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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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.
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
Mike is three years older than Jim.
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
Six less than three times a number is 21.
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
Write your let statement
Write your equation
Solve
Check
Write an answer sentence
A cell phone company charges $39 a month plus $.15 per text message sent. If Jan sends 35 text messages this month, how much does she owe before taxes are added?
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
A rental car company ABC charges $25 per day plus $.15 per mile. Rental car company XYZ charges $18 per day plus $.25 per mile. If you plan to drive 50 miles, who is the cheaper rental company?
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
Evaluate if s = 4
4(4)
16
4 + 4
8
5  4
1
12 ÷ 4
3
Evaluate if s = 6
7(6)
42
3 + (6)
3
7 – (6)
13
18 ÷ (6)
3
Evaluate if n = 3 and r = 5
3² + 7(5)
9 +12
21
9(3)  5²
27 – 25
2
2(3)(5) + 6(3)
30 + 18
48
Evaluate if p = 12 and q = 8
12 + (8) + 6
4 + 6
2
12 – (8) + 3
20 + 3
23
12 – (8) + (8)²
20 + 64
84
Evaluate if a = 2 and b = 6
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
Terms of an Expression
3x + 5y – 8 has 3 terms.
Like Terms
8x²+2x²+5a +a
8x²and 2x² are like terms
5a and a are like terms
LIKE terms: Yes or No?
3x + 7x
Yes  Like
5x + 5y
No  Unlike
4c + c
Yes  Like
4d + 4
No  Unlike
LIKE terms: Yes or No?
3ab – 6b
No  Unlike
2a – 5a
Yes  Like
x andx²
No  Unlike
Yes  Like
6 and 10
Identify the LIKE terms
3m – 2m + 8 – 3m + 6
5x + b – 3x + 4 + 2x – 1 – 3b
6y + 4yz + 6x² + 2yz – 4y + 2x²  5
Coefficients
Example: 6x
The coefficient is 6.
Example: x
The coefficient is 1.
Simplify
Write an expression:
+
3c + 4c
=7c
Write an expression:

8a  1a
= 7a
Write an expression:
+
5c + 4d
Write an expression:

5a – 4b
This expression cannot be simplified. Why not?
6x
10a
4xy
4½y
7a + 6
4d
2xy + 2x
5cd – 2a
0
¾e
24x
2c
1.–5x – 3x2.8x – 2x
3.–7x – (–3x)4.6x – (–4x)
5.–10x –14x6.–9x – (–x)
7.3x – 8x8.x – (–5x)
9. a² + b² + 2a² + 5b²10. 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
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
Practice #1
3(m  4)
3 • m  3 • 4
3m – 12
Practice #2
2(y + 3)
2 • y + (2) • 3
2y + (6)
2y  6
Simplify the following:
3(x + 6) =
3x + 18
4(4 – y) =
16 – 4y
7(2 + z) =
14 + 7z
5(2a + 3) =
10a + 15
Simplify the following:
6(3y  5) =
18y – 30
3 +4(x + 6) =
4x + 27
2x + 3(5x  3) + 5 =
17x – 4
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
12cd
4x
2a
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
r = 6.4
c = 1 3/7
m = 33/14
b = 2.5
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
Inequality: a mathematical sentence using <, >, ≥ , or ≤.
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
≠
Does not equal…
Example: x ≠ 7 means:
7 is not equal to x
x > 7
y < 5
x < 9
x > 2
n > 3
x < 6
y ≥ 3
n < 4
x ≤ 4