chapter 4 systems of equations and problem solving how are systems of equations solved?
TRANSCRIPT
![Page 1: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/1.jpg)
Chapter 4Systems of Equationsand Problem Solving
How are systems of equations solved?
![Page 2: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/2.jpg)
Activation
• Review Yesterday’s Warm-up
![Page 3: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/3.jpg)
4-1SYSTEMS OF EQUATIONS IN TWO VARIABLES
How do you solve a system of equations in two variables graphically?
![Page 4: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/4.jpg)
Vocabulary Systems of equations: two or more
equations using the same variables Linear systems: each equation has
two distinct variables to the first degree.
Independent system: one solution Dependent system: many solutions,
the same line Inconsistent system: no solution,
parallel lines
![Page 5: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/5.jpg)
Directions:
• Solve each equation for y• Graph each equation• State the point of intersection
![Page 6: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/6.jpg)
Examples:
x – y = 5
and y + 3 = 2x
![Page 7: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/7.jpg)
Examples:
3x + y = 5
and 15x + 5y = 2
![Page 8: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/8.jpg)
Examples:
y = 2x + 3
and -4x + 2y = 6
![Page 9: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/9.jpg)
Examples:
x – 2y + 1 = 0
and x + 4y – 6 =0
![Page 10: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/10.jpg)
• What limitations do you think are affiliated with this procedure?
![Page 11: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/11.jpg)
4-1HOMEWORK
PAGE(S): 161NUMBERS: 2 – 16 even
www.phschool.com code age-0775
![Page 12: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/12.jpg)
Activation
• Review Yesterday’s Warm-up
![Page 13: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/13.jpg)
4-2ASOLVING SYSTEMS OF EQUATIONS —SUBSTITUTION
How do you solve a system of equations in two variables by substitution?
![Page 14: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/14.jpg)
Substitution:1) LOOK FOR A VARIABLE W/O A
COEFFICIENT2) SOLVE FOR THAT VARIABLE3) SUBSTITUTE THIS NEW VALUE INTO THE
OTHER EQUATION
exampl:e:4x + 3y = 42x – y = 7
![Page 15: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/15.jpg)
• Example:2y + x = 13y – 2x = 12
![Page 16: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/16.jpg)
• Examples:5x + 3y = 6x - y = -1
![Page 17: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/17.jpg)
4-2AHOMEWORK
PAGE(S): 166 -167NUMBERS: 1 – 8 all
USING SUBSTITUTION
www.phschool.com code age-0775
![Page 18: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/18.jpg)
Activation
• Review Yesterday’s warm-up
![Page 19: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/19.jpg)
4-2B AND 4-6 SOLVING SYSTEMS OF EQUATIONS —LINEAR COMBINATION—ELIMINATION METHOD
CONSISTENT AND DEPENDENTSYSTEMS
How do you solve a system of equations in two variables by linear combinations?
What makes a system dependent, independent, consistent, or inconsistent?
![Page 20: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/20.jpg)
Combination/Elimination1)LOOK FOR OR CREATE A SET OF
OPPOSITESA) TO CREATE USE THE COEFFICIENT OF THE
1ST WITH THE SECOND AND VICE VERSAB) MAKE SURE THERE WILL BE ONE + & ONE –
2) ADD THE EQUATIONS TOGETHER AND SOLVE
3) SUSTITUTE IN EITHER EQUATION AND SOLVE FOR THE REMAINING VARIABLE
![Page 21: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/21.jpg)
• Example:4x – 2y = 7
x + 2y = 3
![Page 22: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/22.jpg)
• Example:4x + 3y = 42x - y = 7
![Page 23: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/23.jpg)
• Example:3x – 7y = 15 5x + 2y = -4
![Page 24: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/24.jpg)
• Example: 2x - y = 3-2x + y = -3
![Page 25: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/25.jpg)
• Example: 2x - y = 3-2x + y = 9
![Page 26: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/26.jpg)
4-2BHOMEWORK
PAGE(S): 166 -167NUMBERS: 10 – 22 even
USING LINEAR COMBINATIONS
www.phschool.com code age-0775
![Page 27: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/27.jpg)
Activation
• Review Yesterday’s Warm-up
![Page 28: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/28.jpg)
4-3USING A SYSTEM OF TWO EQUATIONS
How do you translate real life problems into systems of equations?
![Page 29: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/29.jpg)
• USE ROPES:–Read the problem–Organize your thoughts in
a chart–Plan the equations that
will work–Evaluate the Solution–Summarize your findings
![Page 30: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/30.jpg)
• Example: The sum of the first number and a second
number is -42. The first number minus the second is 52. Find the numbers
1st number x
2nd number y
x + y = -42 x - y = 52
![Page 31: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/31.jpg)
• Example: Soybean meal is 16% protein and corn meal is
9% protein. How many pounds of each should be mixed together to get a 350 pound mix that is 12% protein?
Soybean meal x .16
Corn meal y .09
x + y = 350.16x + .09y = .12 • 350
![Page 32: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/32.jpg)
• Example: A total of $1150 was invested part at 12% and
part at 11%. The total yield was $133.75. How much was invested at each rate?
12% investment x .12
11% investment y .11
x + y = 1150.12x + .11y = 133.75
![Page 33: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/33.jpg)
• Example: One day a store sold 45 pens. One kind cost
$8.75 the other $9.75. In all, $398.75 was earned. How many of each kind were sold?
