Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
1
Multiplying binomials:
We have a special way of remembering how to multiply binomials called FOIL:
F:
first x  x = x2
(x + 7)(x + 5)
O:
outer x  5 = 5x
I:
inner 7  x = 7x
x2 + 5x +7x + 35 (then simplify)
L:
last
7  5 = 35
x2 + 12x + 35
1) (x - 5)(x + 4)
2) (x - 6)(x - 3)
3) (x + 4)(x + 7)
4) (x + 3)(x - 7)
5) (3x - 5)(2x + 8)
6) (11x - 7)(5x + 3)
7) (4x - 9)(9x + 4)
8)(x - 2)(x + 2)
9) (x - 2)(x - 2)
10) (x - 2)2
11) (5x - 4) 2
12) (3x + 2)2
Factoring using GCF:
Take the greatest common factor (GCF) for the numerical coefficient. When choosing the GCF
for the variables, if all the terms have a common variable, take the one with the lowest exponent.
ie) 9x4 + 3x3 + 12x2
GCF: coefficients: 3
Variable (x) : x2
GCF: 3x2
What’s left? Division of monomials:
9x4/3x2
3x3 /3x2
12x2/3x2
3x2
x
4
Factored Completely:
3x2 (3x2 + x+ 4)
Factor each problem using the GCF and check by distributing:
9
7
5
4
3 2
2 3
4
1) 14x - 7x + 21x
2) 26x y - 39x y + 52x y - 13xy
6
5
11
10
4
5 2
3) 32x - 12x - 16x
9
8
5
5) 24b + 4b -6b + 2b
3 3
2 2
4 4 4
2 4
5
3 3
5
6) 96a b + 48a b - 144ab
5
4
8) 75x + 15x -25x3
7) 11x y + 121x y - 88xy
5 4 3
4 3
4) 16x y - 8x y + 24x y - 32xy
3 4 5
9) 132a b c - 48a b c + 72a b c
5
5
10) 16x + 12xy - 9y
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
HOW TO FACTOR TRINOMIALS
Remember your hints:
A. When the last sign is addition
x2 - 5x + 6
1)Both signs the same
2) Both minus (1st sign)
B. When the last sign is subtraction
x2 + 5x – 36 1) signs are different
(x - )(x - )
(x - )(x + )
3) Factors of 6 w/ a sum
of 5. (3 and 2)
(x - 3)(x - 2)
2) Factors of 36 w/ a
difference of 5 (9
and 4)
3) Bigger # goes 1st sign, +
(x - 4)(x + 9)
FOIL Check!!!!!
Factor each trinomial into two binomials check by using FOIL:
1) x2 + 7x + 6
2) t2 – 8t + 12
3) g2 – 10g + 16
2
2
4) r + 4r - 21
5) d – 8d - 33
6) b2 + 5b - 6
7) m2 + 16m + 64
8) z2 + 11z - 26
9) f2 – 12f + 27
2
2
10) x - 17x + 72
11) y + 6y - 72
12) c2 + 5c - 66
13) z2 – 17z + 52
14) q2 – 22q + 121
15) w2 + 8w + 16
2
2
16) u + 6u - 7
17) j – 11j - 42
18) n2 + 24n + 144
19) t2 + 2t -35
20) d2 – 5d - 66
21) r2 – 14r + 48
2
2
22) p + p - 42
23) s + s - 56
24) b2 – 14b + 45
25) f2 + 15f + 36
26) n2 + 7n - 18
27) z2 + 10z - 24
2
2
28) h + 13h + 24
29) w + 29w + 28
30) v2 – 3v – 18
31) y2 - 9
32) g2 – 36
33) t2 – 121
2
2
34) 9k – 25
35) 144m – 49
36) 64e2 – 81
37) a2 + 100
38) w2 – 44
39) d2 – d – 9
Factor using GCF and then factor the trinomial (then check):
40) 4b2 + 20b + 24
41) 10t2 – 80t + 150
42) 9r2 + 90r - 99
43) 3g3 + 27g2 + 60g
44) 12x6 + 72x5 + 60x4
45) 8c9 + 40c8 - 192c7
2
2
46) 12d – 12
47) 25r – 100
48) 5z5 – 320z3
2
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Case II Factoring
Factoring a trinomial with a coefficient for x2 other than 1
Factor:
1)
2)
3)
4)
5)
6)
7)
6x2 + 5x – 4
Look for a GCF:
a. There is no GCF for this trinomial
b. The only way this method works is if you take out the GCF (if there is
one.)
