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What you need to remember from Algebra 1 for Geometry
VOCABULARY OF ALGEBRAIC EXPRESSIONS
TERMS OF AN ALGEBRAIC EXPRESSION:
Addition signs separate algebraic expressions into parts called
terms. There are two types of terms: variable terms and constant
terms. A variable term contains a variable, whereas a constant term
only contains a number.
COEFFICIENTS OF A TERM:
Every term except a constant term has a number and a variable part.
The number is called the coefficient of the term. We refer to the
other part as the variable part.
TERMS VS FACTORS:
Terms are separated by addition or subtraction signs. Factors are
separated by multiplication signs.
(i.e. in 3x  5 , 3x and –5 are terms, 3 is a factor of the first term)
EXAMPLE: For the following expression, fill in the table:
 3x 3  65x 2  4 xy  14
TERMS
COEFFICIENT
-3x3
-3
VARIABLE
PART
X3
LIKE TERMS:
Two terms that have the same exact variable term are considered
like terms. Constant terms are considered like terms.
EXAMPLE: Match up the like terms:
6 x 2 y , 9xy 2 , 9 , 67 pq ,  23 , 5xy 2 , x 2 y
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COMBINING LIKE TERMS
To add two objects, they must be of the same units. We can’t add
feet and inches because they don’t match. The same goes for terms.
They must be like terms.
3x  7 x  3  7x  10x
5xy  8xy  5  8xy  3xy
To add two like terms:
Combine their coefficients and leave the variable part alone.
EXAMPLE: Simplify each expression by combining like terms.
a.) 17 x 2  42 x 2
b.)  y  3 y  5 y
c.)  3  3 y  5 x  2  x
d.) 4x  5  32x  4
d.) 2(4 x  3)  3( x  1)
e.) (6 x  1)  (4 x  5)
f.) (2 y 2  3 y  1)  (5 y 2  6 y  3)
g.) ( x 2  2 x  1)  ( x 2  7 x  3)
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Properties…
Reflexive - For any real number a, a = a.
–7y = –7y
Symmetric - For any real numbers a and b, if a = b then b = a.
If 10 = y, then y = 10.
Transitive - For real numbers a, b and c, if a = b and b = c, then a = c.
If 3x + 2 = y and y = 8, then 3x + 2 = 8.
Substitution - If a = 2 and 3a = b then 3(2) = b.
Distributive - Used on a factor in front of two terms. Multiplication distributes over addition.
a  b  c   ab  ac
Match each of the following properties with the examples at the right.
1.
Reflexive Property
_______ 2x=2x
2.
Distributive Property
_______ 4x - 2 = (2x – 1)2
3.
Substitution Property
_______ If 3 = x, then x = 3.
4.
Symmetric Property
_______ If a + b = c and c = d, then a + b = d.
5.
Transitive Property
_______ If 3x = 9y and x = 6, then 3(6) = 9y.
Identify the Property Being Illustrated.
1. 4(3  a )  12  4a
2. If n = 3 and 2n-3=y then 2(3)-3=y
3. If y = 4 and 4 = x, then y = x
4. If 3 = y, then y = 3
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Solving Equations
Solving equations
without
parentheses or
fractions
Solving equations
with parentheses
and/or fractions
1. On each side of the equation,
collect like terms if possible.
2. Add or subtract terms on both
sides of the equation in order to get
all terms with the variable on one
side of the equation.
3. Add or subtract a value on both
sides of the equation to get all
terms not containing the variable
on the other side of the equation.
4. Divide both sides of the
equations by the coefficient of the
variable.
5. If possible, simplify solution.
6. Check your solution by
substituting the obtained value into
the original equation.
Solve for X:
5 x  2  2 x  10  4 x  3
7 x  2  7  4 x
7 x  (4 x)  2  7  4 x  (4 x)
3 x  2  7
3 x  2  (2)  7  (2)
3 x  9
3 x 9

3
3
x  3
Check: Is x  3 a solution
5  3  2  2  3  10  4(3)  3
1. Remove any parentheses.
2. Simplify, if possible.
3. If fractions exist, multiply all
terms on both sides by the lowest
common denominator of all the
fractions.
4. Now follow the remaining steps
of solving an equation without
parentheses or fractions.
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 6 y  4   48
4
3
15 y  20  y  1  48
2
3
15 y  20  y  47
2
3 
2 15 y   2  20   2  y   2  47 
2 
30 y  40  3 y  94
15  2  6  10  (12)  3
13  6  22  3
19  19
5 3 y  4 
30 y  3 y  40  3 y  3 y  94
27 y  40  94
27 y  40  40  94  40
27 y  54
27 y 54

