Geometry
(Algebra Review)
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Geometry
Algebra Review: 0-5 Linear Equations
If the same number is added to or subtracted from each side of an equation the resulting equation is true.
Example 1:
a. 𝑥 + 8 = −5
b. 𝑛 − 15 = 3
c. 𝑝 + 27 = 12
If each side of an equation is multiplied or divided by the same number, the resulting equation is true.
Example 2:
a. 5𝑔 = 35
𝑐
b. − = 8
c.
6
4𝑥
7
= −3
To solve equations with more than one operation, often called multi-step equations, undo operations by working backward.
Example 3:
a. 9𝑝 + 8 = 35
b. 8𝑥 + 2 = 14𝑥 − 7
When solving equations that contain grouping symbols, first use the Distributive Property to remove the grouping symbols.
Example 4:
a. 4(𝑥 − 7) = 8𝑥 + 6
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b. (18 + 12𝑥) = 6(2𝑥 − 7)
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Geometry
Algebra Review: 0-7 Ordered Pairs
Points in the coordinate plane are named by ordered pairs of the form (x, y). The first number, or x-coordinate, corresponds to a
number on the x-axis. The second number, or y-coordinate, corresponds to a number on the y-axis.
Example 1: Write the ordered pair for each point.
a. Point C
b. Point D
The x-axis and y-axis separate the coordinate plane into four regions, called quadrants.
The point at which the axis intersect is called the origin. The axes and points on the
axes are not located in any of the quadrants.
Example 2: Graph and label each point on a coordinate plane. Name the quadrant in which each point is located.
a. P(4, 2)
b. M(−2, 4)
c. N(−1, 0)
Example 3:
Graph a polygon with vertices P(−1, 1), Q(3, 1), R(1, 4), and S(−3, 4).
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Remember lines have infinitely many points on them. So when you are asked to find points on a line, there are many answers.
*Make a table. Choose values for x. Evaluate each value of x to determine the y. Plot the ordered pairs.
Example 4:
Graph four points that satisfy the equation 𝑦 = −𝑥 − 2
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Geometry
Algebra Review: 0-8 Systems of Linear Equations
Two or more equations that have common variables are called system of equations. The solution of a system of equations in two
variables is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have zero, one, or an
infinite number of solutions. There are three methods by which systems of equations can be solved: graphing, elimination, and
substitution.
Example 1: Solve each system of equations by graphing. Then determine whether each system has no solution, one solution, or
infinitely many solutions.
a. 𝑦 = −3𝑥 + 1
𝑦 =𝑥−3
b. 𝑦 = 2𝑥 + 3
−4x + 2𝑦 = 6
It is difficult to determine the solution of a system when the two graphs intersect at noninteger values. There are algebraic methods by
which an exact solution can be found. One such method is substitution.
Example 2: Use substitution to solve the system of equations.
a. 𝑦 = 3𝑥
−2𝑦 + 9𝑥 = 5
b. 3𝑥 + 2𝑦 = 10
2𝑥 + 3𝑦 = 10
Sometimes adding or subtracting two equations together will eliminate one variable. Using this step to solve a system of equations is
called elimination.
Example 3: Use elimination to solve the system of equations.
a. −3𝑥 + 4𝑦 = 12
3𝑥 − 6𝑦 = 18
b. 3x + 7𝑦 =15
5𝑥 + 2𝑦 = −4
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Geometry
Algebra Review: 0-9 Square Roots and Simplifying Radicals
A radical expression is an expression that contains a square root. The expression is in simplest form when the following three
conditions have been met.
- No radicands have perfect square factors other than 1.
- No radicands contain fractions.
- No radicals appear in the denominator of a fraction.
The Product Property states that for two numbers a and b ≥0, √𝑎𝑏 = √𝑎 ∙ √𝑏.
Example 1: Simplify the radical. Leave your answer in radical form.
a. √50
b. √8 ∙ 2√4
For radical expressions in which the exponent of the variable inside the radical is even and the resulting simplified exponent is odd,
you must use absolute value to ensure nonnegative results.
Example 2: Simplify the radical. Leave your answer in radical form.
a. √18𝑎5 𝑏 4 𝑐 7
b. √20𝑥 3 𝑦 5 𝑧 6
𝑎
√𝑎
𝑏
√𝑏
The Quotient Property states that for any numbers a and b, where a ≥ 0 and b ≥ 0, √ =
.
Example 3: Simplify the radical. Leave your answer in radical form.
a. √
49
36
25
b.√
16
Rationalizing the denominator of a radical expression is a method used to eliminate radicals from the denominator of a fraction. To
rationalize the denominator, multiply the expression by a fraction equivalent to 1 such that the resulting denominator is a perfect
square.
