1. Algebra 1
Section 7.5

2. Systems of Equations
Each of the three methods has its advantages and disadvantages.
Therefore, it is best to be familiar with each of them, so you can decide which method to use.

3. Example 1
Since the second equation is easily solved for x, do that first and then use substitution.
x – 3 = -2
x = 1
x + y = -3
1 + y = -3
y = -4
(1, -4)

4. Example 2 The elimination method should work well here. 2( ) -3( )
2( )
-3( )
3x + 4y = 3
2x – 7y = 31
6x + 8y = 6
-6x + 21y = -93
Add:
29y = -87
y = -3

5. Example 2 3x + 4y = 3 2x – 7y = 31 y = -3 3x + 4(-3) = 3 3x – 12 = 3
Solution:
(5, -3)

6. Example 3
Since both equations are in slope-intercept form, graphing may be acceptable.
y = 3x + 5
y = x – 3

7. Example 3 y
y = 3x + 5
y = x – 3
Solution:
(-4, -7) x

8. Definitions
Systems of equations can be classified according to the number of their solutions.
Inconsistent systems have no solutions; no ordered pair makes both equations true.

9. Definitions
Consistent systems have one or more ordered-pair solutions.
If there is just one solution, it is called an independent system.
If there are infinitely many solutions, it is called a dependent system.

10. Graphs of Systems Inconsistent systems will have parallel lines.
Consistent independent systems have intersecting lines.
Consistent dependent systems have coinciding lines.

11. Example 4
If a system of equations is inconsistent, the variable terms can be eliminated, and the remaining equation is a false statement, such as 0 = 2.
If graphed, we see parallel lines.

12. Example 4
Both lines have the same slope, but different y-intercepts.

13. Example 5
If a system of equations is consistent dependent, the variable terms can be eliminated, and the remaining equation is a true statement, such as 6 = 6.
If graphed, we see coinciding lines.

14. Example 5
Both lines have the same slope and the same y-intercept, and if written in slope-intercept form, are identical.

15. Consistent Independent
One ordered pair solution
Graph: Intersecting lines
Slopes are unequal
A single ordered pair (x, y) results from substitution or elimination

16. Consistent Dependent
All the points on the line (infinite number of solutions)
Graph: Coinciding lines
Slopes and y-intercepts are both equal
A true statement results from substitution or elimination

17. Inconsistent No solution Graph: Parallel lines
Slopes are equal; y-intercepts are unequal
A false statement results from substitution or elimination

18. Example 6 m1 = -1 b1 = 3 m2 = -1 b2 = 4 m1 = -1 b1 = 4
Equation 1
m1 = -1
b1 = 3
Equation 2
m2 = -1
b2 = 4
Equation 3
m1 = -1
b1 = 4
Equations 2 and 3 describe the same line, so they are a consistent dependent system.

19. Example 6 m1 = -1 b1 = 3 m2 = -1 b2 = 4 m1 = -1 b1 = 4
Equation 1
m1 = -1
b1 = 3
Equation 2
m2 = -1
b2 = 4
Equation 3
m1 = -1
b1 = 4
Equations 1 and 3 also describe parallel lines, so they are an inconsistent system.
Equations 1 and 2 describe parallel lines, so they are an inconsistent system.

20. Homework: pp