Transcript Document

BY GRAPHING
Y = 2X + 1
Y = -X + 4
(1,3) IS THE
SOLUTION
Example #1:
y = 4x
3x + y = -21
Step 1: Solve one equation for one variable.
y = 4x
(This equation is already solved for y.)
Step 2: Substitute the expression from step one into
the other equation.
3x + y = -21
3x + 4x = -21
Simplify and solve the equation.
7x = -21
x = -3
y = 4x
3x + y = -21
Step 3: Substitute what you solved into Step 1
y = 4x
y = 4(-3)
y = -12
Solution to the system is (-3, -12).
y = 4x
3x + y = -21
Step 4: Check the solution in both equations.
Solution to the system is (-3,-12).
y = 4x
-12 = 4(-3)
-12 = -12
3x + y = -21
3(-3) + (-12) = -21
-9 + (-12) = -21
-21= -21
Example #2:
Solve using “Substitution”
x + y = 10
5x – y = 2
Step 1: Solve one equation for one variable.
x + y = 10
y = -x +10
Step 2: Substitute the expression from step one into
the other equation.
5x - y = 2
5x -(-x +10) = 2
x + y = 10
5x – y = 2
Simplify and solve the equation.
5x -(-x + 10) = 2
5x + x -10 = 2
6x -10 = 2
6x = 12
x=2
x + y = 10
5x – y = 2
Step 3: Substitute back what you solved into step 1
y = -x + 10
y = -(2) + 10
y=8
Solution to the system is (2,8).
Like variables
must be lined
under each
other.
We need to
eliminate
(get rid of)
a variable.
The x’s will
be the
easiest. So,
we will add
the two
equations.
Solve:
by ELIMINATION
x + y = 12
-x + 3y = -8
Divide by 4
4y = 4
y=1
THEN----
Substitute
your answer
into either
original
equation and
solve for the
second
variable.
X +Y = 12
x + 1 = 12
-1 -1
x = 11
(11,1)
Answer
Now check our answers
in both equations------
X + Y =12
11 + 1 = 12
12 = 12
-x + 3y = -8
-11 + 3(1) = -8
-11 + 3 = -8
-8 = -8
Like variables
must be lined
under each
other.
We need to
eliminate
(get rid of)
a variable.
The y’s be
will the
easiest.So,
we will add
the two
equations.
Solve:
by ELIMINATION
5x - 4y = -21
-2x + 4y = 18
Divide by 3
3x = -3
x = -1
THEN----
Substitute
your answer
into either
original
equation and
solve for the
second
variable.
5X - 4Y = -21
5(-1) – 4y = -21
-5 – 4y = -21
5
5
-4y = -16
y=4
(-1, 4)
Now check our answers
in both equations-----Answer
5x - 4y = -21
5(-1) – 4(4) = -21
-5 - 16 = -21
-21 = -21
-2x + 4y = 18
-2(-1) + 4(4) = 18
2 + 16 = 18
Like variables
must be lined
under each
other.
We need to
eliminate
(get rid of)
a variable.
The y’s will
be the
easiest. So,
we will add
the two
equations.
Solve:
by ELIMINATION
2x + 7y = 31
5x - 7y = - 45
Divide by 7
7x = -14
x = -2
THEN----
Substitute
your answer
into either
original
equation and
solve for the
second
variable.
2X + 7Y = 31
2(-2) + 7y = 31
-4 + 7y = 31
4
4
7y = 35
y=5
(-2, 5)
Now check our answers
in both equations-----Answer
2x + 7y = 31
2(-2) + 7(5) = 31
-4 + 35 = 31
31 = 31
5x – 7y = - 45
5(-2) - 7(5) = - 45
-10 - 35 = - 45
- 45 =- 45
Like variables
must be lined
under each
other.
We need to eliminate
(get rid of) a variable.
To simply add this
time will not eliminate
a variable. If one of the
x’s was negative, it
would be eliminated
when we add. So we
will multiply one
equation by a – 1.
Solve:
by ELIMINATION
x + y = 30
x + 7y = 6
X + Y = 30
X + Y = 30
( X + 7Y = 6 ) -1
-X – 7Y = - 6
-6Y = 24
Now add the two
equations and
solve.
-6
-6
Y=-4
THEN----
Substitute
your answer
into either
original
equation and
solve for the
second
variable.
X + Y = 30
X + - 4 = 30
4
4
X = 34
(34, - 4)
Now check our answers
in both equations-----Answer
x + y = 30
34 + - 4 = 30
30 = 30
x + 7y = 6
34 + 7(- 4) = 6
34 - 28 = 6
6=6
Like variables
must be lined
under each
other.
We need to eliminate
(get rid of) a variable.
To simply add this
time will not eliminate
a variable. If there was
a –2x in the 1st
equation, the x’s
would be eliminated
when we add. So we
will multiply the 1st
equation by a – 2.
Solve:
by ELIMINATION
x+ y=4
2x + 3y = 9
( X + Y = 4 ) -2
2X + 3Y = 9
-2X - 2 Y = - 8
2X + 3Y = 9
Y=1
Now add the two
equations and
solve.
THEN----
Substitute
your answer
into either
original
equation and
solve for the
second
variable.
X+Y=4
X +1=4
- 1 -1
X=3
(3,1)
Now check our answers
in both equations-----Answer
x+y=4
3+1=4
4=4
2x + 3y = 9
2(3) + 3(1) = 9
6+3=9
9=9