View
7.473
Download
0
Embed Size (px)
DESCRIPTION
Citation preview
Solving Equations Solving Equations
1) open sentence2) equation3) solution
Translate verbal expressions into algebraic expression and equations and vice versa. Solve equations using the properties of equality.
Solving Equations Solving Equations
A mathematical sentence (expression) containing one or more variables is called an open sentence.
A mathematical sentence stating that two mathematical expressions are equalis called an _________.equation
Open sentences are neither true nor false until the variables have been replaced by numbers.
Each replacement that results in a true statement is called a ________ of theopen sentence.
solution
To solve equations, we can use properties of equality.Some of these equivalence relations are listed in the following table.
Properties of Equality Property Symbol Example
Reflexive
Symmetric
Transitive
Substitution
For any real number a,
a = a,
For all real numbers a and b,
If a = b, then
For all real numbers a, b, and c.
If a = b, and b = c, then
If a = b, then a may be replacedby b and b may be replaced by a.
b = a
a = c
– 5 + y = – 5 + y
If 3 = 5x – 6, then 5x – 6 = 3
If 2x + 1 = 7 and 7 = 5x – 8
then, 2x + 1 = 5x – 8
If (4 + 5)m = 18
then 9m = 18
Solving Equations Solving Equations
Sometimes an equation can be solved by adding the same number to each side or bysubtracting the same number from each side or by multiplying or dividing each side by the same number.
Addition and Subtraction Properties of Equality
For any real numbers a, b, and c, if a = b, then
a = b+ c + c a = b- c - c
Example:
If x – 4 = 5, then x – 4 = 5+ 4 + 4
If n + 3 = –11, then n + 3 = –11– 3 – 3
Solving Equations Solving Equations
Sometimes an equation can be solved by adding the same number to each side or bysubtracting the same number from each side or by multiplying or dividing each side by the same number.
Multiplication and Division Properties of Equality
For any real numbers a, b, and c, if a = b, then
a = b· c · c a = b
Example:
4 4
c c
then64m
If 6 4m then6y3 If - 6 3 y
-3 -3
Solving Equations Solving Equations
• What will we discuss?What are the parts of an equationWhat does it mean to solve an equationHow do we use inverse operations to solve
equationsHow to solve simple and complex equations
Chapter 5: Solving Equations
What Does it Mean to Solve an What Does it Mean to Solve an Equation?Equation?
To solve an equation means to find every number that makes the equation true.
We do this by adding or subtracting to each side of the equation … but always keep it balanced!
What are the parts of an equation?What are the parts of an equation?
Let’s first take a look at an Let’s first take a look at an equation and identify its partsequation and identify its parts
3 1 2 3 6x x
Coefficient
ConstantVariable
So do we just use trial and error to So do we just use trial and error to find the right value?find the right value?
No. No. We can use We can use inverse operationsinverse operations to to
isolate, or solve for, the variable’s isolate, or solve for, the variable’s value.value.
Inverse operations? Think about it …Inverse operations? Think about it …The inverse operation of addition is The inverse operation of addition is subtractionsubtraction. And the inverse . And the inverse operation of multiplication is operation of multiplication is divisiondivision..
Solving 1 Step EquationsSolving 1 Step Equations
How much does the suitcase weigh
in terms of blocks?
B=Blocks S=Suitcase
Equation: 6B + S = 9B
-6B
-6B
3BS =What is the weight of the suitcase if each
block has a weight of 2lbs. ?
S = 3 (2) = 6 lbs.
So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?
Let’s take a look at a simple equationLet’s take a look at a simple equation
Step 1:- 13
Answer:
2113 x- 13
8x
Now that we have solved the equation, let’s check the solution:
2121
2113 x
21138
So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?
Let’s take a look at a simple equationLet’s take a look at a simple equation
Step 1:+ 5
Answer:
125 y+ 5
17y
Now that we have solved the equation, let’s check the solution:
1212
125 y
12517
So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?
Let’s take a look at a simple equationLet’s take a look at a simple equation
Step 1:
Step 2:
25
Answer:
7525 W25
25
75W
3W
Now that we have solved the equation, let’s check the solution:
7525 W
75)3(25
7575
So how do we solve equations with So how do we solve equations with inverse operations?inverse operations?
Let’s take a look at a simple equationLet’s take a look at a simple equation
Step 1:
Step 2:
(16)
Answer:
416
A
(16)
)16(4A
64A
Now that we have solved the equation, let’s check the solution:
44
416
A
416
64
1 Step Equation1 Step Equation
X + 11 = 9 X - 37 = 52 3X = 72-11 -
11X = -2
3 3
X = 24
1
1
20 + h = 41 17 - s = 27
37 37
52 37 - X
89X
20-20-
41 h 20
21h1717
2717
s
10 sThis is the same as -
1S=10
101
10
1
s
1 Step Equations Continued…1 Step Equations Continued…6X = 42
25
P = 34
6 6
1X=7
or x = 7
1
1
47
x
1
74
1
7
7
x
28x
213
s
1
213
s1321 sCross Multiply
21
3s
7
1
Multiply by the
reciprocal of 2/5
2
5
4
3
2
5
5
2 p
8
151 p
Multi Step Equations
Solve:
8m – 10 = 36
423
31176w
8m – 10 = 36
8m = 468 8
m =
+ 10 + 10 1717
31176
w
416w
1
64
1
6 16
w
84w
Multi Step EquationsMulti Step Equations
5x 2 = x + 4 Solve:
5x 2 = x + 4
Notice that there are variables on both sides
5x = x + 6
Get rid of the -2 on the left side
Simplify
5x = x + 6Get rid of the x on the right side4x = 6Get rid of the coefficient of x
4 4
23x = Simplify
Simplify
+ 2 + 2
– x– x
Solving a ProportionSolving a Proportion
Solve the proportion belowSolve the proportion below
60
126
C)12()60(6 C
C12360 12 12
C30
Solving a ProportionSolving a Proportion
Solve the proportion belowSolve the proportion below
852
13 a )(52)8(13 a
a52104 52 52
a2
Checking the Solution to a Checking the Solution to a ProportionProportion
Let’s check the solution to the Let’s check the solution to the proportion we solved on the last slideproportion we solved on the last slide
852
13 a
852
13 2
4
1
4
1
Using Proportions to Solve Using Proportions to Solve ProblemsProblems
You get 46 miles to a gallon of gas. You get 46 miles to a gallon of gas. How far can you go on 16 gallons of How far can you go on 16 gallons of gas?gas?
m
16
46
1 )16(461 m
736m
Multi-step SolutionsMulti-step Solutions
Let’s take a look at our original equationLet’s take a look at our original equation
3 1 2 3 6x x
3 2 4x x
x 6
4 2 4x
Step 1:-12 -12
Step 2:+x +x
Step 3:4 4
Answer:
Multi-step SolutionsMulti-step Solutions(involving distribution)(involving distribution)
Consider the following equationConsider the following equation
Step 1:
Step 2:
Step 3:
Answer:
42)5(6 x
+3042306 x
+30
726 x6 6
12x
Finding Variations of FormulasFinding Variations of Formulas
Solve the formula for Solve the formula for rr..
trs 2
s trs 2
s s
tr 2
2s
tr
22