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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 3 x−5 , 3 x and –5 are terms, 3 is a factor of the first term)

EXAMPLE: For the following expression, fill in the table:

TERMS COEFFICIENT VARIABLE PART

-3x3 -3 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 x2 y , 9 xy

2, 9 , 67 pq , −23 , 5 xy

2, x

2 y

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COMBINING LIKE TERMSTo 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.

3 x+7x=(3+7 ) x=10 x 5 xy−8 xy= (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 x2−42x2b.) − y+3 y−5 y

c.) −3+3 y−5 x+2+x d.) 4 ( x+5 )−3 (2x−4 )

d.) e.)

f.) g.)

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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.

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. 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

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:

Check: Is a solution

Solving equations with parentheses and/or fractions

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.

*Remember to check your solution (see

previous example)

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Solve the Following Equations.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

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Isolating ‘y’

Also known as solving for ‘y’ or putting in slope-intercept form.

y = mx +bSOLVE FOR Y means “get y by itself.”

y + 2x = 3 – 2 x –2 x

y = 3 – 2x

and then y = -2x + 3

Your goal is 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 b. 33x - 11y = 99

TRY IT!

1. 4x - y = 6 2. 2x + 3y = -9

SLOPE Y-INTERCEPT

Step 1: Move the mx

Step 2: Rearrange

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Simplifying Radicals

About Radicals

n√b , n is the index, b is the radicand, and √ is the radical sign.

If n is 2, it is usually omitted: √9 is read the square root of 9. 3√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: √a⋅√b=√a⋅bEx. √2⋅√3=√6

(Rule: When multiplying radicals, multiply the numbers under the radical to find the product)

Quotient Property of Square Roots: √ ab=√a√b

Ex. √ 29=√2

√9=√2

3(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.

4 is used very often

9“I’m Perfect”

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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:

Simplify the expression. Show or explain your work.

(1) (2) =

(3) √1649

(4) =

(5) √50 = (6) =

ac

b

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Pythagorean Theorem Handout

Example 1

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?

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).

a2 + b2 = c2

5 cm

9 cmx

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Example 3

Try These Pythagorean Theorem Problems Directions: Find the measure of the third/missing side. Simplify Radical Answers.

1. 2. 3.

4. 5. 6.

3

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9 16

12

5 13

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5

10

4

a

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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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