Solving an Equation
description
Transcript of Solving an Equation
![Page 1: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/1.jpg)
Solving an Equation
• Finding ALL possible values that make an equation true.
• You are done solving an equation when you can look at an equation and “know” what values make it true.
• If you can’t look at an equation and “know” what values make it true, try rearranging the equation to make it easier to look at.
![Page 2: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/2.jpg)
Examples
xxxxx
28)1(4732
132
![Page 3: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/3.jpg)
Examples
x=2
x=1/3
2x=10; x=5
xxxxx
28)1(4732
132
![Page 4: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/4.jpg)
Solving an Equation
• Finding ALL possible values that make an equation true.
• You are done solving an equation when you can look at an equation and “know” what values make it true.
• If you can’t look at an equation and “know” what values make it true, try rearranging the equation to make it easier to look at.
![Page 5: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/5.jpg)
Ways to rearrange equations
• Properties of arithmetic– Associative, Commutative, and Distributive– Use these to get rid of parentheses and combine like terms– These change the shape of one side of the equation, but not its value.
![Page 6: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/6.jpg)
Doing the same thing to both sides
• Adding, Subtracting, Multiplying, Dividing, a number from both sides of the equation.– Changes the value of both sides, but not the equality.
![Page 7: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/7.jpg)
WARNING• Each side is a number. When multiplying (or dividing) multiply
(or divide) the whole number.
![Page 8: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/8.jpg)
Rearranging an equation to solve
2 42
244
2844
28)1(4
22
44
xx
xx
xxxx
xx
![Page 9: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/9.jpg)
Solve for K: 2(K-7) = 4 - 3(K+2)
A) K=12/5B) K=5/24C) K=24/5D) None of the above
![Page 10: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/10.jpg)
Solve for K: 2(K-7) = 4 - 3(K+2)
A
![Page 11: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/11.jpg)
Solve an inequality
• Method 1: Convert to an equality, solve, then test.– Slower– Works for complex problems later in the course
• Method 2: Solve the inequality like an equality– Faster– Easier to make mistakes– Does not work for complicated problems like
quadratic inequalities.
![Page 12: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/12.jpg)
Method 1• Start with inequality: 8<4-2x• Convert to an equality: 8=4-2x• Solve: -2=x• Draw a number line <------------|------------------> x• x<-2 -2 x>-2• Your possible answers are x<-2 and x>-2. Pick a number on each
side to test which is correct.• Test (x<-2) with x=-3: 8<4-2(-3) 8<10 TRUE• Test (x>-2) with x=0: 8<4-2(0) 8<4 FALSE• ANSWER: x<-2
![Page 13: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/13.jpg)
Method 2• Start with inequality:
8<4-2x• Solve by doing the same thing to both sides 8+2x<4•
2x<4-8• 2x<-4• x<-2
![Page 14: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/14.jpg)
PROBLEM
• Using method 2, if I do this problem two different ways, I get two different answers.
DON’T MATCH!
![Page 15: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/15.jpg)
Which is Correct?
• I know x<-2 is correct, because when I did method 1, I plugged in to check my work.
CORRECT INCORRECT
![Page 16: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/16.jpg)
How did it happen?
• When I turned -2x into +2x, I moved it to the other side of the <. When I divided by -2, I turned negative into positive, but didn’t move it to the other side.
Moved x
Never moved x.
![Page 17: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/17.jpg)
How to fix it
• When I divide by a negative, I can have the same effect as “moving to the other side” by “flipping the sign”
Moved x
Flipped the sign
Answers Match
![Page 18: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/18.jpg)
NEW RULE
• When multiplying or dividing an inequality by a negative number, flip the sign of the inequality
• Or, just never multiply by a negative.
![Page 19: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/19.jpg)
Solve the following inequality:
-3(x+4) > 53
A) x > 65/3 B) x > -65/3C) x < 65/3D) x < -65/3E) None of the above
![Page 20: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/20.jpg)
Solve the following inequality:
-3(x+4) > 53
D
![Page 21: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/21.jpg)
How to set up word problems
![Page 22: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/22.jpg)
How to set up a word problem(with equation given)
• A jewelry maker has total revenue for necklaces given by R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces made and sold. How many necklaces must be produced and sold in order to break even?
![Page 23: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/23.jpg)
Step 1: Define the variables• A jewelry maker has total revenue for necklaces given by
R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces made and sold. How many necklaces must be produced and sold in order to break even?
