solving equations and problems
DESCRIPTION
SOLVING EQUATIONS AND PROBLEMS. CHAPTER 3. Section 3-1 Transforming Equations: Addition and Subtraction. Addition Property of Equality. If a, b, and c are any real numbers, and a = b, then a + c = b + c and c + a = c + b. Subtraction Property of Equality. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/1.jpg)
SOLVING SOLVING EQUATIONS AND EQUATIONS AND
PROBLEMSPROBLEMS
SOLVING SOLVING EQUATIONS AND EQUATIONS AND
PROBLEMSPROBLEMSCHAPTER 3CHAPTER 3
![Page 2: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/2.jpg)
Section 3-1 Section 3-1 Transforming Transforming Equations: Equations:
Addition and Addition and SubtractionSubtraction
Section 3-1 Section 3-1 Transforming Transforming Equations: Equations:
Addition and Addition and SubtractionSubtraction
![Page 3: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/3.jpg)
Addition Property of Equality
If a, b, and c are any real numbers, and a = b, then
a + c = b + c andc + a = c + b
![Page 4: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/4.jpg)
Subtraction Property of
EqualityIf a, b, and c are any real numbers, and a = b, then
a - c = b - c andc - a = c - b
![Page 5: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/5.jpg)
Equivalent Equations
Equations having the same solution set over a given domain.
-5 = n + 13 and -18 = n are equivalent
![Page 6: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/6.jpg)
Transforming an Equation into an
Equivalent Equation
![Page 7: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/7.jpg)
Transformation by Substitution
Substitute an equivalent
expression for any expression in a given equation.
![Page 8: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/8.jpg)
Transformation by Addition
Add the same real number to each side of a given equation.
![Page 9: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/9.jpg)
Transformation by Subtraction
Subtract the same real number from
each side of a given equation.
![Page 10: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/10.jpg)
EXAMPLES
Solve:x – 8 = 17Add 8
x – 8 + 8 = 17 + 8x = 25
![Page 11: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/11.jpg)
EXAMPLES
Solve:-5 = n + 13Subtract 13 -5 -13 = n + 13 – 13
-18 = n
![Page 12: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/12.jpg)
EXAMPLES
Solve:x + 5 = 9Subtract 5
x + 5 – 5 = 9 - 5 x = 4
![Page 13: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/13.jpg)
Section 3-2Section 3-2 Transforming Equations: Transforming Equations:
Multiplication and Multiplication and DivisionDivision
Section 3-2Section 3-2 Transforming Equations: Transforming Equations:
Multiplication and Multiplication and DivisionDivision
![Page 14: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/14.jpg)
Multiplication Property of Equality
If a, b, and c are any real numbers, and a = b, then
ca = cb andac = bc
![Page 15: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/15.jpg)
Division Property of Equality
If a and b are real numbers, c is any nonzero real number, and a = b, then
a/c = b/c
![Page 16: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/16.jpg)
Transformation by Multiplication
Multiply each side of a given equation by the same nonzero
real number.
![Page 17: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/17.jpg)
Transformation by Division
Divide each side of a given equation by the same nonzero
real number.
![Page 18: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/18.jpg)
EXAMPLES
Solve:• 6x = 222• 8 = -2/3t• m/3 = -5
![Page 19: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/19.jpg)
Section 3-3 Section 3-3 Using Several Using Several
TransformationTransformationss
Section 3-3 Section 3-3 Using Several Using Several
TransformationTransformationss
![Page 20: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/20.jpg)
Inverse Operations
For all real numbers a and b,
(a + b) – b = a and(a – b) + b = a
![Page 21: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/21.jpg)
Inverse Operations
For all real numbers a and all nonzero real numbers b
(ab) b = a and(a b)b = a
![Page 22: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/22.jpg)
EXAMPLESSolve:1. 5n – 9 = 712. 1/5x + 2 = -13. 40 = 2x + 3x4. 8(w + 1) – 3 = 48
![Page 23: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/23.jpg)
3-4 Using 3-4 Using Equations to Equations to
Solve ProblemsSolve Problems
3-4 Using 3-4 Using Equations to Equations to
Solve ProblemsSolve Problems
![Page 24: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/24.jpg)
EXAMPLESThe sum of 38 and twice a number is 124. Find the number.
![Page 25: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/25.jpg)
EXAMPLESThe perimeter of a trapezoid is 90 cm. The parallel bases are 24 cm and 38 cm long. The lengths of the other two sides are consecutive odd integers. What are the
lengths of these other two sides?
![Page 26: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/26.jpg)
Solution
38
24
x + 2 x
![Page 27: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/27.jpg)
3-5 Equations with 3-5 Equations with Variables on Both Variables on Both
SidesSides
3-5 Equations with 3-5 Equations with Variables on Both Variables on Both
SidesSides
![Page 28: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/28.jpg)
EXAMPLES
•6x = 4x + 18•3y = 15 – 2y•(4 + y)/5 = y•3/5x = 4 – 8/5x•4(r – 9) + 2 = 12r + 14
![Page 29: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/29.jpg)
3-6 Problem 3-6 Problem Solving: Using Solving: Using
ChartsCharts
3-6 Problem 3-6 Problem Solving: Using Solving: Using
ChartsCharts
![Page 30: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/30.jpg)
PROBLEMA swimming pool that is 25 m long is 13 m narrower than a pool that is 50 m long. Organize in chart form.
![Page 31: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/31.jpg)
SOLUTION
Length Width
1st pool
25 w -13
2nd pool
50 w
![Page 32: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/32.jpg)
PROBLEMA roll of carpet 9 ft wide is 20 ft longer than a roll of carpet 12 ft wide. Organize in chart form.
![Page 33: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/33.jpg)
SOLUTION
Width Length
1st roll 9 x + 20
2nd roll
12 x
![Page 34: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/34.jpg)
PROBLEMAn egg scrambled with butter has one more gram of protein than an egg fried in butter. Ten scrambled eggs have as much protein as a dozen fried eggs.
How much protein is in
one fried egg?
![Page 35: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/35.jpg)
SOLUTION
Protein per egg
Number of eggs
Total Protein
Scrambled egg
x + 1 10 10(x + 1)
Fried egg
x 12 12(x)
![Page 36: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/36.jpg)
3-7 Cost, Income, 3-7 Cost, Income, and Value and Value ProblemsProblems
3-7 Cost, Income, 3-7 Cost, Income, and Value and Value ProblemsProblems
![Page 37: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/37.jpg)
Formulas•Cost = # of items x price/item•Income = hrs worked x wage/hour•Total value = # of items x value/item
![Page 38: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/38.jpg)
PROBLEMTickets for the senior class play cost $6 for adults and $3 for students. A total of 846 tickets worth $3846 were sold. How many student tickets were sold?
![Page 39: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/39.jpg)
SOLUTION
number Price per ticket
Total Cost
Student
x 3 3x
Adult 846 - x 6 6(846-x)
![Page 40: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/40.jpg)
PROBLEM
Marlee makes $5 an hour working after school and $6 an hour working on Saturdays. Last week she made $64.50 by working a total of 12 hours. How many hours did she work on Saturday?
![Page 41: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/41.jpg)
SOLUTION
hours
wages
Income
Saturdays
x $6 6x
Weekdays
12-x $5 5(12-x)
![Page 42: SOLVING EQUATIONS AND PROBLEMS](https://reader037.vdocuments.mx/reader037/viewer/2022102800/568132af550346895d99655a/html5/thumbnails/42.jpg)
THE ENDTHE ENDTHE ENDTHE ENDThe EndThe End