Algebra Notes. Writing Algebraic Expressions. Let Statement: math sentence used to define a variable to represent the unknown quantities. Laura has twice as much homework as Ann. The Bills won five more games than they lost. Seven more than three times a number is 25.
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The Bills won five more games than they lost.
Seven more than three times a number is 25.
The length of a rectangle is 3 cm more than the width.
Let Ann = a
Let Laura = 2a
Let games lost = g
Let Bills won = 5 + g
Let Yankees = y
Let Tigers = 3y
Let width = w
Let length = 3 + w
Eight more than twice a number is 32
Seven more than three times a number is 25.
Twice a number increased by four is 16.
Let Jim = j
Let Mike = 3+ j
Let number = n
2n + 8 = 32
Let number = n
3n + 7
Let number = n
2n + 4 = 16
Fifteen less than twice a number is 25.
Sixtysix is eleven more than five times a number.
Let number = n
3n – 6 = 21
Let number = n
2n – 15 = 25
Let number = n
66 = 5n + 66
The Bills won five more games than they lost.
Let text message = t
Jam owes $44.25
39 + 0.15t
t = 44.25
Let text message = s
12 + 2s
s = 4
4 snacks
Joe attends a carnival. The admission is $48. Tickets for rides cost $4 each. Joe needs one ticket for each ride. Write an equation Joe can use to determine the number of ride tickets, r, he can buy if he has $200 before he pays the admission fee.
Let miles = m
ABC: 25 + 0.15m
$32.50
XYZ: 18 + 0.25
$30.50
XYZ is cheaper
Let number or rides = r
48 + 4r = 200
r = 38
38 rides
3² + 7(5)
9 +12
21
9(3)  5²
27 – 25
2
2(3)(5) + 6(3)
30 + 18
48
12 + (8) + 6
4 + 6
2
12 – (8) + 3
20 + 3
23
12 – (8) + (8)²
20 + 64
84
3(2)² + 5(6)²
3(4) + 5(36)
192
4(2)² + 3(6)
4(8) + 18
14
7(2)²  (6²/3)
7(4) – (36/3)
28 – 12
16
3x + 5y – 8 has 3 terms.
8x²+2x²+5a +a
8x²and 2x² are like terms
5a and a are like terms
3m – 2m + 8 – 3m + 6
5x + b – 3x + 4 + 2x – 1 – 3b
6y + 4yz + 6x² + 2yz – 4y + 2x²  5
Example: 6x
The coefficient is 6.
Example: x
The coefficient is 1.
6x
10a
4xy
4½y
7a + 6
4d
2xy + 2x
5cd – 2a
0
¾e
24x
2c
3. –7x – (–3x) 4. 6x – (–4x)
5. –10x –14x 6. –9x – (–x)
7. 3x – 8x 8. x – (–5x)
9. a² + b² + 2a² + 5b² 10. 7h² + 3 – 2h² + 4
8x
6x
4x
10x
8x
24x
5x
6x
5h² + 7
3a + 6b²
11. 3x + 3y + x + y + z 12. 5b +5b + 6b²  10 – 3b
13. Find the perimeter of the rectangle:
A 4x + 3y
B 8x + 6y
C 12xy
D 4x²+ 3y²
4x + 4y +z
6a² + 7b  10
Example: Add 2x² + 6x + 5 and 3x²  2x – 1
2x²  3x²
6x – 2x
5 – 1
5x²
4x
4
5x² + 4x + 4
Change the subtraction sign to addition and reverse the sign of each term that follows
Then add as usual
Example: Subtract 5y² + 2xy  5 and 3x²  2x – 1
Start with: 5y² + 2xy  5 2y²  3xy + 3
Place like terms together: _______+ ________+ ________
Add the like terms: _________+ __________+ _________
Final answer:


+
+
5y²  2y²
2xy – 3xy
5 + 3
3y²
xy
2
3y²  xy  2
1. (2x + 3y) + (4x + 9y) 2.(3a + 5b + 7c)  (5a – 2b + 9c)
2a + 7b – 2c
6x + 12y
11x
11a + 3b – 2c
3x³ + 2x²  x  7
3m + 9n 17
7.Subtract 8a + 5b – 6c from 10a + 8b + 7c
(10a + 8b + 7c)  (8a + 5b – 6c)
8. (4x + 8y + 9z – 7a + 5b) – (4b + 5x + 7y + 3z + 2a)
9. (– 3x2 + 4x – 11) – (–6x2 – 8x + 10) .
10. (7e² + 3e +2) + (9 – 6e + 4e²) + (9e + 2 – 6e²)
2a – 3b + 13c
x – y + 6z – 9a + b
3x² + 12x  21
5e² + 6e + 13
Some of the measures of the polygons are given. P represents the measure of the perimeter. Find the measure of the other side or sides.
x²  15x + 3
2x + y
4x  3
14x²  4x + 7
a(b + c) = ab + ac
Example:
5(x + 7) =
5x + 35
5•x
+
5•7
2(4 + 9x) 2. 7(x + 1) 3. 12(a + b + c)
8 + 18x
7x  7
12a + 12b + 12c
3w – 3x + 2z
7a + 7c + 7b
70x  50
x – 2
25x + 25
6 + 6x²y³ + 9y²
y + yx
36x² + 36x
81x + 8y
4. 12a – 6h 5. 3x + 9 6. 12x + y
7. 24a – 4 8. 72a + 9n 9. 8a  8v
3(x + 3)
6(2a – h)
Cannot be simplified
4(6a – 1)
9(8a + n)
8(a – v)
2n – 10 = 50
+10
+10
2n = 60
2n = 60
2
2
n
= 30
2n – 10 = 50
2 (30) – 10 = 50
60 – 10 = 50
50 = 50
n = 10
n = 15
n = 42
h = 9/5
n = 4
z = 5.8
x = 22
a = 22
x = 22
x = 22
x = 30
a = 22
c = 22
a = 22
x = 12.3
b = 22
14. 2(b – 2) + b + 3
4(n – 5)  7 = 9 + 2n – 4n
4n – 20  7 = 9 + 2n – 4n
4n – 20  7 = 9 + 2n – 4n
4n – 27 = 9 – 2n
4n – 27 = 9 – 2n
+27
+27
4n = 36 – 2n
4n = 36 – 2n
+2n
+2n
6n = 36
6n = 36
6
6
n = 6
4(n – 5)  7 = 9 + 2n – 4n
4(6 – 5)  7 = 9 + 2(6) – 4(6)
4(1)  7 = 9 + 12 – 24
4 – 7 = 21  24
3 = 3
r = 2
n = 3
y = 11
v = 12
Example: 3 + y> 8.
Inequalities use symbols like <and> which means less than or greater than.
They also use the symbols ≤ and ≥which means less than or equal to and greater than or equal to.
Inequalities
X > 2
X ≥ 2 1/2