Geometry Fundamentals
Summer Assignment 2015
The following packet contains topics and definitions that you will be required to know in order to succeed in
Geometry Fundamentals this year. You are advised to be familiar with each of the concepts and to complete
the included problems by September 3, 2015. All of these topics were discussed in either Algebra I or Math 8
and will be used frequently throughout the year. All problems are expected to be completed.
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Geometry Fundamentals Summer Assignment
The following topics will begin your study of Geometry. These topics are considered to be a review of your
previous math courses and will not be covered in length during the start of the school year.
Note: You should expect to purchase a scientific calculator for this course.
Section 1: Fractions
To multiply fractions:




Multiply the numerators of the fractions
Multiply the denominators of the fractions
Place the product of the numerators over the product of the denominators
Simplify the fraction
Example: Multiply




2
9
and
3
12
Multiply the numerators (2*3=6)
Multiply the denominators (9*12=108)
Place the product of the numerators over the product of the denominators (
Simplify the fraction (
6
1
=
)
108
18
6
108
To divide fractions:





Invert (i.e. turn over) the denominator fraction and multiply the fractions
Multiply the numerators of the fractions
Multiply the denominators of the fractions
Place the product of the numerators over the product of the denominators
Simplify the fraction
2
3
Example: Divide 9 and 12





3
1) 12 x 4 =
4)
20
3
4
=
2
3
2
12
Invert the denominator fraction and multiply ( 9 ÷ 12 = 9 * 3 )
Multiply the numerators (2*12=24)
Multiply the denominators (9*3=27)
24
Place the product of the numerators over the product of the denominators ( )
24
8
2)
1
5
5)
1
10
27
Simplify the fraction ( 27 = 9 )
x
10
4
÷
=
3
5
=
2
3)
2
7
6)
2
5
21
x 30 =
÷
8
10
=
)
Section 2: Simplifying Algebraic Expressions
The difference between an expression and an equation is that an expression doesn’t have an equal sign.
Expressions can only be simplified, not solved. Simplifying an expression often involves combining like terms.
Terms are like if and only if they have the same variable and degree or if they are constants. Simplifying
expressions also refers to substituting values to get a resultant value of the expression.
Simplify the following expressions by combining like terms.
7) 3 + 2𝑦 2 − 7 − 5𝑥 − 4𝑦 3 + 6𝑥
8) 𝑥 2 + 𝑥 2 + 𝑥 + 𝑥
9) 4(3𝑥 − 2𝑥 3 + 5) − 6𝑥
10) 𝑥(2𝑥 − 3𝑥 4 + 2𝑦 − 5𝑥𝑦)
11) 8𝑎 − (7𝑏 − 4𝑎) − 3(4𝑎 + 2𝑏)
Evaluate the following expressions by substituting the given values for the variables.
12) 6𝑎2 − 2𝑏 + 4𝑎𝑏 − 5𝑎 ; 𝑎 = −3 𝑎𝑛𝑑 𝑏 = 4
13) – 𝑘 2 + 4𝑚 − 2𝑘𝑚 − (3𝑘 + 2𝑚); 𝑘 = −2 𝑎𝑛𝑑 𝑚 = 3
14) 3(4𝑐 − 2𝑑) + 𝑑(𝑑𝑐 2 + 7); 𝑐 = −2 𝑎𝑛𝑑 𝑑 = 3
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Section 3: Solving Equations
When solving an equation, remember to combine like terms first. Terms are like if and only if they have the same
variable and degree or if they are constants. Then, take steps to isolate the variable by following the order of
operations backwards and doing the inverse operation.
Solve each equation and check your answer.
15) 3n + 2 = 17
16) 4 – 2y = 8
17) 3(n – 4) = 15
18) 6 – (3t + 4) = -17
19) 5k + 2(k + 1) = 23
20)
1
21) − 2 𝑚 − 3 = 1
23) 7 =
5
𝑥
5
𝑝
7
− 10 = 30
22) (w + 5) – (2w + 5) = 5
24)
1
(2𝑥
3
4
− 1) =
3
(𝑥
4
+ 2)
Section 4: Graphing Linear Equations
A linear function is a function where the highest power of x is 1. You have seen these functions in many forms.
Some of the common forms are y = mx + b (slope-intercept form) and Ax + By = C (standard form). Notice in both
forms that the exponent for x is 1.
Every linear function has an x and y intercept.


