1. Solving Linear Systems
3-1,2
Solving Linear Systems
Warm Up (Slide #2-3)
Objective and Standards (Slide #4)
Vocab (Slide #5–8)
Lesson Presentation (Slide #9–31)
Text Questions (NONE)
Worksheets 3.1A, 3.2A (Slide #32)
Lesson Quiz (Slide #33-34)
Holt Algebra 2

2. Solving Linear Systems
3-1,2
Solving Linear Systems
Warm Up
Use substitution to determine if (1, –2) is an element of the solution set of the linear equation. no
yes
1. y = 2x + 1
2. y = 3x – 5
Write each equation in slope-intercept form.
4. 4y – 3x = 8
3. 2y + 8x = 6
y = –4x + 3

3. Solving Linear Systems
3-1,2
Solving Linear Systems
Warm Up
Determine if the given ordered pair is an element of the solution set of
2x – y = 5
3y + x = 6
2. (–1, 1) no
1. (3, 1)
yes
Solve each equation for y.
3. x + 3y = 2x + 4y – 4
y = –x + 4
4. 6x y = 3y + 2x – 1
y = 2x + 3

4. Solving Linear Systems
3-1,2
Solving Linear Systems
Objectives
1. Solve systems of linear equations with:
Graphs and tables
Substitution
Elimination
2. Determine whether there will be one, none, or an infinite number of solutions by noting characteristics of each equation.

5. Solving Linear Systems
3-1,2
Solving Linear Systems
Vocabulary
system of equations
linear system
substitution
elimination
linear combinations

6. Solving Linear Systems
3-1,2
Solving Linear Systems
A system of equations is a set of two or more equations containing two or more variables. A linear system is a system of equations containing only linear equations.
A line is an infinite set of points that are solutions to a linear equation. The solution of a system of equations is the set of all points that satisfy each equation.

7. Solving Linear Systems
3-1,2
Solving Linear Systems
There are two aspects of substitution:
In one, a possible solution (ordered pair) is given and you simply substitute its x and y values into each equation to see if that point satisfies both.
In the other, you substitute the equivalent expression for a variable from one equation into the other equation, solve for one variable, then use that value to solve for the other variable.
(I know…it sounds all so confusing, but it’s really easy.)

8. Solving Linear Systems
3-1,2
Solving Linear Systems
You can also solve systems of equations with the elimination method.
With elimination, you get rid of one of the variables by adding or subtracting equations.
You may have to multiply one or both equations by a number to create variable terms that can be eliminated.
The elimination method is sometimes called the addition method or linear combinations.
Reading Math

9. Solving Linear Systems
3-1,2
Solving Linear Systems
Points to remember about linear equations and systems:
On the graph of the system of two equations, the solution is the set of points where the lines intersect.
A point is a solution to a system of equation if the x- and y-values of the point satisfy both equations.

10. Example 1A: Verifying Solutions of Linear Systems
3-1,2
Solving Linear Systems
Example 1A: Verifying Solutions of Linear Systems
To see if a given point is a solution to a linear system, substitute the (x,y) values into both equations.
For example… “Is (1,3) the solution to this linear system?”
(1, 3);
x – 3y = –8
3x + 2y = 9
Ans: YES

11. Solving Linear Systems
3-1,2
Solving Linear Systems
Check It Out! Example 1b
Is (5,3) an element of the solution set for the system of equations?
(5, 3);
6x – 7y = 1
3x + 7y = 5
Ans: NO

12. Solving Linear Systems
3-1,2
Solving Linear Systems
Example 2A: Solving Linear Systems by Using Graphs and Tables on your graphing calculator
Solve the system. Check your answer.
First, solve each equation for y, then graph both:
2x – 3y = 3
y + 2 = x
On the graph, the lines appear to intersect at the ordered pair (3, 1)
Use the calculator’s
“Table” or “Trace”
function to verify.
Sol’n: (3,1)

13. Solving Linear Systems
3-1,2
Solving Linear Systems
Check It Out! Example 2b
Use a graph and a table to solve the system. Check your answer.
First, solve each equation for y.
x + y = 8
2x – y = 4
Sol’n: (4, 4)

14. Solving Linear Systems
3-1,2
Solving Linear Systems
Different Slopes
will have
ONE solution
Same Slopes
and Same Y-int.
will have
INFINITE Sol’ns.
Same Slopes
but Different Y-int.
will have
NO Solutions.

15. Solving Linear Systems
3-1,2
Solving Linear Systems
An identity, such as 0 = 0, 8 = 8, -7 = -7, etc…
is always true and indicates infinite solutions.
A contradiction, such as 1 = 3, 5 = 9, -8 = 8, etc…
is never true and indicates no solution.
Remember!

16. Example 4: Summer Sports Application
3-1,2
Solving Linear Systems
Example 4: Summer Sports Application
One golf course charges $20 to rent golf clubs plus $55 per hour for golf cart rental.
A different course charges $35 to rent clubs plus $45 per hour to rent a cart.
Both places allow rentals in ½hr. increments.
Q: For what number of hours is the cost of renting clubs and a cart the same for each course?

