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BY GRAPHING
Y = 2X + 1
Y = -X + 4
(1,3) IS THE
SOLUTION
Graphing is not the only way to
solve a system of equations. It is
not really the best way because it
has to be graphed perfectly and
some answers are not integers.
SOOOO
We need to learn another way!!!!
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