Take the coefficient for x2 (6) and multiply it with the last term (4):
6x2 + 5x – 4
6 * 4 = 24
x2 + 5x – 24
Factor the new trinomial:
x2 + 5x – 24
(x + 8)(x – 3)
Take the coefficient that you multiplied in the beginning (6) and put it back in the
parenthesis (only with the x):
(x + 8)(x – 3)
(6x + 8)(6x – 3)
Find the GCF on each factor (on each set of parenthesis):
(6x + 8)
 2(3x+ 4)
(6x – 3)
 3(2x – 1)
Keep the factors left in the parenthesis:
(3x + 4)(2x – 1)
FOIL CHECK
Extra Problems: (Remember... GCF
1st)
1) 7x2 + 19x – 6
(3x + 4)(2x – 1)
2) 36x2 - 21x + 3
3) 12x2 - 16x + 5
6x2 –3x + 8x – 4
4) 20x2 +42x – 20
5) 9x2 - 3x – 42
6x2 + 5x – 4
6) 16x2 - 10x + 1
7) 24x2 + x – 3
8) 9x2 + 35x – 4
9) 16x2 + 8x + 1
10) 48x2 + 16x – 20
3
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Factor each trinomial and FOIL Check:
1) x2 – 6x – 72
2) x2 + 14x + 13
3) x2 – 19x + 88
4) x2 + 2x – 63
5) x2 – 196
6) x2 – 1
7) x2 + 20x + 64
8) x2 + 11x - 12
9) x2 - 12x + 35
10) x2 - 17x + 70
11) x2 + 14x - 72
12) x2 + 5x – 36
13) x2 - 20x + 96
14) x2 - 24x + 144
15) x2 + 10x + 25
Factor using the GCF:
10
9
8
16) 24x - 144x + 48x
17) 64x5y3 – 40x4y4 + 32x3y4 – 8x2y3
Factor using the GCF and then factor the quadratic:
18) x4 – 15x3 + 56x2
19) 4x2 + 24x – 240
20) 5x3 – 5x2 –360x
21) 12x2 – 243
22) 16x2 – 16
23) 8x17 – 512x15
Mixed Problems:
24) 49x2 – 25
25) 4x2 – 121
26) x4 – 36
27) x16 – 64
28) x100 – 169
29) 48x8 – 12
30) 25x2 – 100
31) 36x4 – 9
32) 100x2 – 225
33) x2 + 64
34) x2 – 48
35) x2 – 2x + 24
36) x2 + 11x – 30
37) 5x2 + 20
38) 7x2 – 7x - 84
4
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
5
1-Step Factoring: Factor each quadratic. If the quadratic is unable to be
factored, your answer should be PRIME.
Examples:
(last sign +)
(last sign - )
x2 – 10x + 24
x2 + x – 12
Same sign, both Different Signs
Factors of 24, sum of 10
Factors of 12, diff. of 1
(x – 6)(x – 4)
(x + 4)(x – 3)
1) x2 + 5x + 4
4) g2 + 5g – 50
7) s2 – 9s + 20
10) x2 – 6x – 7
13) g2 – 5g – 84
16) p2 – 81
19) z2 + 9z – 36
22) b2 – 5b – 36
25) y2 – 4y – 60
28) x2 + 61x + 60
31) a2 + 4a – 96
34) t2 + 21t + 108
2) a2 – 12a + 35
5) t2 – 2t + 48
8) j2 + 7j + 12
11) n2 -25
14) z2 + 17z + 72
17) w2 – w – 132
20) h2 + 12h + 36
23) x2 – 36
26) v2 + 16v – 60
29) g2 – 23g + 60
32) y2 – y – 110
35) w2 – 64
(D.O.T.S)
x2 – 49
Diff of Two Sq.
