1. Solving Systems of Equations Algebraically

2. ACT WARM-UP
Which statement best describes the relationship between the graphs of y = 2 and x = 2?
A) The two lines have the same slope.
B) The lines are perpendicular.
C) The lines are parallel.
D) The lines intersect at (2, 0).
E) None of the above.
y = 2 is a horizontal line and x = 2 is a vertical line. Therefore, the answer is B) perpendicular.

3. Objectives Solve systems of equations by substitution.
Solve systems of equations by elimination.

4. How do you solve a system of linear equations by substitution?
Essential Question
How do you solve a system of linear equations by substitution?

5. Substitution Method
The substitution method is used to eliminate one of the variables by replacement when solving a system of equations.  
Think of it as "grabbing" what one variable equals from one equation and "plugging" it into the other equation.

6. Elimination method
Simultaneous equations written in standard form got you baffled?   Relax!  You can do it! 
Think of the adding or combining like terms method as simply "eliminating" one of the variables to make your life easier.

7. Solving Systems
You will need paper and pencil or a whiteboard to solve the following problems. 

8. Use substitution to solve the systems of equations.
Substitute 26 – 4y for x in the second equation and solve for y.
Second equation
Substitute 26 – 4y for x.
Subtract 26 from each side.
Divide each side by –9.

9. Now substitute the value for y in either of the original equations and solve for x.
First equation
Replace y with 4
Simplify
Subtract 16 from each side.
Answer: The solution of the system is (10, 4).

10. Answer: (5, 1)
Example 2-1b

11. Use the elimination method to solve the system of equations.
In each equation, the coefficient of x is 1. If one equation is multiplied by -1, the variable x will be eliminated when the two equations are added together.
Example 2-3a

12. Now find x by substituting 4 for y in either original equation.
Second equation
Replace y with 4.
Subtract 4 from each side.
Answer: The solution is (2, 4).
Example 2-3a

13. Use the elimination method to solve the system of equations.
Answer: (17, –4)
Example 2-3b

14. Use the elimination method to solve the system of equations.
Multiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable.
Multiply by 2.
Multiply by 3.
Example 2-4a

15. Replace x with 3 and solve for y.
First equation
Replace x with 3.
Multiply.
Subtract 6 from each side.
Divide each side by 3.
Answer: The solution is (3, 2).
Example 2-4a

16. Use the elimination method to solve the system of equations.
Answer: (–5, 4)
Example 2-4b

17. Use the elimination method to solve the system of equations.
Use multiplication to eliminate x.
Multiply by 2.
Answer: Since there are no values of x and y that will make the equation true, there are no solutions for the system of equations.
Example 2-5a

18. Use the elimination method to solve the system of equations.
Answer: There are no solutions for this system of equations.
Example 2-5b

19. Solve Systems
Solve these problems algebraically, using either the substitution or addition/
combination method.
a.       A) x + 4y =26, x- 5y = - 10
B) x –3y = 2, x + 7y = 12
C)  3x –y = 7, x + 4y = 11
D) 2x –3y = 11, 2x +2y = 6
E) x + 2y = 10, x + y = 6
F)    x +3y = 5, x +5y = -3
G) 2x + 3y = 12, 5x – 2y = 11

20. Let x = represent skis, y =represent snowboards
Skiing trip: 40 members of South Aiken High School went on a one-day ski trip. They can rent skis for $10 per day or snowboards for $12 per day. All members paid a total of $420. Write a system of equations that represents the number of members who rented the two types of equipment. How many members rented skis and how many rented snowboards?
Let x = represent skis, y =represent snowboards
# snowboards
plus
# skis is
Members y + x = 40
Example 1-2a

21. 30 members rented skis and 10 members rented snowboards.
$ Amount is
$ cost per day
for skis
plus
for Snowboard.
420 =
10x
12y +
Answer:
x + y = 40
10x + 12y = 420
30 members rented skis and 10 members rented snowboards.
Example 1-2a

22. Mr. Talbot is writing a test for his science classes
Mr. Talbot is writing a test for his science classes. The test will have true/false questions worth 2 points each and multiple-choice questions worth 4 points each for a total of 100 points. He wants to have twice as many multiple-choice questions as true/false. Write a system of equations that represents the number of each type of question. How many true/false questions and multiple-choice questions will be on the test?
Answer:
2x +4y = 100
y = 2x
(10(T/F), 20(M/C) )

23. Let x = first number, y = the second number
Three times one number added to five times another number is 54. The second number is two less than the first. Find the numbers
Let x = first number, y = the second number
3 times #
plus
5 times other # is
total 3x + 5y = 54
Example 1-2a

24. Other # is First # 2 less than y = x - 2 Answer: 3x + 5y = 54
(8,6)
Example 1-2a

25. The average of two numbers is 7
The average of two numbers is 7. Find the numbers if three times one number is one half of the other number.
Answer:
(x + y)/2 = 7
3x= 1/2y
(2, 12)
Example 1-2b

26. How do you solve a system of linear equations by substitution?
Essential Question
How do you solve a system of linear equations by substitution?
Solve one equation for one variable in terms of the other. Then, this expression is substituted for the variable in the other equation.

27. Solving Systems Decisions
Graphing – when a finding a rough solution or determining if a solution exists
Substitution – when one or both equations are given in the slope-intercept form
Elimination – when the equations are in standard form. Multiply the equations in order to get opposite coefficients so variables will cancel when you add the equations together