Type 1 x 8.75
Type 2 y 9.75
x + y = 458.75x + 9.75y = 398.75
![Page 34: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/34.jpg)
4-3HOMEWORK
PAGE(S): 171 -173NUMBERS: 4 – 24 by 4’s
www.phschool.com code age-0775
![Page 35: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/35.jpg)
Activation
• Review Yesterday’s Warm-up
![Page 36: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/36.jpg)
4-4SYSTEMS OF EQUATIONS IN THREE VARIABLES
How do you solve a system of equations in three variables? How is it similar to solving a system in two equations?
![Page 37: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/37.jpg)
Find x, y, z2x + y - z = 53x - y + 2z = -1 x - y - z = 0
![Page 38: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/38.jpg)
Find x, y, z2x - y + z = 4 x + 3y - z = 114x + y - z = 14
![Page 39: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/39.jpg)
Find x, y, z2x + z = 7 x + 3y + 2z = 54x + 2y - 3z = -3
![Page 40: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/40.jpg)
4-4HOMEWORK
PAGE(S): 178 - 179NUMBERS: 4 – 24 by 4’s
www.phschool.com code age-0775
![Page 41: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/41.jpg)
Activation
• Review Yesterday’s Warm-up
![Page 42: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/42.jpg)
4-5USING A SYSTEM OF THREE EQUATIONS
How do you translate word problems into a system of three equations?
![Page 43: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/43.jpg)
• Example: The sum of three numbers is 105. The third is 11 less
than ten times the second. Twice the first is 7 more than three times the second. Find the numbers.
1st number x
2nd number Y
3rd number z
x + y + z = 105 z = 10y – 11
2x = 7 + 3y
![Page 44: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/44.jpg)
• Example:Sawmills A, B, C can produce 7400 board feet of lumber per day. A and B together can produce 4700 board feet, while B and C together can produce 5200 board feet. How many board feet can each mill produce?
Mill A x
Mill B y
Mill C z
x + y + z = 7400 x + y = 4700 y + z = 5200
![Page 45: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/45.jpg)
4-5HOMEWORK
PAGE(S): 181 - 182NUMBERS: 4, 8, 12, 16
www.phschool.com code age-0775
![Page 46: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/46.jpg)
Activation
• Review Yesterday’s Warm-up
![Page 47: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/47.jpg)
4-7SYSTEMS OF INEQUALITIES
How do you solve a system of linear inequalities?
![Page 48: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/48.jpg)
Vocabulary:
Feasible region: the area of all possible outcomes
![Page 49: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/49.jpg)
Directions:
• Solve each equation for y• Graph each equation• Shade each with lines• Shade the intersecting lines a
solid color
![Page 50: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/50.jpg)
Examples x – 2y < 6 y ≤ -3/2 x + 5
![Page 51: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/51.jpg)
y ≤ -2x + 4 x > -3
![Page 52: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/52.jpg)
y < 4y ≥ |x – 3|
![Page 53: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/53.jpg)
3x + 4y ≥ 12
5x + 6y ≤ 301 ≤ x ≤ 3
![Page 54: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/54.jpg)
4-7HOMEWORK
PAGE(S): 192NUMBERS: 4 – 32 by 4’s
www.phschool.com code age-0775
![Page 55: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/55.jpg)
REVIEW
PAGE(S): 200 NUMBERS: all
![Page 56: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/56.jpg)
Activation
• Review yesterday’s warm-up
![Page 57: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/57.jpg)
4-8USING LINEAR PROGRAMMING
EQ: What is linear programming?
![Page 58: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/58.jpg)
VOCABULARY:
• Linear programming– identifies minimum or maximum of a given situation
• Constraints—the linear inequalities that are determined by the problem
• Objective—the equation that proves the minimum or maximum value.
![Page 59: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/59.jpg)
Directions:• Read the problem• List the constraints• List the objective• Graph the inequalities finding the
feasible region• Solve for the vertices (the points
of intersection)• Test the vertices in the objective
![Page 60: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/60.jpg)
Example:What values of y
maximize P givenConstraints: y≥3/2x -3 y ≤-x + 7 x≥0 y≥0Objective:
P = 3x +2y
x y P
![Page 61: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/61.jpg)
You are selling cases of mixed nuts and roasted peanuts. You can order no more than a total of 500 cans and packages and spend no more than $600. If both sell equally well, how can you maximize the profit assuming you will sell everything that you buy?
x y P
![Page 62: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/62.jpg)
Partner Problem (sample was #8)• A florist has to order roses and carnations for Valentine’s Day. The florist needs to decide
how many dozen roses and carnations should be ordered to obtain a maximum profit. Roses: The florist’s cost is $20 per dozen, the profit over cost is $20 per dozen. Carnations: The florist’s cost is $5 per dozen, the profit over cost is $8 per dozen. The florist can order no more than 60 dozen flowers. Based on previous years, a minimum of 20 dozen carnations must be ordered. The florist cannot order more than $450 worth of roses and carnations. Find out how many dozen of each the florist should order to max. profit!
Cost Total ordered Profit
x y P=20x + 8y
![Page 63: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/63.jpg)
Sample of what must be handed in for Partner
problem
![Page 64: Chapter 4 Systems of Equations and Problem Solving How are systems of equations solved?](https://reader035.vdocument.in/reader035/viewer/2022081422/551c4be65503467b488b5029/html5/thumbnails/64.jpg)
4-8PARTNER PROJECT
See worksheet