(x + 7)(x – 7)
3) f2 – 3f – 18
6) x2 – 100
9) k2 + 2k – 24
12) c2 – 13c – 40
15) q2 – 3q + 18
18) x2 + 13x – 48
21) r2 + 5r + 36
24) m2 – 20m + 36
27) r2 + 7r – 60
30) b2 – 121
33) x2 + x + 90
36) x2 – 14x + 49
2-Step Factoring: Factor using the GCF and then try to factor what’s left.
Example:
6x2 – 18x + 12
6(x2 – 3x + 2)
6(x – 2)(x – 1)
37) 5x2 + 10x - 120
40) 6d2 + 60d + 150
43) 7f2 + 84f + 252
46) 5g2 - 245
38) 3w2 -33w +90
41) 9x2 - 36
44) 2x2 – 2x - 180
47) 9k2 – 99k + 252
39) 8t2 – 32t - 256
42) 10z2 + 50z - 240
45) 4s2 - 144
48) 25k2 – 225
Case II: Factor using your steps for Case II factoring. Remember GCF is always the 1
step of any type of factoring!!!
Example:
49) 2x2 – 7x - 30
52) 18y2 + 19y + 5
55) 12s2 – 22s - 20
58) 40x2 + 205x + 25
st
6x2 – 5x – 4 (mult. 1st by last)
x2 – 5x – 24 (factor)
(x – 8)(x + 3) (put the 1st #, 6, back in)
(6x – 8)(6x + 3) (reduce: take 2 out of the 1st factor
and 3 out of the 2nd)
(3x – 4)(2x + 1)
50) 12s2 + 19s + 4
53) 15f2 – 14f + 3
56) 24d2 – 6d - 30
59) 100z2 + 10z - 20
51) 18c2 + 9c - 2
54) 15k2 + 7k - 8
57) 21w2 + 93w + 36
60) 24r2 – 90r + 21
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Simplifying and Combining Like Terms
Exponent
Coefficient
4x
2
* Write the coefficients, variables, and exponents of:
a) 8c2
b) 9x
c) y8
Variable (or Base)
d) 12a2b3
Like Terms: Terms that have identical variable parts {same variable(s) and same exponent(s)}
When simplifying using addition and subtraction, combine “like terms” by keeping the "like
term" and adding or subtracting the numerical coefficients.
Examples:
3x + 4x = 7x
13xy – 9xy = 4xy
12x3y2 - 5x3y2 = 7x3y2
Why can’t you simplify?
4x3 + 4y3
11x2 – 7x
6x3y + 5xy3
Simplify:
1) 7x + 5 – 3x
2) 6w2 + 11w + 8w2 – 15w
3) (6x + 4) + (15 – 7x)
4) (12x – 5) – (7x – 11)
5) (2x2 - 3x + 7) – (-3x2 + 4x – 7)
6) 11a2b – 12ab2
WORKING WITH THE DISTRIBUTIVE PROPERTY
Example:
3(2x – 5) + 5(3x +6) =
Since in the order of operations, multiplication comes before addition and subtraction, we must
get rid of the multiplication before you can combine like terms. We do this by using the
distributive property:
3(2x – 5) + 5(3x +6) =
3(2x) – 3(5) + 5(3x) + 5(6) =
6x - 15 + 15x + 30 =
Now you can combine the like terms:
6x + 15x = 21x
-15 + 30 = 15
Final answer: 21x + 15
6
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
7
Solving Equations
Golden Rule of Algebra:
“Do unto one side of the equal sign as you will do to the other…”
Whatever you do on one side of the equal sign, you MUST do the same exact thing
on the other side. If you multiply by -2 on the left side, you have to multiply by -2 on the other.
If you subtract 15 from one side, you must subtract 15 from the other. You can do whatever you
want (to get the x by itself) as long as you do it on both sides of the equal sign.
Solving Single Step Equations:
To solve single step equations, you do the opposite of whatever the operation is. The
opposite of addition is subtraction and the opposite of multiplication is division.
Solve for x:
2) x – 11 = 19
5) (x/-5) = 3
1) x + 5 = 12
4) 5x = -30
3) 22 – x = 17
6) ⅔ x = - 8
Solving Multi-Step Equations:
3x – 5 = 22
To get the x by itself, you will need to get rid of the 5 and the 3.
+5 +5
We do this by going in opposite order of PEMDAS. We get rid
of addition and subtraction first.
