Download Basic College Mathematics (ALEKS) Chapter Eleven Solving

Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
Section 11 – 1 Properties of Real Numbers
Variables - Letters or Symbols that represents any number.
(If you can do it with numbers, you can do it with variables.)
Constants – values that never change. ߨ = 3.14159 …
Expression – a collection of variables, constants, numbers, operators.
2x, 4 + y,
3T – 7,
௬
଼
Write an expression that represents:
Turkey costs $6.99 per pound.
What is the cost of “p” pounds?
$6.99p
The width if a basketball court is
44 feet shorter than the length.
W = L - 44
6 tons of gravel was carried in “n”
number of trucks.
6
݊
EVALUATING EXPRESSIONS
-4Y2 when Y = -3 -4(-3)2 = -4(-3)(-3) = -36
ଵ
ଵ
-6T + 4V when t = -1 and V = - -6(-1) + 4(- ) = 6 +(-2) = 4
ଶ
ଶ
Commutative Property
X+3=X+3
Z(2) = 2Z
-12 + P = P – 12
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Chapter 11
Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
AB = BA
Associative Property
2(-8W) = (2)(-8)(W) = -16W
6.2 + (14.7 + X) = (6.2 + 14.7) + X = 20.9 + X
OR X + 20.9
1
1
− ሺ3ܻሻ = ൬− ൰ ሺ3ሻሺܻሻ = −ܻ
3
3
Distributive Property
4(2 + M) = (4)(2) + (4) (M) = 8 + 4M or
4M + 8
6(5P – 3Q + 1) = (6)(5P) – (6)(3Q) + (6)(1) = 30P – 18Q + 6
ହ
ଶ
ଵ
ହ
ହ
ଵ
ቀ4ܶ + ቁ = ቀ ቁ ሺ4ܶሻ + ቀ ቁ ቀ ቁ = 10ܶ +
ଷ
ଶ
ଶ
ଷ
ହ
଺
-4(6 – 10X) = (-4)(6) – (-4)(10X) = -24 +40X or 40X – 24
-(2X – 3Y + 4Z) = (-1)(2X) – (-1)(3Y) + (-1)(4Z) = -2X + 3Y- 4Z
Section 11 – 2 Simplifying Expressions
TERM – a number, or a product or quotient of numbers and variables.
-8X3
xy
-40
Constant term – just numbers:
5PQ
௔
௕
2, 6, 50
Coefficient of term – the number that is multiplied by a variable:
4XY 4
-8X -8
Like Term – have same variables raised to the same powers:
-4x and 6x both variables x like term
-4x and 6y variables x and y unlike term
2
Chapter 11
Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
18ab and 4ba variable ab like term
18ab and 4a variables ab and a unlike terms
ଵ
-4.8Y + 8.1Y2 - ܻ – 8
ଶ
ଵ
terms
-4.8Y, 8.1Y2, - ܻ, −8
ଶ
ଵ
Coefficients -4.8, 8.1, - , −8
ଶ
Like terms -4.8Y, -ଵ ܻ
ଶ
Combining Like Terms
-4W + 11W = -W-W-W-W+W+W+W+W+W+W+W+W+W+W =
W+W+W+W+W+W+W = 7W
-4W + 11W = W(-4 +11) = W(7) = 7W
Z – 4Z + 22Z = (1 – 4 + 22)Z = 19Z
4A – 10B – A + 16B + 9 = (4A – A) + (-10B + 16B) + (9) =3A + 6B + 9
ହ
ଽ
ଶ
ହ
ସ
ହ
ସ
ଶ
ହ
− ܹ + ܹ − = ( − ሻ + ቀ− ܹ + ܹቁ =
ଷ
ଷ
ଽ
ଽ
ଽ
ଷ
ଷ
ଵ
ଽ
+ ܹ=ܹ+
ଵ
ଽ
6.3X – 4.1Y – 2.4 + 2.1Y + 1.1 = (6.3X) + (2.1Y - 4.1Y) + (1.1 – 2.4)
6.3X – 2Y – 1.3
Parentheses
8 – 6(W + 4) = 8 – 6W – 24 = -6W – 16
-5(10 – M) – 2(M + 1) = -50 + 5M – 2M – 2 = 3M - 52
Section 11 – 3 Addition and Subtraction Properties of Equality
Equation – statement that TWO quantities are equal. NEEDS an = sign.
3
Chapter 11
Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
-2W = 6
Equation
4=T–3
Equation
X+8
Expression
Solution – makes the equations TRUE.
Z+3=8
Z=6
(6) + 3 = 9 NOT 8
So not a solution.
