to have opposite coefficients.
(Hint: Think like you’re finding an LCD).
Multiply EVERY term in the equation by
the factor needed to get the opposites.
Follow the same steps for elimination with
addition or subtraction.
Note: There are several different paths to
the same correct answer with these
problems. (not only one way to do them)
-2x – 6y = -16
2x – 5y = -17
-11y = -33
y=3
Substitute for y in the original, first equation
and find x. (Or you could substitute for y
in th esecond equation…your choice.)
x + 3(3) = 8
x+9=8
x = -1
Solution: (-1,3)
G) -2(x + 3y = 8)
2x – 5y = -17
ONE SOLUTION – Multiply top equation by 2
and bottom equation by 8 and then add.
H) 7x + 8y = -2
3x – 2y = 10
2(7x + 8y = -2)
8(3x – 2y = 10)
14x + 16y = -4
24x – 16y = 80
38x
= 76
x=2
Substitute for x and find y.
3(2) – 2y = 10
6 – 2y = 10
-2y = -4 so
y = 2;
Solution: (2,-2)
3
Special Cases
More Examples
As shown in the previous examples, many
systems of equations have one point or
ordered pair that is the solution.
However, there are other systems that
have no solution or infinitely many
solutions.
For these special cases, while working the
problem two things can happen:
1) You get a false statement. (no solution)
2) You get a true statement. (infinitely
many solutions)
Solution: Multiply the top equation by
4 and then add.
8x + 4y = 24
-8x – 4y = -24
0=0
This is a true statement, which tells us
that ANY ordered pair is a solution for
this system. Therefore, the answer is
INFINITELY MANY SOLUTIONS.
J) 4(2x + y = 6)
-8x – 4y = -24
Solution: Multiply the top equation by
-2 and then add
K) -2(4x – 3y = 8)
8x - 6y = 14
-8x + 6y = -16
8x - 6y = 14
0 + 0 = -2
This is a false statement. Therefore, the
answer is NO SOLUTION.
J) 2x + y = 6
-8x – 4y = -24
More Examples
K) 4x – 3y = 8
8x - 6y = 14
POSSIBLE SOLUTIONS for a System
of Linear Equations
1.
2.
3.
Answer is an ordered pair, (x, y).
If variables cancel out and you get a
true statement, the solution is
infinitely many solutions. (Graph would
be same line.)
If variables cancel out and you get a
false statement, there is no solution.
(Graph would be parallel lines.)
4