3x
3
= 27
3
Then, we get rid of multiplication and division.
x =9
We check the answer by putting it back in the original equation:
3x - 5 = 22, x = 9
3(9) - 5 = 22
27 - 5 = 22
22 = 22 (It checks)
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Simple Equations:
1) 9x - 11 = -38
2) 160 = 7x + 6
3) 32 - 6x = 53
4) -4 = 42 - 4x
5) ¾x - 11 = 16
6) 37 = 25 - (2/3)x
7) 4x – 7 = -23
8) 12x + 9 = - 15
9) 21 – 4x = 45
10) (x/7) – 4 = 4
11) (-x/5) + 3 = 7
12) 26 = 60 – 2x
Equations with more than 1 x on the same side of the equal sign:
You need to simplify (combine like terms) and then use the same steps as a multi-step equation.
Example:
9x + 11 – 5x + 10 = -15
4x + 21 = -15
Now it looks like a multistep eq. that we did in the 1 st
-21 -21
Use subtraction to get rid of the addition.
4x
= -36
4
4
Now divide to get rid of the multiplication
9x – 5x = 4x and
11 + 10 = 21
x
13) 15x - 24 - 4x = -79
= -9
14) 102 = 69 - 7x + 3x
15) 3(2x - 5) - 4x = 33
16) 3(4x - 5) + 2(11 - 2x) = 43
17) 9(3x + 6) - 6(7x - 3) = 12
18) 7(4x - 5) - 4(6x + 5) = -91
19) 8(4x + 2) + 5(3x - 7) = 122
Equations with x's on BOTH sides of the equal sign:
You need to "Get the X's on one side and the numbers on the other." Then you can solve.
Example: 12x – 11 = 7x + 9
-7x
-7x
5x – 11 = 9
+11 +11
5x
= 20
5
5
x
Move the x’s to one side.
Now it looks like a multistep equation that we did in the 1 st section.
Add to get rid of the subtraction.
Now divide to get rid of the multiplication
=4
20) 11x - 3 = 7x + 25
21) 22 - 4x = 12x + 126
23) ¾x - 12 = ½x -6
24) 5(2x + 4) = 4(3x + 7)
25) 12(3x + 4) = 6(7x + 2) 26) 3x - 25 = 11x - 5 + 2x
8
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
9
Solving Quadratic Equations
Solving quadratic equations (equations with x2 can be done in different ways. We will use two
different methods. What both methods have in common is that the equation has to be set to = 0.
For instance, if the equation was x2 – 22 = 9x, you would have to subtract 9x from both sides of
the equal sign so the equation would be x2 – 9x – 22 = 0.
Solve by factoring: After the equation is set equal to 0, you factor the trinomial.
x2 – 9x – 22 = 0
(x-11) (x+2) = 0
Now you would set each factor equal to zero and solve. Think about it, if the product of the two
binomials equals zero, well then one of the factors has to be zero.
x2 – 9x – 22 = 0
(x-11) (x+2) = 0
x – 11 = 0
x+2=0
+11 +11
-2
x = 11
x = -2
or
-2
* Check in the ORIGINAL equation!
Solving Quadratics by Factoring:
20) x2 - 5x - 14 = 0
21) x2 + 11x = -30
22) x2 - 45 = 4x
23) x2 = 15x - 56
24) 3x2 + 9x = 54
25) x3 = x2 + 12x
26) 25x2 = 5x3 + 30x
27) 108x = 12x2 + 216
29) 10x2 - 5x + 11 = 9x2 + x + 83
28) 3x2 - 2x - 8 = 2x2
30) 4x2 + 3x - 12 = 6x2 - 7x - 60
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
10
Solve using the quadratic formula:
When ax2 + bx + c = 0
x=
-b ± √b2 – 4ac .
2a
a is the coefficient of x2 b is the coefficient of x c is the number (third term)
Notice the ±
is what will give your two answers (just like you had when solving by factoring)
x2 – 9x – 22 = 0
a=1
x=
-b ± √b2 – 4ac .
2a
b= - 9
c = -22
x=
-(-9) ± √ (-9)2 – 4(1)(-22)
2(1)
x=
9 ± √81 + 88
2
x=
9 ± √169 .