x – intercept: Where a function crosses the x – axis. The coordinate is represented by (x, 0).
y – intercept: Where a function crosses the y – axis. The coordinate is represented by (0, y).
A key concept to consider when thinking of linear functions is slope.
𝐴
Slope is the “m” in the y = mx + b and is defined to be - 𝐵 for standard form of a line. Here are some definitions of
slope:
𝑠𝑙𝑜𝑝𝑒 = 𝑚 =
𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒
𝑟𝑖𝑠𝑒 𝑦2 − 𝑦1
=
=
ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑟𝑢𝑛 𝑥2 − 𝑥1
Positive slopes increase from left to right.
Negative slopes decrease from left to right.
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If a line has a slope of zero, it is horizontal and the equation is y = b.
*All points on the line have the same y-coordinate.
If a line has an undefined slope, it is vertical and the equation is x = c.
*All points on the line have the same x-coordinate.
Parallel lines have the same slopes.
Perpendicular lines have slopes that are opposite reciprocals.
Formulas for equations of a line:
Slope-Intercept: y = mx + b
Point-Slope Form: y – y1=m(x – x1)
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Graph and label each linear equation. (Note: You may need to put the equation in slope-intercept form.)
Are the lines parallel, perpendicular or neither? Explain your answer using the equations.
25) 𝑦 = 3𝑥 − 2
6𝑥 − 2𝑦 = −4
26) 𝑦 =
−1
𝑥
3
+4
𝑦 = 3𝑥 − 1
______________________
______________________
______________________
______________________
27) 𝑦 – 2𝑥 = −5
1
𝑦 = 2𝑥
28) 𝑦 + 3 =
−3
𝑥
4
𝑦– 2 =
______________________
______________________
______________________
______________________
7
1
𝑥
3
Writing Linear Equations
The Point-Slope Form of the equation of a nonvertical line that passes through the point (𝑥1 , 𝑦1 ) and has a slope
of “m” is 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ). You can use the Point-Slope Form to write the equation in Slope-Intercept Form.
𝑦 = 𝑚𝑥 + 𝑏.

Write the equation of the line that has a slope of -3 and passes through the point (-1,7).
Step 1: Write an equation containing one variable, and solve it.
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Start with the Point-Slope Form.
𝑦 − (7) = −3(𝑥 − (−1)
Substitute (7) for 𝑦1 , (-1)for 𝑥1 , and (-3) for m.
𝑦 − 7 = −3(𝑥 + 1)
Eliminate double signs by distributing.
𝑦 − 7 = −3𝑥 − 3
Distribute (-3).
𝑦 = −3𝑥 + 4
Add (7) to both sides of the equation.
Check the answer:
𝑦 = −3𝑥 + 4
7 = −3(−1) + 4
7=3+4
7=7

Start with one equation.
Substitute “-1” for “x” and 7 for “y”.
Simplify.
Simplify.
Use the Point-Slope Form to write an equation of a line in Slope-Intercept Form that goes through the
given point and has the given slope.
29) (3,-4); m=6
30) (-2,-7); m=-2
31) (1,-8); m=-1
32) (-6,1); m=3
33) (-3,-2); m=-5
33) (4,-2); m=−
2
8
3
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Solving Systems of Equations Using Substitution
You can solve a system of equations by replacing one variable with an equivalent expression containing the other
variable. This method is known as the Substitution Method. Once the replacement is made, you create a onevariable equation that can be solved by using the following steps.

