January 31, 2014

31

Today:
Warm-Up: (4)

Review Systems of Equations
New Solving Techniques

Monday: Review for Test Tuesday
4th Period: Only 10 Notebooks submitted
Please leave notebooks again before you

leave.

Warm the
a. Write the equation ofUp line
b. Write the inequality an the line. for a line
1. Write of equation
perpendicular to 2x -4y = -2
2. Solve for a: 9a – 2b = c + 4a

4. Write the systems of equations
shown by the graph below.

Review: Solve Systems of Equations by Graphing
1 =

1+0

2 +

( + 0)
1 = 3

(2,1)

Step 1: Put both equations in
slope - intercept form.

Step 2: Graph both equations on
the same coordinate plane.

Step 3: Plot the point where the
graphs intersect.

Step 4: Check to make sure your
solution makes both equations true.

Review: Solve Systems of Equations by Elimination
(addition or subtraction)
 Elimination is easiest when the equations
are in standard form.
1: Put the equations in
Standard Form.
Step 2: Determine which

variable to eliminate.
Step 3: Add or subtract

the equations.
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.


2x + 7y = 31
5x - 7y = - 45
7x + 0 = -14

x = -2

Like variables must be lined under each other.

THEN----


2X + 7Y = 31
2(-2) + 7y = 31
-4 + 7y = 31
Substitute your
4
4
answer into either
original equation
and solve for the
second variable.

7y = 35; y = 5
Solution

(-2, 5)

Now check our answers in both
equations------

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

Solve Systems of Equations by Elimination
(Multiplying)

Like variables
must be lined
under each
other.

x + +y1y 4 4
1x = =
2x + 3y = 9

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.

(Multiplying)

( X + Y = 4) -2
2X + 3Y = 9

-2X - 2 Y = - 8
2X + 3Y = 9

Now add the two
equations and solve.

THEN----

Y=1

(Multiplying)

X+Y=4

X +1=4
- 1 -1
X=3
Solution

Substitute your
answer into either
original equation
and solve for the
second variable.

(3,1)

Now check our answers in both equations--

x+y=4
3+1=4
4=4

2x + 3y = 9
2(3) + 3(1) = 9
6+3=9

9=9

(Multiplying)
3x – 2y = -7
2x -5y = 10

Can you multiply either equation
by an integer in order to eliminate
one of the variables?

Here, we must multiply both
equations by a (different)
number in order to easily
eliminate one of the variables.

 Eliminate
 Plug back in
 Solve for other
variable

Multiply the top
equation by 2, and the
bottom equation by -3

Write your solution as
an ordered pair
(-5,-4)
Plug both solutions into
original equations

3x – 2y = -7
-15 – (-8) = -7
-7 = - 7

2x - 5y = 10

-10 – (-20) = 10
10= 10

Solve: By Substitution
Recall that when we 'solve' a point-slope formula,
we end up in slope-intercept form. In much the
same way, the substitution method is closely
related to the elimination method.
After eliminating one variable and solving for the other,
we substitute the value of the variable back into the
equation.
For example: Solve 2x + 3y = -26 using elimination
4x - 3y = 2
What is the
value of x ?

-4

At this point we substitute -4 for
x, and solve for y. This is exactly
what the substitution method is
except it is done at the beginning.

Solve: By Substitution
Example 1: y = 2x
4x - y = -4

Since the first equation tells us
that y = 2x, replace the y with 2x
in the second equation.

4x - 2x = -4; 2x = -4; x = -2

Then, substitute -2 for x in the first equation:
y = 2(-2); y = -4
Finally, plug both values in and check for equality.
-4 = 2(-2); True;
4(-2) - (-4) = -4; -8 + 4 = -4; True

January 31, 2014

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January 31, 2014