2
-4(1)(-22) = 88
Split and do the + side and - side
9 – 13
2
9 + 13
2
x = 11
or
x = -2
* Check in the ORIGINAL equation!
Solving Quadratics Using the Quadratic Formula:
31) 2x2 - 6x + 1 = 0
32) 3x2 + 2x = 3
33) 4x2 + 2 = -7x
34) 7x2 = 3x + 2
35) 3x2 + 6 = 5x
36) 9x - 3 = 4x2
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Proportions and Percents
Proportions:
A proportion is a statement that two ratios are equal. When trying to solve proportions we use
the Cross Products Property of Proportions.
A = C
A(D) = B(C)
B
D
Example:
6__ = x__
x + 5__ =
1.5___
11
121
12
6
6(121) = 11x
6(x + 5) = 12(1.5)
726 = 11x
6x + 30 = 18
-30
-30
6x
= -12
6
6
x
= -2
726 = 11x
11
11
66 = x
1)
x
14
_
=
16
35
2)
x–3 _ =
x+3
12
30
_
Percents:
Is
=
%___
Of
100
Example:
What number is 20% of 50?
Is:
?x
x
=
20 .
Of:
of 50
50
100
%:
20%
100: 100
100x = 20(50)
100x = 1,000
100x = 1,000
100
100
x = 10
a) What number is 40% of 160?
b) 48 is what percent of 128?
c) 28 is 75% of what number?
d) What number is 36% of 400?
11
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
12
Part I:
1)
x . =
12
18
54
.
2) -
13
x
. =
65
90
.
3) x + 4 . =
9
4) - 16 . =
8 .
6x-2
11
6) What is 20% of 32?
5)
14
. =
16
7) 72 is 40% of what number?
8) 21.56 is what percent of 98?
9) - 31 is what percent of -124?
6x .
18
3x
.
3x + 3
10) What is 62% of 140?
Part II:
1)
x . =
12
13
78
4)
- 16 . =
5x-2
8 .
11
.
2)
- 13
x
5)
x+5
x-3
. =
195
150
. =
x
9
.
.
3) x + 4 . =
9
6x .
18
6)
9 _
x+8
x-4 _ =
12
7) 12 is 40% of what number?
8) 21.56 is what percent of 98?
9) 45 is what percent of 180?
10) What is 62% of 70?
Part III:
1)
23
x
4)
x+1
x+6
.
=
.
57.5
45
=
.
2
x
.
2) 3x – 5 . =
13
5x + 1 .
52
5) 2x – 4 . =
x+5
x-2 .
x+1
10) What is 80% of 850?
8) 128 is 32% of what number?
9) 72 is what percent of 120?
10) What is 80% of 850?
3) 5x -1
10x+5
6) x + 7
2x – 1
= 33 .
45
= x+6 .
x-2
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
13
Mixed Equations: Figure out what type of equation you have and then pick a strategy
to solve.
1) 20 - (5/8)x = 40
2) 6(7x - 2) = 8(4x + 1)
3) 2(5x - 4) - 3(4x + 3) = -43
4) x2 + 44 = 15x
5) 3x2 + 18x = 81
6) 3x2 = 5x + 5
7) 11x - 5 = 7x - 53
8) 6(3x + 1) + 5(10 - 4x)= 39
9) ¼x - 33 = -49
10) 7x2 - 1 = 3x
11) 9(3x + 1) = 8(5x + 6)
12) 15x = x2 – 16
13) x2 + 8x = 12
14) 9(4x + 7) - 6(7x + 10) = -54
15) 44 = 20 - 2x
16) 4x2 - 128 = 16x
17) 3x2 - 8x + 6 = x + 6
18) 7(6x + 2) = 10(3x + 5)
19) 3x2 + 13x - 12 = 9x2 - 11x - 12
21)
24)
14 . =
8x - 4
10 . =
7x + 2
35
50
8 .
5x + 4
.
22)
25)
20) 2x2 - 14 = 10x
x+5
x-4
. =
x-6
. =
2x - 3
x
32
.
x + 12 .
x+4
23) x - 10_ =
12
6 _
x-4
26) 2x - 3 =
x+1
x-3 _
x+3
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
14
Solving Systems of Equations
In order to solve for two variables, you need to have two equations. If you only have one
equation there are an infinite amount of ordered pairs (x,y) that will work. For example:
4x – 2y = 16 you can have x = 4 and y = 0 (4,0) and (2, -2) and (0, -4) and an infinite amount of
others. To be able to solve for a single ordered pair, you need a second equation.
