1. Do Now 1/18/12
In your notebook, explain how you know if two equations contain one solution, no solutions, or infinitely many solutions.
Provide an example of each

2. Infinitely many
solutions
No solution
One solution

3. Objective
practice identifying the number of solutions of a linear system

4. Section 7.5 “Solve Special Types of Linear Systems”
consists of two or more linear equations in the same variables.
Types of solutions:
(1) a single point of intersection – intersecting lines
(2) no solution – parallel lines
(3) infinitely many solutions – when two equations represent the same line

5. + “Solve Linear Systems by Elimination” Multiplying First!!”
Eliminated
x (2)
4x + 5y = 35
8x + 10y = 70
Equation 1 +
x (-5)
15x - 10y = 45
-3x + 2y = -9
Equation 2
23x = 115
“Consistent Independent
System”
x = 5
4x + 5y = 35
Equation 1
Substitute value for
x into either of the original equations
4(5) + 5y = 35
20 + 5y = 35
y = 3
4(5) + 5(3) = 35
35 = 35
The solution is the point (5,3).
Substitute (5,3) into both equations to check.
-3(5) + 2(3) = -9
-9 = -9

6. “Solve Linear Systems with No Solution”
Eliminated
Eliminated
3x + 2y = 10
Equation 1 _ +
-3x + (-2y) = -2
3x + 2y = 2
Equation 2
This is a false statement,
therefore the system has no
solution.
0 = 8
“Inconsistent
System”
No Solution
By looking at the graph, the lines
are PARALLEL and therefore will
never intersect.

7. “Solve Linear Systems with Infinitely Many Solutions”
Equation 1
x – 2y = -4
Equation 2
y = ½x + 2
Use ‘Substitution’ because we know what y is equals.
Equation 1
x – 2y = -4
x – 2(½x + 2) = -4
x – x – 4 = -4
This is a true statement,
therefore the system has infinitely many solutions.
-4 = -4
“Consistent
Dependent
System”
Infinitely Many Solutions
By looking at the graph, the lines are the SAME and therefore intersect at every point, INFINITELY!

8. How Do You Determine the Number of Solutions of a Linear System?
First rewrite the equations in slope-intercept form.
Then compare the slope and y-intercepts.
y -intercept
slope
y = mx + b
Number of Solutions
Slopes and y-intercepts
One solution
Different slopes
No solution
Same slope
Different y-intercepts
Infinitely many solutions
Same y-intercept