𝑦 = −4𝑥 + 8
Solve using substitution. {
𝑦 =𝑥−7
Step 1: Write an equation containing one variable, and solve it.
𝑦 = −4𝑥 + 8
Start with one equation.
𝑥 − 7 = −4𝑥 + 8 Substitute “x+7” for “y”.
5𝑥 − 7 = 8
Add “4x” to each side of the equation.
5𝑥 = 15
Add 7 to each side of the equation.
𝑥=3
Divide each side of the equation by 5.
Step 2: Solve for the other variable in either equation.
𝑦 = −4𝑥 + 8
Start with one equation.
𝑦 = −4(3) + 8
Substitute “3” for “x”.
𝑦 = −12 + 8
Simplify.
𝑦 = −4
Simplify.
Since x=3 and y=-4, the solution is (3,-4).
Check the answer:
𝑦 = −4𝑥 + 8
−4 = −4(3) + 8
−4 = −12 + 8
−4 = −4
Check the answer:
𝑦 =𝑥−7
−4 = (3) − 7
−4 = −4
Start with one equation.
Substitute “3” for “x” and -4 for “y”.
Simplify.
Simplify.
Start with the other equation.
Substitute “3” for “x” and -4 for “y”.
Simplify.
Since x=3 and y=-4 for both equations in the system of equations, the solution is correct.
34) Solve {
𝑦 =𝑥+1
𝑦 = 2𝑥 − 1
35) Solve {
𝑦 = 4𝑥 − 8
𝑦 = 2𝑥 + 10
36) Solve {
𝑦 = −3𝑥 − 6
𝑦 =𝑥−4
37) Solve {
3𝑥 − 6𝑦 = 30
𝑦 = −6𝑥 + 34
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Section 5 - Multiplying Polynomials
FOIL Method
The FOIL method is a special case of a more general method for multiplying algebraic expressions using the
distributive law.




First (“first” terms of each binomial are multiplied together)
Outer (“outside” terms are multiplied—that is, the first term of the first binomial and the second term of
the second)
Inner (“inside” terms are multiplied—second term of the first binomial and first term of the second)
Last (“last” terms of each binomial are multiplied)
The general form is:
Once you have multiplied by using the FOIL method, you must combine any like terms.

Use FOIL to multiply (x + 3)(x + 2)
"first": (x)(x) = x2
"outer": (x)(2) = 2x
"inner": (3)(x) = 3x
"last": (3)(2) = 6
So: (x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
38) Multiply (x + 5) (x + 7)
39) Multiply (y – 3) (y – 5)
40) Multiply (4x + 2) (4x – 2)
41) Multiply (2a + 3) (3a – 4)
42) Multiply (a + 4)(a – 4)
43) Multiply (5t + 4)2
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44) Multiply (3y – 2)(3y + 2)
45) Multiply (w2 + 2)(w2 – 2)
46) Multiply (2x – 5y)(3x + y)
47) Multiply (5x + 3)(–x + 2)
Distribution Method
Sometimes you will have to multiply one multi-term polynomial by another multi-term polynomial. This type of
multiplication will use a form of distribution for one of the polynomials to the other polynomial.

Simplify (x + 3)(4x2 – 4x – 7)
(x + 3) (4x2 – 4x – 7)
= (x)(4x2 – 4x – 7) + (3)(4x2 – 4x – 7)
= 4x2(x) – 4x(x) – 7(x) + 4x2(3) – 4x(3) – 7(3)
= 4x3 – 4x2 – 7x + 12x2 – 12x – 21
= 4x3 – 4x2 + 12x2 – 7x – 12x – 21
= 4x3 + 8x2 – 19x – 21
48) Multiply (w + 2)(w2 + 2w – 1)
49) Multiply (x – 1)(x2 + x+ 1)
Thank you for taking the time to complete this assignment.
Your Geometry Fundamentals class will begin with these questions next year.
Have a great summer!!!
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