27
27
y  2
*Remember to check your solution (see
previous example)
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Solve the Following Equations.
1. 4 x  6  x  9
2. 4  7 x  1  6 x
3. 4 x  3  6 x  9
4. 41  2n  2  n
5. 6(2  y)  3(3  y)
6. 4 y  2( y  5)  2
7. 6 x  9 x  4  2 x  2
8. 2(2 x  3)  4( x  1)  2
9. 3  6a  9  6a
10. 9 x  6   x  4
11. (3)  (2  m)  (4m  1)  9
12. (60)  (2 x)  (2 x  40)  180
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Isolating ‘y’
Also known as solving for ‘y’ or putting in slope-intercept form.
Y-INTERCEPT
SLOPE
y = mx +b
SOLVE FOR Y means “get y by itself.”
y + 2x = 3
– 2x
–2x
y = 3 – 2x
and then
Your goal is
y = -2x + 3
Step 1: Move the mx
Step 2: Rearrange
y = mx + b
Step 1: MOVE the mx term to the right side of the “=” by adding or subtracting it.
Step 2: REARRANGE terms on the right side. Put the linear term first plus or minus the constant
Step 3: DIVIDE by the coefficient of y.
Step 4: SIMPLIFY and reduce fractions
Examples :
a. -4x + 2y = 8
2 y  8  4x
2 y  4x  8
y  2x  4
b. 33x - 11y = 99
11 y  99  33 x
11 y  33 x  99
y  3x  9
TRY IT!
1.
4x - y = 6
2. 2x + 3y = -9
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Simplifying Radicals
About Radicals
n
is the radical sign.
b , n is the index, b is the radicand, and
If n is 2, it is usually omitted: 9 is read the square root of 9.
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8 is read the cube root of 8. It means, what number multiplied by
itself 3 times is 8?
(The answer is 2.)
Radical Properties
Product Property of Square Roots:
Ex. 2  3  6
a  b  ab
(Rule: When multiplying radicals, multiply the numbers under the
radical to find the product)
Quotient Property of Square Roots:
a
a

b
b
Ex.
2
2
2


9
3
9
(Rule: The square root a fraction is the same as the square root of
the numerator divided by the square root of the denominator)
Simplifying Square Roots
To simplify square root radicals, find factors of the radicand which
are perfect squares, other than 1.
Perfect Squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, etc.
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“I’m Perfect”
4 is used very often
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A Radical is completely Simplified if…
(1) The radicand has no perfect square factors (other than 1).
Ex: 33 Since 33 has no perfect square factors (other than 1),
33 is simplified.
Ex: 20 20 has a factor of 4 which is a perfect square:
20  4  5  2 5
(2) The radicand is not a fraction.
If it is a fraction, use the Quotient Property.
Ex:
1
1 1


4
4 2
Simplify the expression. Show or explain your work.
(1)
24 
(3)
16
49
(5)
50 =
(2)
425 =
(4) 3 81 =
(6) 2 120 =
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Pythagorean Theorem Handout
The Pythagorean Theorem is used
to find the lengths of sides of
RIGHT Triangles.
In a Right Triangle
 The side across from the right
angle is the hypotenuse (c).
 The other two sides are called
the legs (a and b).
c
a
a2 + b2 = c2
b
Example 1
a 2  b2  c2
6 2  82  x 2
36  64  x 2
100  x 2
100  x
10  x
Example 2
A wall is supported by a brace 10 feet long, as shown in the diagram below. If one end of the brace is
placed 6 feet from the base of the wall, how many feet up the wall does the brace reach?
a 2  b2  c 2
x 2  62  102
c
a
x 2  36  100
x 2  64
x  64
x 8
b
Example 3
a 2  b2  c2
9 cm
x
x 2  52  9 2
x 2  25  81
x 2  56
5 cm
x  56  2 2 2 7  2 14
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Try These Pythagorean Theorem Problems
Directions: Find the measure of the third/missing side. Simplify Radical Answers.
1.
1
2
2.
1
2
1
6
9
3
3.
4
4.
5
5.
1
3
5
6.
1
0
4
5
7. Firefighters have a 17 foot extension ladder. In order to reach 15 feet up a building,
how far away from the building should the foot of the ladder be placed?
8. George rides his bike 9 KM south and then 12 KM east. How far is he from his starting
point?
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