When we introduce the second equation, we will be able to solve for a single ordered pair that
will work in both equations. There are two ways to solve a system of equations (algebraically
and graphically). We will focus on solving algebraically. There are two methods of solving
algebraically (substitution and elimination). The key to both of them is changing one (or both)
equations so there is only one variable to solve for. Then you follow all the rules of solving for
the one variable. Then plug the value back into one of the original equations to find the value of
the second variable. Always state your answer as an ordered pair.
SUBSTITUTION
Example: x = 3y + 8
5x + 2y = 6
5(3y + 8) + 2y = 6 Substitute 3y +8 for the x in the 2nd equation
15y + 40 + 2y = 6 Distribute and solve.
17y + 40 = 6
17y = -34
y = -2
x = 3(-2) + 8
substitute the value for y back in to find x.
x= -6 + 8
x=2
(2, -2)
Check in BOTH ORIGINAL EQUATIONS!
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
15
Solve each system and check (in both equations):
a) x = 2y + 1
c) 5x – y = 7
b) y = 3x + 4
5x – 6y = 13
9x + 2y = -37
d) x + 3y = 11
e) 7x + 9y = -74
6x – 5y = 20
4x + 2y = 28
f) 10x – y = 1
4x + y = -5
8x + 3y = -8
Solving Systems with Linear Combinations (“Elimination”):
Sometimes solving a system of equations using substitution can be very difficult. For these
problems we solve using Linear Combinations (or Elimination). With elimination you solve by
eliminating one of the variables. This is accomplished by adding the 2 equations together.
Before you can add the equations together, you need one of the two variables to have two things:
1) Same Coefficient
2) Different Signs (one positive and one negative)
When you add terms with the same coefficient and different signs, the term drops out. You then
solve for the variable that is left. After you have solved for one variable, you plug the value into
one of the original equations and solve for the 2nd variable (just like Substitution). Then, you
check the solution in both original equations. The only difference between Substitution and
Elimination is how you solve for the 1st variable. After that they are the same.
Examples:
A) Sometimes it works out that the 2 equations already have a variable with the same coefficient
and different signs. You can then just add the equations:
3x + 4y = 10 (The +4y and -4y cancel out
5x – 4y = -58 leaving you with just 8x.)
8x
= -48
8
8
x
= -6
Plug x = -6 in:
3(-6) + 4y = 10
-18 + 4y = 10
+18
+18
4y = 28
4
4
y=7
Final Solution: (-6, 7) CHECK IN BOTH!!!!
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
B) Sometimes (usually) the equations do not have same coefficient and different signs, so we
have a little bit of manipulating to do.
3x + 8y = 25
With this system, nothing will drop out if we just add the
5x + 4y = 23
equations. So we will multiply the bottom one by (-2).
-2(5x + 4y = 23)
Now the y’s have the same coefficient with different signs.
- 10x -8y = -46
3x + 8y = 25
Now plug x = 3 in:
- 10x -8y = -46
3(3) + 8y = 25
- 7x
= -21
9 + 8y = 25
-7
-7
-9
-9
8y = 16
x
=3
8
8
y=2
Final Solution: (3,2) CHECK IN BOTH!!!!
C. Sometimes we need to manipulate both equations. We can do this by “criss crossing the
coefficients.”
6x + 7y = 11
This is different than Example B, because no coeffcient
5x – 6y = -50
goes into another evenly.
-5(6x + 7y = 11)
You need the negative sign to change the 6x to negative
6(5x – 6y = -50)
so the signs will be different.
You can also use 5 and -6. You can also “criss cross”
the y coefficients.
-30x – 35y = -55
30x – 36y = -300
- 71y = -355
-71
-71
y=5
Plug in y = 5
5x – 6(5) = -50
5x – 30 = -50
+30
+30
5x
= -20
5
5
x = -4
Final Solution: (-4, 5) CHECK IN BOTH!!!!
