The document discusses different methods for solving linear systems, including graphing, substitution, addition, subtraction, and multiplication. It explains that a linear system can have one solution (consistent independent), no solution (inconsistent), or infinitely many solutions (consistent dependent), depending on whether the lines are parallel, intersect at a single point, or are the same line. Examples are provided to illustrate determining the number of solutions based on the slopes and y-intercepts when the equations are in slope-intercept form.

1. Summary Methods for Solving Linear Systems
Method Example When to use
Graphing When you want to see the lines
that the equations
represent
Substitution y = 4 – 2x When one equation is already
4x + 3y = 8 solved for x or y
Addition 4x + 7y = 15 When the coefficients of one
6x – 7y = 5 variable are opposites
Subtraction 3x + 5y = -13 When the coefficients of one
3x + y = -5 variable are the same
Multiplication 9x + 2y = 38 When no corresponding
3x – 5y = 7 coefficients are the same or
opposites

3. • Consistent Independent System
a linear system that has
exactly one solution
• Inconsistent system
a linear system with no solution
• Consistent dependent system
a linear system with infinitely many
solutions

4. Example 1: A Linear System with no solution
Show that the linear system has no solution.
3x + 2y = 10 Equation 1
3x + 2y = 2 Equation 2
Solution
Method 1 Graphing
Graph the linear system.
Method 2 Elimination
Subtract the equations.

5. Example 2: A Linear System with infinitely many solutions
Show that the linear system has infinitely many solutions.
x – 2y = -4 Equation 1
y = ½ x + 2 Equation 2
Solution
Method 1 Graphing
Graph the linear system.
Method 2 Substitution

6. IDENTIFYING THE NUMBER OF SOLUTIONS when the
equations of the linear system are written in slope-
intercept form.
Slopes and y- Number of
intercepts solutions
Different slopes One solution
Same slope No solution
Different y-intercepts
Same slope Infinitely many
Same y-intercept solutions

7. Try on your own:
Tell whether the linear system has no solution or
infinitely many solutions.
Explain your answer.
1. 5x + 3y = 6 2. y = 2x – 4
-5x -3y = 3 -6x + 3y = -12

8. Textbook p. 408 – 410 # 3 – 10 odd
15 – 20, 26 – 28,
35 – 38 all