• x=number of necklaces• R(x)=number of $ earned for x necklaces• C(x)=number of $ spend to make x necklaces
![Page 24: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/24.jpg)
Step 2: Interpret the question• A jewelry maker has total revenue for necklaces given by
R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces made and sold. How many necklaces must be produced and sold in order to break even?
• x=number of necklaces• R(x)=number of $ earned for x necklaces• C(x)=number of $ spend to make x necklaces• x=?• R(x)=C(x)
![Page 25: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/25.jpg)
Step 3: Substitute• A jewelry maker has total revenue for necklaces given by
R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces made and sold. How many necklaces must be produced and sold in order to break even?
• x=number of necklaces• R(x)=number of $ earned for x necklaces• C(x)=number of $ spend to make x necklaces• x=?• R(x)=C(x)• 90.75x=24.50x+4770
![Page 26: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/26.jpg)
DONE!• A jewelry maker has total revenue for necklaces given by
R(x)=90.75x, and incurs a total cost of C(x)=24.50x+4770, where x is the number of necklaces made and sold. How many necklaces must be produced and sold in order to break even?
• x=number of necklaces• R(x)=number of $ earned for x necklaces• C(x)=number of $ spend to make x necklaces• x=?• R(x)=C(x)• 90.75x=24.50x+4770 solve for x.
![Page 27: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/27.jpg)
How to set up a word problem(without an equation)
• Suppose that in a certain business, there were 30 employees in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
![Page 28: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/28.jpg)
Step 1: Identify your variables• Suppose that in a certain business, there were 30 employees
in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
• P=number of employees• t=number of years
![Page 29: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/29.jpg)
Step 2: Identify your Points• Suppose that in a certain business, there were 30 employees
in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
• P=number of employees• t=number of years [Put independent variable first]• (t,P): (1920,30), (1940,38)
![Page 30: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/30.jpg)
Step 3: Interpret your question• Suppose that in a certain business, there were 30 employees
in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
• P=number of employees• t=number of years• (t,P): (1920,30), (1940,38)• t=?• P>40?
![Page 31: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/31.jpg)
Step 4: Choose your generic formula
• Suppose that in a certain business, there were 30 employees in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
• P=number of employees• t=number of years• (t,P): (1920,30), (1940,38)• t=?• P>40?• (y-b)=m(x-a)
![Page 32: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/32.jpg)
Step 5: Match variables and plug in numbers
• Suppose that in a certain business, there were 30 employees in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
• P=number of employees• t=number of years [Make independent variable x]• (t,P): (1920,30), (1940,38)• t=?• P>40?• (y-b)=m(x-a)• (P-30)=[(38-30)/(1940-1920)](t-1920)
![Page 33: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/33.jpg)
Step 6: Rearrange and substitute
• Suppose that in a certain business, there were 30 employees in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
• P=number of employees• t=number of years [Make independent variable x]• (t,P): (1920,30), (1940,38)• t=?• P>40?• y=m(x-a)+b• (P-30)=[8/20](t-1920)• P=[8/20](t-1920)+30>40
![Page 34: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/34.jpg)
DONE!
• Suppose that in a certain business, there were 30 employees in the year 1920, and 38 employees in the year 1940. Assuming linear growth, In what years will the business have more than 40 employees?
• P=number of employees• t=number of years [Make independent variable x]• (t,P): (1920,30), (1940,38)• t=?• P>40?• y=m(x-a)+b• (P-30)=[(38-30)/(1940-1920)](t-1920)• P=[8/20](t-1920)+30>40, solve for t.
![Page 35: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/35.jpg)
A plumbing store’s monthly profit from the sale of PVC pipe can be described by P(x)=5.25x-1150 dollars, where x is the number of feet of PVC pipe sold. Set up an inequality that would help determine what level of monthly sales is necessary to incur positive profit.
A) 5.25x >1150B) 5.25x+1150>0C) 5.25x+1150=0D) Both A & BE) All of the above
![Page 36: Solving an Equation](https://reader036.fdocuments.us/reader036/viewer/2022062411/568168cd550346895ddfbc9a/html5/thumbnails/36.jpg)
A plumbing store’s monthly profit from the sale of PVC pipe can be described by P(x)=5.25x-1150 dollars, where x is the number of feet of PVC pipe sold. Set up an inequality that would help determine what level of monthly sales is necessary to incur positive profit.
Define VariablesP(x)=number of dollars of profitx=number of feet of pipe sold
Interpret Questionx=?P(x)>0
Substitute5.25x-1150>0
A) 5.25x>1150