1. 3.1 Systems of Linear Equations
Using graphs and tables to solve systems
Using substitution and elimination to solve systems
Using systems to model data
Value, interests, and mixture problems
Using linear inequalities in one variable to make predictions

2. Using Two Models to Make a Prediction
When will the life expectancy of men and women be equal?
L = W(t) = 0.114t
L = M(t) = 0.204t
Equal at approximately 87 years old in 2064.
100
Years of Life 80
(84.11, 87.06) 60 20 40 60 80
100
120
Years since 1980

3. System of Linear Equations in Two Variables (Linear System)
Two or more linear equations containing two variables
y = 3x + 3
y = -x – 5

4. Solution of a System
An ordered pair (a,b) is a solution of a linear system if it satisfies both equations.
The solution sets of a system is the set of all solutions for that system.
To solve a system is to find its solution set.
The solution set can be found by finding the intersection of the graphs of the two equations.

5. Find the Ordered Pairs that Satisfy Both Equations
Graph both equations on the same coordinate plane
y = 3x + 3
y = -x – 5
Verify
(-3) = 3(-2) + 3
-3 =
-3 = -3
(-3) = -(-2) – 5
-3 = 2 – 5
Only one point satisfies
both equations
(-2,-3) is the solution set of the system
Solutions for
y = 3x + 3
Solutions for
y = -x – 5
Solution for both (-2,-3)

6. Example ¾x + ⅜y = ⅞ y = 3x – 5 Solve first equation for y
6x -6x + 3y = 7 – 6x
3y = -6x + 7
y = -2x + 7/3

7. y = 3x – 5 -.6 = 3(1.45) – 5 -.6 = 4.35 – 5 -.6 ≈ -.65 y = -2x + 7/3
-.6 = -2(1.45) + 7/3
-.6 = /3
-.6 ≈ -.57
(1.45,-.6)

8. Inconsistent System A linear system whose solution set is empty
Example…Parallel lines never intersect
no ordered pairs satisfy both systems

9. Dependent System
A linear system that has an infinite number of solutions
Example….Two equations of the same line
All solutions satisfy both lines
y = 2x – 2
-2x + y = -2
-2x +2x + y = -2 +2x

10. One Solution System
There is exactly one ordered pair that satisfies the linear system
Example…Two lines
that intersect in only
one point

11. Solving Systems with Tablesx 1 2 3 4
y = 4x – 6 -6 -2 6 10
y = -6x + 14 14 8 -4
-10
Since (2,2) is a solution to both equations, it is a solution of the linear system.

12. 3.2 Using Substitution
Isolate a variable on one side of either equation
Substitute the expression for the variable into the other equation
Solve the second equation
Substitute the solution into one of the equations

13. Example 1 y = x – 1 3x + 2y = 13 3x + 2(x – 1) = 13 3x + 2x – 2 = 13
x = 3
y = 3 – 1
y = 2
Solution set for the linear system is (3,2).

14. Example 2 2x – 6y = 4 3x – 7y = 8 2x – 6y +6y = 4 +6y 2x = 6y + 4
x = 3y + 2
3(3y + 2) – 7y = 8
9y + 6 – 7y = 8
2y = 8 -6
2y = 2
y = 1
x = 3(1) + 2
x = 5
The solution set is (5, 1).

15. Using Elimination Adding left and right sides of equations
If a=b and c=d, then a + c = b + d
Substitute a for b and c for d
a + c = a + c
both sides are the same

16. Using Elimination
Multiply both equations by a number so that the coefficients of one variable are equal in absolute value and opposite sign.
Add the left and right sides of the equations to eliminate a variable.
Solve the equation.
Substitute the solution into one of the equations and solve.

17. Example 1 5x – 6y = 9 + 2x + 6y = 12 7x + 0 = 21 7x = 21 7 7 x = 3
x = 3
2(3) + 6y = 12
y = 12 -6
6y = 12
y = 2
Solution set for the system is (3,2)

18. Example 2 3x + 7y = 29 6x – 12y = 32 -2(3x + 7y) = -2(29)
y = 1
3x + 7(1) = 29
3x = 29 -7
3x = 22
x = 22/3
Solution (22/3, 1)

19. Example 3 6x + 15y = 42 + 35x – 15y = 40 2x + 5y = 14 41x + 0 = 82
x = 2
2(2) + 5y = 14
y = 14 -4
5y = 10
y = 2
2x + 5y = 14
7x – 3y = 8
3(2x + 5y) = 3(14)
6x + 15y = 42
5(7x – 3y) = 5(8)
35x – 15y = 40
Solution (2,2)

20. Using Elimination with Fractions
2x – 5y
3x – 2y
6x – 5y = -3
9( ) = ( )9
-2(6x – 5y) = (-3)-2
-12x + 10y = 6
15( ) = ( )15
3x – 10y = 21
+___________
-9x = 27
x = -3
6(-3) – 5y = -3
– 5y =
-5y = 15
y = -3

21. Inconsistent Systems
If the result of substitution or elimination of a linear system is a false statement, then the system is inconsistent.
y = 3x + 3
-1(y) = -1(3x + 3)
-y = -3x – 3
+ y = 3x – 2
0 = 0 – 2
0 ≠ -2 False
y = 3x + 3
y = 3x – 2
3x + 3 = 3x – 2
3x -3x + 3 = 3x -3x – 2
3 ≠ - 2 False

22. Dependent Systems
If the result of applying substitution or elimination to a linear system is a true statement, then the system is dependent.
y -3x = 3x -3x – 4
-3(-3x + y) = (-4)-3
9x + -3y = 12
+ -9x + 3y = -12
0 = 0
True
y = 3x – 4
-9x + 3y = -12
-9x + 3(3x – 4) = -12
-9x + 9x -12 = -12
-12 = -12
True

23. Solving systems in one variable
To solve an equation A = B, in one variable, x, where A and B are expressions,
Solve, graph or use a table of the system
y = A
y = B
Where the x-coordinates of the solutions of the system are the solutions of the equation A = B

24. Systems in one variable
y = 4x – 3
y = -x + 2
4x – 3 = -x + 2
4x +x – 3 = -x +x + 2
5x – 3 +3 = 2 +3
5x = 5
x = 1
(1,1)

25. Graphing to solve equations in one variable
-2x + 6 = 5/4x – 3
(3,1)
-2x + 6 = -4
(6,-4)
(3,1)
(6,-4)

26. Using Tables to Solve Equations in One Variable
-2x + 7 = 4x – 5
Solution (2,3) x y -2 11
-13 -1 9 -9 7 -5 1 5 2 3 4

27. 3.3 Systems to Model Data
Predict when the life expectance of men and women will be the same.
L = W(t) = .114t
L = M(t) = .204t
.114t -.114t = .204t t
= .09t
7.57 = .09t
t ≈ 84.11

28. Solving to Make Predictions
In 1950, there were 4 women nursing students at a private college and 84 men. If the number of women nursing students increases by 13 a year and the number of male nursing students by 6 a year. What year will the number of male and female students be the same?
A = W(t) = 13t + 4
A = M(t) = 6t + 84

29. Using substitution 13t + 4 = 6t + 84 13t + 4 -4 = 6t + 84 -4
t ≈ 11.43
Equal in 1961~1962