Practice:
1) 7x + 3y = 10
5x – 6y = 56
2) 11x + 5y = 27
4x + 6y = 60
3) 9x + 7y = 126
7x – 9y = -32
4) 12x – 5y = 63
8x + 3y = 23
5) 5x + 9y = 14
6x + 11y = 18
6) 10x – 9y = 36
4x + 3y = -12
7) 5x + 6y = 42
3x + 14y = 20
8) 7x – 5y = -42
8x + 3y = -48
9) 4x – 3y = 19
8x + 5y = 159
16
Name
Factoring, Solving Equations, and Algebraic Systems
Solve each system algebraically:
1) 5x - 2y = -9
7x + 2y = -27
2) -4x + 2y = -16
5x – 3y = 19
3) x = 2y -6
5y –3x = 11
4) 5x – 6y = -74
7x + 5y = 17
5) 4x – 5 = y
7x + 5y = 83
6) 7x + 4y = -11
5x + 2y = - 13
7) 5x – 6y = -17
3x + 8y = -16
8) x = 6 + 2y
6x – 5y = 15
9) 6x + 5y = 23
11x + 4y = 6
10) y = 3x + 4
8x – 9y = 59
11) 12x – 7y = 46
4x + 3y = -6
12) 9x – 4y = -88
2x + 5y = 4
13) 24x – 6y = -66
12x – 3y = -33
14) 5x – 6y = 42
15x – 18y = 54
15) 7x + 6y = -12
5x + 2y = -20
16) 13x – 3y = 78
4x + 6y = -66
17) 2y – 5 = x
4x – 11y = -38
18) 3x – 7y = -10
5x + 12y = -64
19) 6x – 17y = -104
4x – 7y = -39
20) 9x – 5y = -43
3x + 11y = 87
21) 9x = 11y + 25
5x – 12y = 8
22) 6y = 5x - 38
7x + 9y = 1
23) 6x + 5y = 33
5x + 37 = 3y
24) y = 3x + 5
12x – 7y = 1
Algebra 1 Summer School
17
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Answer Key to Algebraic Systems (page 4):
1) (-3,-3)
7) (-4, -.5)
13) many sol.
19) (2.5, 7)
2) (5,2)
8) (0,-3)
14) no sol.
20) (-1/3, 8)
3) (8,7)
9) (-2,7)
15) (-6,5)
21) (4,1)
4) (-4,9)
10) (-5,-11)
16) (3,-13)
22) (4,-3)
5) (4,11)
11) (1.5, -4)
17) (7,6)
23) (-2,9)
6) (-5,6)
12)(-8, 4)
18) (-8, -2)
24) (-4,-7)
1) 6x – 5y = -7
11x + 5y = 58
2) 5x + 4y = -69
5x -7y = 52
3) 6x + 7y = -28
5x – 14y = -182
4) 11x – 4y = 53
7x – 8y = 1
5) 3x – 7y = 42
2x + 5y = 57
6) 9x – 4y = 177
6x – 5y = 111
7) 8x – 11y = 77
6x + 4y = -28
8) 13x – 2y = 72
9x + 5y = -14
9) 12x = 20- 8y
5x – 6y = -57
10) 5y = 8x + 97
10x + 7y = 51
Answer Key for this sheet:
1) (3, 5)
2) (-5, -11)
3) (-14, 8)
4) (7,6)
5) (21,3)
6) (21, 3)
7) (0, -7)
8) (4, -10)
9) (-3,7)
10) (-4, 13)
18
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
1) x = 7y - 5
9x – 8y = 65
2) y = 9 – 4x
6x - 7y = -97
3) -x = 12 – 4y
8x + 7y = -18
4) 6x – 5y = -63
8x + 10y = 66
5) 14x + 5y = 18
7x + 11y = 111
6) 4x – 9y = -56
6x + 7y = -2
7) 12x + 11y = -72
11x + 14y = -66
8) 10x – 13y = 31
12x + 7y = -8
9) 3x + 11y = -2
7x + 6y = 74
10) 8x + 5y = -77
5x – 6y = 34
11) 10x + 5y = 55
12x + 7y = 74
12) 9x + 7y = 35
8x – 11y = -210
13) 6x + 11y = 18
5x + 9y = 14
14) 2x = 5y – 26
3x – 7y = -34
15) 8y = -6 – 11y
6x – 13y = 153
16) 4x – 7y = 52
9x + 10y = -89
17) 6x = 7y + 93
5x – 11y = 93
18) 10x = 7y + 5
4y = 3x - 11
19) 12x + 7y = -42
8x + 5y = -28
20) 6x + 11y = -13
y = 5x + 21
21) x = 15 – 2y
7x – 8y = 39
22) 4x – 13y = -2
7x + 5y = -59
23) 11x = 5y -1
6y = 9x + 18
24) x = 5y - 32
4x + 6y = 2
19
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
20
The Pythagorean Theorem:
a2 + b2 = c2
For all right triangles:
a= 3
b= 4
(where c is the hypotenuse)
c= ?