Z=5
(5) + 3 = 8
A Solution.
2 + 3X = 23 when X = 7
2 + 3(7) = 23
2 + 21 = 23
23 = 23 Yes
-4X + 1 = 9 when X = 2
-4(2) + 1 = 9
-8 + 1 = 9
7 = 9 NO
LINEAR EQUATION, with one variable, can be written in the form
AX + B = 0
EQUIVALENT EQUATIONS are two or more equations that have the
same solution.
Are these equivalent equations
X = 5 and
2X = 10
First X = 5
Second 2(__) = 10 clear 2x5 = 10
YES both have a solution of 5
Addition Property of Equality
A = B then
4
Chapter 11
Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
A + C = B + C (add C to both sides)
Subtraction Property of Equality
A = B then
A – C = B – C (subtract C from both sides)
(Just DO the opposite math operation)
Y – 4 = 12
(note that the problem show a subtraction of 4)
Y – 4 + 4 = 12 + 4
(the opposite math operation is to add 4)
Y + 0 = 16
Y = 16
7.9 = -3.2 + P
7.9 + 3.2 = -3.2 + 3.2 + P
11.1 = P
or P = 11.1
M + 8 = 21
(note adding 8)
M + 8 – 8 = 21 – 8
(opposite is subtraction of 8)
M = 13
-16 = 1 + Z
-16 – 1 = 1 – 1 + Z
-17 = Z
or Z = -17
4
1
+ ‫ =ܥ‬−
5
4
5
Chapter 11
Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
ସ
ହ
−
ସ
ହ
ଵ
+ ‫ =ܥ‬− −
ସ
ଵ
ହ
ସ
ହ
ସ
ଵ
ସ
ସ
C = ቀ− ∗ ቁ + ቀ− ∗ ቁ = −
ସ
ହ
ହ
ସ
= − + ሺ− ሻ
ସ
ହ
ହ
ଶ଴
ଵ଺
+ ቀ− ቁ = −
ଶ଴
ଶଵ
ଶ଴
-9.13 = Y – 2.27
-9.13 +2.27 = Y -2.27 + 2.27
-6.86 = Y
or
Y = -6.86
Section 11 – 4 Multiplication and Division Properties of Equality
A=B
A*C = B*C
A=B
஺
஼
=
஻
஼
4X = 32
ସ௑
ସ
(note 4 times X)
=
ଷଶ
ସ
(opposite is divide by 4
X=8
18 = -2W
ଵ଼
ିଶ
=
ିଶௐ
ିଶ
-9 = W
or W = -9
1.9 = -M
1.9 * (-1) = (-1)*(-M)
-1.9 = M
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Chapter 11
Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
4.1Z = 28.29
ସ.ଵ௓
ସ.ଵ
ଶ଼.ଶଽ
=
ସ.ଵ
Z = 6.9
ହ
ଵ
− ܲ=
ଽ
ହ
(times − )
ଷ
ଽ
ହ
ହ
− ܲ ÷ ቀ− ቁ =
ଽ
ଽ
ହ
ଽ
ቀ− ቁ ‫ ݔ‬ቀ− ቁ ܲ =
ଽ
P=−
஼
ଵଶ
ହ
ଵ
ଷ
ହ
÷ ሺ− ሻ
ଽ
ଵ
ଷ
‫ݔ‬−
ଽ
ହ
ଷ
ହ
=3
(divide by 12)
஼
12( ሻ = 12 ∗ 3
ଵଶ
C = 36
ܶ
= −8
5
்
5 * = −8 ∗ 5
ହ
T = -40
-4.6 + X = 12.9
-4.6 + 4.6 + X = 12.9 + 4.6
X = 17.5
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Chapter 11
Basic College Mathematics (ALEKS) Chapter Eleven Solving Equations
5 = -2P
ହ
ିଶ
ିଶ௉
=
ିଶ
ହ
− =ܲ
ଶ
ଵ
Z+ =
ଷ
ହ
଼
ଵ
ଵ
ହ
Z+ − = −
ଷ
ଷ
଺
ଵ
ହ
ଵ
ଷ
ଶ
Z=( ሻ−ቀ ∗ ቁ=
଺
3
ܺ=1
5
ଷ
ହ ଷ
ହ
( )( )X = ( )1
ଷ ହ
ଷ
ଶ
ܺ=
ହ
଺
−
ଶ
଺
=
ଷ
଺
=
ହ
ଷ
2X + 4 = 10
2X + 4 – 4 = 10 - 4
2X = 6
ଶ௑
ଶ
=
଺
ଶ
X=3
8
Chapter 11
ଵ
ଶ