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![Page 1: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/1.jpg)
![Page 2: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/2.jpg)
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.
![Page 3: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/3.jpg)
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Open sentences are neither true nor false until the variables have been replaced by numbers.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
![Page 4: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/4.jpg)
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
![Page 5: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/5.jpg)
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
![Page 6: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/6.jpg)
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
![Page 7: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/7.jpg)
• 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
![Page 8: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/8.jpg)
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!
![Page 9: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/9.jpg)
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
![Page 10: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/10.jpg)
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..
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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.
![Page 12: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/12.jpg)
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
![Page 13: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/13.jpg)
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
![Page 14: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/14.jpg)
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
![Page 15: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/15.jpg)
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
![Page 16: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/16.jpg)
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
![Page 17: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/17.jpg)
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
![Page 18: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/18.jpg)
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
![Page 19: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/19.jpg)
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
![Page 20: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/20.jpg)
Solving a ProportionSolving a Proportion
Solve the proportion belowSolve the proportion below
60
126
C)12()60(6 C
C12360 12 12
C30
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Solving a ProportionSolving a Proportion
Solve the proportion belowSolve the proportion below
852
13 a )(52)8(13 a
a52104 52 52
a2
![Page 22: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/22.jpg)
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
![Page 23: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/23.jpg)
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
![Page 24: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/24.jpg)
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:
![Page 25: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/25.jpg)
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
![Page 26: Solving Equations 1) open sentence 2) equation 3) solution Translate verbal expressions into algebraic expression and equations and vice versa. Solve.](https://reader035.fdocuments.in/reader035/viewer/2022062322/5697c0121a28abf838ccc4d6/html5/thumbnails/26.jpg)
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