Pythagorean Triples
3,4,5
5,12,13
7,24,25
8,15,17
9, 40, 41
c
32 + 42 = c2
9 + 16 = c2
25
= c2
25 =  c2
5
=c
3
4
Right Triangle Trigonometry: SOH-CAH-TOA
SOH:
sin = opposite
hypotenuse
B
CAH: cos = adjacent
hypotenuse
TOA: tan = opposite
adjacent
B
hypotenuse
adjacent
hypotenuse
opposite
A
C
opposite
Based on angle B
A
C
adjacent
Based on angle C
The hypotenuse is always the side opposite the right angle. The opposite and adjacent depend on
which angle you are looking at.
opposite: The side directly across from the angle (does not touch the angle).
adjacent: (Next to) The side that touches the angle, but is not the hypotenuse.
When deciding when to use the Pythagorean Theorem or Trigonometry, the rule of thumb is that
if there is an angle involved in the problem, use the trigonometry. If you are dealing only with
missing sides (no angles), use the Pythagorean Theorem.
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Use the Pythagorean Theorem to solve for the missing side:
1)
2)
3)
x
x
11
x
39
36
18
x
17
Use the SIN to find the missing side:
* When printing off the school website, all the info on angles will be missing.*
4)
5)
9
x
15
x
21
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Use the COS to find the missing side:
* When printing off the school website, all the info on angles will be missing.*
6)
7)
x
21
x
14
Use the TAN to find the missing side:
8)
11
9)
x
x
13
Use SIN, COS, and TAN to find the missing angle:
10)
11)
18
9
12)
14
13
13
17
22
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
Mixed Problems: Find x (You will have to choose whether to use Pythagorean Theorem or
trig.). * When printing off the school website, all the info on angles will be missing.*
Round all sides to the nearest tenth and all angles to the nearest degree:
1)
2)
x
3)
x
10
15
14
24
x
4)
5)
10
6)
x
11
x
16
23
7)
31
8)
9)
12
x
10
19
37
7
10)
x
11)
12)
17
20
x
x
x
18
23
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
24
1) A 25-foot ladder is leaning against a building. The base of the ladder is 7 feet away from
the base of the building. How far up the building is the top of the ladder resting?
2) A bird is perched in a tree. It is 35 feet off the ground. It sees a piece of bread on the
ground 15 feet from the base of the tree. How far will the bird have to fly to get the bread
(round to the nearest hundredth)?
3) The dimensions of a rectangle are 12m and 4m. Find the length of the diagonal to the
nearest tenth.
4) A boy is sitting on the floor flying a kite. He has 260 feet of string. If the kite is 195 feet
high, find the angle of elevation.
5) A ladder is leaning against a building. If the angle of depression is 58º and the base of the
ladder is 9 feet from the base of the building? How long is the ladder?
Name
Factoring, Solving Equations, and Algebraic Systems
Algebra 1 Summer School
25
6) A boy is sitting on the floor flying a kite. He has 450 feet of string. The angle of elevation is
64°. How high is the kite?
7) An apple tree casts a 23-foot shadow. If the angle of depression is 36°, how tall is the tree?
8) A bird is perched on top of a 22-foot oak tree. It sees a piece of bread on the ground, 45 feet
from the base of the tree. How far will the bird have to fly to get the piece of bread?
9) A 15-foot ladder is leaning against a building. The base of the ladder is 4 feet away from the
base of the building. Find the angle of elevation created by the ladder.
10) An eagle flies 16 miles north and then 12 miles west. How far is the bird from the staring
point?