17. Solving Linear Systems
3-1,2
Solving Linear Systems
Example 4 Continued
Step 1 Write an equation for the cost of renting clubs
and a cart at each golf course.
Let x represent the number of hours and y represent
the total cost in dollars.
City Park Golf Course: y = 55x + 20
Sea Vista Golf Course: y = 45x + 35
Sol’n: 1.5hrs

18. Solving Linear Systems
3-1,2
Solving Linear Systems
Check It Out! Example 4
Ravi is comparing the costs of long distance calling cards. To use card A, it costs $0.50 to connect and then $0.05 per minute. To use card B, it costs $0.20 to connect and then $0.08 per minute. For what number of minutes does it cost the same amount to use each card for a single call?
Step 1 Write an equation for the cost for each of the different long distance calling cards.
Let x represent the number of minutes and y represent
the total cost in dollars.
Card A: y = 0.05x
Card B: y = 0.08x

19. Solving Linear Systems
3-1,2
Solving Linear Systems
Check It Out! Example 4
Card A: y = 0.05x
Card B: y = 0.08x
Step 2 Here’s a situation where, since both are equal to “y”, you can set the two equations equal to each other.
.05x = .08x + .20
-.05x x
.30 = .03x
÷.03 ÷.03
10 = x
So, both plans are the same for a 10minute call

20. Example 1A: Solving Linear Systems by Substitution
3-1,2
Solving Linear Systems
Example 1A: Solving Linear Systems by Substitution
Use variable substitution to solve the system:
y = x – 1
x + y = 7
Step 1: Substitute the equivalent expression for “y” from the first equation in place of “y” in the second equation and solve for “x”.
x + y = 7
x + (x – 1) = 7
2x – 1 = 7
2x = 8
x = 4
Then…

21. Solving Linear Systems
3-1,2
Solving Linear Systems
Example 1A Continued
Step 2: Substitute the x-value into one of the original equations to solve for y.
y = x – 1
y = (4) – 1
y = 3
The solution is the ordered pair (4, 3).

22. Solving Linear Systems
3-1,2
Solving Linear Systems
Example 1A Continued
Check A graph or table supports your answer.

23. Example 1B: Solving Linear Systems by Substitution
3-1,2
Solving Linear Systems
Example 1B: Solving Linear Systems by Substitution
Use substitution to solve the systems of equations.
2y + x = 4
3x – 4y = 7
y = 2x – 1
3x + 2y = 26
5x + 6y = –9
2x – 2 = –y
Sol’n: (3, 1/2)
Sol’n: (4,7)
Sol’n: (3,-4)

24. Check It Out! Example 1a Continued
3-1,2
Solving Linear Systems
Check It Out! Example 1a Continued
Check A graph or table supports your answer.

25. Example 2A: Solving Linear Systems by Elimination
3-1,2
Solving Linear Systems
Example 2A: Solving Linear Systems by Elimination
3x + 2y = 4
4x – 2y = –18
3x + 5y = –16
2x + 3y = –9
4x + 7y = –25
–12x –7y = 19
Sol’n: (–2, 5)
Sol’n: (3,-5)
Sol’n: (.75, -4)

26. Solving Linear Systems
3-1,2
Solving Linear Systems
Check It Out! Example 2b
Use elimination to solve the system of equations.
5x – 3y = 42
8x + 5y = 28
Sol’n: (6,-4)

27. Example 3: Solving Systems with Infinitely Many or No Solutions
3-1,2
Solving Linear Systems
Example 3: Solving Systems with Infinitely Many or No Solutions
3x + y = 1
56x + 8y = –32
6x + 3y = –12
2y + 6x = –18
7x + y = –4
2x + y = –6
Sol’n: NONE
Sol’n: INFINITE
Sol’n: NONE

28. Example 4: Zoology Application
3-1,2
Solving Linear Systems
Example 4: Zoology Application
A veterinarian needs 60 pounds of dog food that is 15% protein. He will combine a beef mix that is 18% protein with a bacon mix that is 9% protein. How many pounds of each does he need to make the 15% protein mixture?
Let x present the amount of beef mix in the mixture.
Let y present the amount of bacon mix in the mixture.
Write one equation based on the amount of dog food
Write another equation based on the amount of protein
THEN…

29. Solving Linear Systems
3-1,2
Solving Linear Systems
Example 4 Continued
x + y = 60
0.18x +0.09y = 9
Solve the system.
Sol’n: (40, 20)

30. Solving Linear Systems
3-1,2
Solving Linear Systems
Check It Out! Example 4
A coffee blend contains Sumatra beans which cost $5/lb, and Kona beans, which cost $13/lb. If the blend costs $10/lb, how much of each type of coffee is in 50 lb of the blend?
Let x represent the amount of the Sumatra beans in the blend.
Let y represent the amount of the Kona beans in the blend.
Write one equation based on the amount of each bean
Write another equation based on cost of the beans:
THEN…

31. Check It Out! Example 4 Continued
Solving Linear Systems
3-1,2
Check It Out! Example 4 Continued
x + y = 50
5x + 13y = 500
Solve the system.
Sol’n: (18.75, 31.25)

32. Solving Linear Systems
3-1,2

33. Solving Linear Systems
3-1,2
Solving Linear Systems
Lesson Quiz: Part I
Use substitution to determine if the given ordered pair is an element of the solution set of the system of equations.
x + 3y = –9
x + y = 2
2. (–3, –2)
1. (4, –2)
y – 2x = 4
y + 2x = 5
Solve the system using a table and graph. Check your answer.
x + y = 1 3.
3x –2y = 8

34. Solving Linear Systems
3-1,2
Solving Linear Systems
Lesson Quiz: Part II
Which system has NO solution and which has INFINITE solutions.
–4x = 2y – 10
y + 2x = –10 4. 5.
y + 2x = –10
y + 2x = –10
6. Kayak Kottage charges $26 to rent a kayak plus $24 per hour for lessons. Power Paddles charges $12 for rental plus $32 per hour for lessons. Both places allow rentals in 15min (1/4hr) For what number of hours is the cost of equipment and lessons the same for each company?