Download Step 1

objective
• I Can
systems.
• I Can
state the first step for solving
solve systems of equations
by graphing, substitution or
elimination.
BY GRAPHING
Y = 2X + 1
Y = -X + 4
(1,3) IS THE
SOLUTION
Solving a system of equations by graphing.
Let's summarize! There are 3 steps to solving
a system using a graph.
Step 1: Graph both equations.
Graph using slope and y – intercept
or x- and y-intercepts. Be sure to use
a ruler and graph paper!
Step 2: Do the graphs intersect?
This is the solution! LABEL the
solution!
Step 3: Check your solution.
Substitute the x and y values into
both equations to verify the point is a
solution to both equations.
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!!!!
Solving a system of equations by elimination using
multiplication.
Step 1: Put the equations in
Standard Form.
Step 2: Determine
which variable to
eliminate.
Step 3: Multiply the
equations and solve.
Step 4: Plug back in to
find the other
variable.
Step 5: Check your
solution.
Standard Form: Ax + By =
C
Look for variables that have
the
same coefficient.
Solve for the variable.
Substitute the value of the
variable
into the
equation.
Substitute
your
ordered pair
into
BOTH equations.
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