SOLVING EQUATIONS

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SOLVING EQUATIONS 4x 3 9
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Solution of the equation The number ( value) that
makes the equation true Properties of
Equality Reflexive a a anything is equal to
itself Symmetric If a b then b a if a
and b are equal they can be interchanged in a
problem to find the solution (substitution) Transi
tive If a b and b c, then a c this says
a,b,and c are all the same value and can be
interchanged to help find a solution to an
equation
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Addition if a b then a c b c this says
we keep equality in an equation if we add the
same quantity to both sides Subtraction If a b
then a c b c this says we keep equality
in an equation if we subtract the same quantity
from both sides Multiplication if a b then ac
bc if we multiply both sides of an equation by
the same amount we keep equality
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Division if a b then a /c b/c c 0 here
again if the two sides of an equation are equal
we can divide both sides by the same number and
still maintain the equality All ADDS
UP WHATEVER YOU DO TO ONE SIDE OF AN EQUATION
YOU MUST DO TO THE OTHER TO MAINTAIN THE
EQUALITY!!!!!!!!
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Lets Look at these (Variable on Both Sides)
8z 12 5z 21 8z 5z 12 5z 5z
21 (Subtraction) 3z 12 - 21 3z
12 12 -21 -12 (Subtraction)
3z -33 3z / 3 - 33 /
3 (Division) z
-11 CHECK 8z 12 5z 21
8(-11) 12 5(-11) 21
-76 -76
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2t 3 9 4t 2t 4t 3 9 4t 4t
(addition) 6t 3 9 6t 3 3
9 3 (addition) 6t
12 6t / 6 12 /6
(division) t 2 CHECK
2t 3 9 4t 2(2)
3 9 4(2)
1 1
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USING THE DISTRIBUTIVE PROPERTY TO SOLVE EQUATIONS
3(K 8) 21
3 K 3 8 21 Use the
distributive property
3k 24 21 Simplify
3k 24 24 21 24 Subtract 24
from both sides
3k -3
Simplify
3k /3 -3/3
Divide each side by 3
K -1
Simplify
Check 3(-1 8) 21
3(7) 21
21 21
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6(t 2) 2(9 2t) 6t 2(6)
2(9) 2(2t) 6t 12 18 4t 6t
4t 12 18 4t 4t 10t 12 18
10t 12 12 18 12 10t
30 10t / 10 30 / 10
t 3
CHECK 6(t 2) 2(9 2t) 6(3 2) 2 (9
2(3)) 6(1) 2(3) 6 6
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Solving a formula for one of its variables Solve
the formula for the area of a triangle for h
(height) A ½ bh
Formula for area of triangle 2(A) ½ bh (2)
Multiply each side by 2 2A bh
Simplify 2A/b
bh/b Since we need the
h by itself
we need to divide both sides
by b 2A/b h
Simplify We can
now use this formula to solve for the height if
we know the area and base of the triangle

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TRY THIS Solve the formula for the perimeter of a
rectangle for the length P 2(l
w) Formula for the perimeter
of a rectangle
P 2l 2w Use
distributive property P 2w 2l 2w - 2w
Subtract 2w from each
side P - 2w 2l
Simplify (P 2w)/2 2l/2
Divide each side by 2 (P 2w)/2 l
Simplify
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TRANSFORMING EQUATIONS Solve y 5x 7 for x
y 5x 7 Original
problem y 7 5x 7 7
Subtract 7 from each side y 7 5x
Simplify (y 7)/5 5x/5
Divide each side by 5 (y
7)/5 x simplify
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Real World Connections
The length of a rectangle pool is 6ft. More than
its width. The perimeter of the rectangle is 24
ft. What is the length and width of the
rectangle?
What is perimeter?
The distance around a figure.
What kind of relationship can we draw from the
problem?
The length is 6 ft. more than the width
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Let w width Let w 6 length
Length is described in terms of the width
Using the perimeter formula P2L 2W we can
find the width of the rectangle
P 2L 2W
24 2(W 6) 2W Substitute 24 for P
and W 6 for L
24 2W 12 2W Use the distributive
prop
24 4w 12 Combine like terms
24 -12 4W 12 12 Subtract 12 from
each side
12 4W Simplify
12/4 4W/4 Divide each side by 4
3 W Simplify
Width is 3 ft. Length is 6 ft. more so length is
9 ft.
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Using Ratios The sides of a triangle are in the
ratio 121315. The perimeter is 120cm. Find
the lengths of the sides of the triangle
15x
12x
13x
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Let 12x shortest side Let 13x second side Let
15x longest side Set up problem 12x 13x 15x
120 40x 120 Combine
like terms 40x/40
120/40 Division property x 3 ARE WE
DONE?
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12x shortest side 12(3) 36cm 13x
second side 13(3) 39 cm 15x longest
side 15(3) 45cm MAKE SURE YOU ANSWER
THE QUESTION
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Round trip travel
Troy drives into the city to buy a cd at the
local music store. Because of traffic
conditions, he averages only 15mph. On his drive
home he averages 35mph. If the total time is 2
hours, how long does it take him to drive to the
computer store? How long was his trip home
Let t the time it took Troy to drive to the
music store. Since the total travel time is 2
hours then 2-t is the time it took Troy to drive
home.
What else can we use from the problem?
He drove 15mph going
It is the same distance to and from the store
He drives 35mph home
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We can put the information into a chart.



Rate
Time
Distance
Part of Troys travel
t
To the Computer store
15mph
15t
35(2 t)
Return home
35mph
2-t
Using the formula D rt and the knowledge that
the distance was the same to the store as it was
on the return trip home we can set up an equation.
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15t 35(2 t) distance travel to
and from the
store are equal
15t 70 35t Use the
distributive prop
15t 35 t 70 35t 35t Add 35t to each side
50t 70 Combine like
terms
50t/50 70/50 Divide each side
by 50
t 1.4 Simplify
It took Troy 1.4 hours to drive to the store and
it took him .6 hours to return home.
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Assignment Page 21 Problems 1 -28 evens 34
35 40 50