All methods (Daisy Crain)

Introduction
  • The systems of equations 
  • Substitution
  • Elimination 
  • Graphing
  • Applying systems of Linear Equations
  • Algebra 1 Advanced 4th Period 
Task
  • Substitution 
  • An office supply company sells two types of notebook laptops. They charge $150 for one of the machines and $225 for the other. If the company sold 22 notebooks for a total of $3900 last month, how many of each type were sold?
  1. X=Notebooks $150(1)    Y=Notebooks$225(2)
  2. X+Y=22
  3. 150X+225Y=3900
  4. X+Y=22
  5. -Y     -Y(Subtracting)
  6. X=22-Y
  7. 150(22-Y)+225Y=3900(Then your going to multiply)
  8. 3300+150Y+225Y=3900
  9. 3300+75Y=3900
  10. 75Y=600
  11. Y=8(Y=Notebooks$225)Finale Answer
  12. X+8=22
  13. X=14

X=14(X=Notebooks$1500)Finale Answer

 

  • Substitution
  • Means replacing the variables in an algebraic expression with numerical or algebraic values.
  • Ex: Find the value of 3b+4  when b=10
  • 3b means 3 times b = 3 times 10 = 30
  • so 3b+ 4= 30 +4= 34

Solving Systems of Equations by substitution

  1. Step 1 Solve for a variable if necessary.
  2. Step 2 Substitute expression into the other equation.
  3. Step 3 Solve that equation for the remaining variable.
  4. Step 4 Substitute value into one of the original equations and solve.
  5. Step 5 Write the values as an ordered pair & check.
  6. y=3x
  7. y=x-2
  8. 3x=x-2
  9. -1x  -x
  10. 2x=-2
  11. 2     2

    x=-1
  12. y=3x
  13. y=3(-1)
  14. y=-3
  15. (-1,-3)
  16. Checking 
  17. y=3x
  18. -3=3(-1)
  19. y=x-2
  20. -3=(-1)-2 
Process
  1. Solving Systems of Equations by Elimination (Addition)
  • where you actually eliminate one of the variables by adding the two equations, it means to mark out, to take out, to make it go away.
  1. Step 1 Align like terms and equal signs, usually in standard form.
  2. Step 2 If needed, create opposite coefficients by multiplying both sides of equations by a number.
  3. Step 3 Add equations, combining like terms and eliminating one variable.
  4. Step 4 Solve remaining equation.
  5. Step 5 Substitute solution back into one of the original equations to find the other value.
  6. Y+3x= -2
  7. 2y-3x=14
  8. 3y=12
  9. 3     3
  10. y=4
  11. y+3x-2
  12. 4+3x=-2
  13. -4       -4
  14. 3x=-6
  15. 3     3
  16. x=-2

Finale Answer(-2,4)

 

Example Problem

  • A hardware store earned $956.50 from renting ladders and power tools last week. The store charged 36 days for ladders and 85 days for power tools. This week the store charged 36 days for ladders , 70 days for power tools , and earned $829. How much does the store charge per day for ladders and power tools?
  1. 36L+85p=956.50
  2. -36L=70p=829
  3. 15p=127.50
  4. 15        15(Dividing)

p=$8.50/ per day(Finale Answer)Ladders

  1. 36L+70p(8.50)=829
  2. 36L+595=829
  3.        -595  -595(Subtracting)
  4. 36L=234
  5. 36    36(Dividing)

L=$6.50/ per day(Finale Answer) Power Tools

  • Sometimes BOTH variables eliminate from the problem!
  • If true statement remains there are infinite solutions.
  • If false statement remains there are no solutions.
  • (IM) infinite solutions Example problem
  1. -2x+y=-4
  2. 2x-y=4

0=0 True

  1. No Solution Example problem
  2. 2x-3y=6
  3. -4x+6y=4
  4. 4x-6y=12
  5. 0=16
  6. False
Evaluation
  • Systems of Linear Equations
  • A collection of linear equations, where the number of equations matches the number of variables, allowing for an algebraic solution.
  • Examples of Linear Systems
  • 2 variables
  1. y=3x+1
  2. y=-2x+7
  • 3 variables
  1. y=2x+3z+10
  2. y=-4x+7z-8
  3. y=-3x+5z+12

Applying Systems of Linear Equations

  • Elimination 
  • Example problem
  1. 3x-y=10
  2. 2x-y=7
  • Solve the equation for y
  1. 3x-y=10
  2. y=-7+2x
  • Eliminate the given value of y into the equation 3x-y=10
  1. 3x-(-7+2x)=10
  • solve the equation for x
  1. x=3
  • Eliminate the given value of x into the equation y=-7+2x
  1. y=-7+2x3
  2. y=-1
  • The possible solution of the system is the ordered pair (x, y)=(3, -1)Finale Answer
  • Check if the given ordered pair is the solution of the system of equations
  • 3x3-(-10=10
  • 2x3-(-1)=7
  • 10=10
  • 7=7

Since all of the equalities are true, the ordered pair is the solution of the system.

Substitution Example problem

  • -x+2y=3
  • 4x-5y=-3
  • Solve the equation for x
  • x=-3+2y
  • 4x-5y=-3
  • Substitute the given value of x into the equation 4x-5y=-3
  • 4(-3+2y0-5y=-3
  • Solve the equation for y
  • y=3
  • Substitute the given value of y into the equation x=-3+2y
  • x=-3+2x-3
  • Solve the equation for x
  • x=3
  • The possible solution of the system is the ordered pair (x, y)=(3, 3)
  • Check if the given ordered pair is the solution of the system of equations
  • -3+2x3=3
  • 4x3-5x3=-3
  • 3=3
  • -3=-3
  • Since all of the equalities are true, the ordered pair is the solution of the system.
Conclusion

Systems of Linear Inequalities 

  • A set of 2 or more inequalities containing 2 or more variables.
  • The solutions of A a system of linear inequalities consists of all the ordered pairs that make the inequalities in the system true.
  • Tell whether the ordered pair is a solution of the given system.
  1.  (x, y)= (2, -2)
  2. Y<x-3
  3. Y>-x=1
  4. y<x-3
  5. -2<2-3
  6. -2<-1(True)
  7. Y>-x+1
  8. -2>-2+1
  9. -2>-1x(False)

NO

  1. (x, y)=(2, 5)
  2. Y>2x
  3. Y>x+2
  4. Y>2x
  5. 5>2(2) True
  6. Y>x+2

5>2+2 True 

YES

 

Credits

Drum roll..........Math Journal and of course Mrs. Taylor 

Teacher Page

Mrs. Taylor 

4th period 

Daisy Crain