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Example 4: Simplify the radical. Leave your answer in radical form.
a.
3
√5
b.
√17𝑥
√20
Sometimes conjugates are used to simplify radical expressions. Conjugates are binomials of the form 𝑝√𝑞 + 𝑟 √𝑡 and 𝑝√𝑞 − 𝑟 √𝑡.
Example 5: Simplify the radical. Leave your answer in radical form.
a.
8
6−√3
b.
3
5−√2
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Geometry
General Algebra Skills Guided Notes
Skill 1: Solving Equations
a) −3(4𝑥 + 3) + 4(6𝑥 + 1) = 43
Skill 2: Simplify by Multiplying
a) (𝑥 − 3)(6𝑥 − 2)
b) (4𝑎 + 2)(6𝑎2 − 𝑎 + 2)
Skill 3: Factoring
a). n2 + 4n – 12
b) 3n2 – 8n + 4
Skill 4: Solving Quadratic Equations
a) 10n2 + 2 = 292
b) (2𝑚 + 3)(4𝑚 + 3) = 0
c) n2 + 3n – 12 = 6
d) 2x2 + 3x – 20 = 0
e) 5x2 + 9x = –4
Skill 5: Simplifying Radicals
a) 18
b)
24
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Geometry
Skills Review Worksheet
Name: _____________________________________
For numbers 1 – 3, solve each equation.
1. 8x – 2 = –9 + 7x
2. 12 = –4(–6x – 3)
3. –5(1 – 5x) + 5(–8x – 2) = –4x – 8x
For numbers 4 – 6, simplify each expression by multiplying.
4. 2x(–2x – 3)
5. (8p – 2)(6p + 2)
6. (n2 + 6n – 4)(2n – 4)
8. b2 + 16b + 64
9. 2n2 + 5n + 2
10. 9n2 + 10 = 91
11. (k + 1)(k – 5) = 0
12. n2 + 7n + 15 = 5
13. n2 – 10n + 22 = –2
14. 2m2 – 7m – 13 = –10
For numbers 7 – 9, factor each expression.
7. b2 + 8b + 7
For numbers 10 – 14, solve each equation.
For numbers 15 – 18, simplify each radical.
15.
72
17. 32
16.
80
18. 90
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Geometry
Algebra Skills Practice
Name: _____________________________________
I. Solving Linear Equations
1. 2x + 5 = 11
2. 3x + 5 = –16
3. 2(x – 3) = 84
4. 5x – 32 = 80
5. 3(2x + 5) – 3x = 6
6. 3x – 4(x – 4) + 4 = 13
II. Solving Systems of Equations by Elimination.
2 x  7 y  3
7. 
4 x  2 y  18
 x  y  39
8. 
 x  y  1785
6 x  4 y  7
9. 
15 x  12 y  1
11x  3 y  39
10. 
6 x  12 y  19
III. Solving Systems of Equations by Substitution
 x  6 y  2
11. 
5 x  30 y  10
9 x  2 y  6
12. 
5 x  4 y  12
2 x  3 y  8
13. 
9 x  3 y  14
10 x  5 y  3
14. 
6 x  30 y  81
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IV. Simplifying Radicals
15.
18.
21.
52
5
15
50
75
16.
6
10
17.
19.
22.
3
12
8
20.
3
16
23.
24
3 5
20
10 10
80
V. Solving Quadratic Equations by the Quadratic Formula
24. x2 – x = 6
25. x2 + 8 = 6x
26. 4x2 = 4x – 1
27. 4x2 – 3x = 7
VI. Solving Quadratic Equations by Factoring (when a = 1)
28. x2 – 2x – 35 = 0
29. x2 – 10x – 24 = 0
30. x2 – 9x = 2x + 12
31. 32x + 240 = –x2
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VII. Solving Quadratic Equations by Factoring (when a ≠ 1)
32. 2x2 + x – 3 = 0
33. 24x – 35 = 4x2
34. 7x + 21 = 14x2
35. –72x2 + 36x + 36 = 0
VIII. Solving Special Cases of Quadratic Equations
36. x2 – 3 = 125
37. 45x2 – 586 = 19,259
38. 12x2 + 420 = 40x2 – 1372
39. 4x2 + 5 = 54
40. 5x2 + 5 = x2 + 25
41. 3x2 – 6x = 11x
IX. Solving Radical Equations and Proportions
42.
3x  2  5
43. 5 x  2  3
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44.
2y
1

y 3 y
45.
x
x8

2x  6
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