1. 5.1 Solving Systems of Linear Equations by Graphing

2. System of Linear Equations
two linear eqns. considered at the same time
Ex. x + y = 5
x – y = 1
solutions to systems of eqns. are all ordered pairs that are solns. to BOTH eqns. (both eqns. give a true stmt. when ordered pair is sub. in)

3. Ex. For the system: x + y = 5 x – y = 1
is (3, 2) a soln?
x + y = 5
3 + 2 = 5
5 = 5
true
x – y = 1
3 – 2 = 1
1 = 1
Since (3, 2) satisfies BOTH eqns, YES, it is a soln to the system
(b) is (-1, 6) a soln?
x + y = 5
= 5
5 = 5
true
x – y = 1
-1 – 6 = 1
-7 = 1
false
Since (-1, 6) DOES NOT satisfy BOTH eqns, NO, it is NOT a soln to the system

4. Solving by Graphing Graph first eqn.
Graph second eqn. on same set of axes
Look for a point of intersection
The point of intersection is the soln.
If there is no point of intersection no solution
If lines intersect everywhereinfinitely many solns.
Check the soln. in BOTH eqns., if necessary

5. Ex. Solve the system by graphing: y = -2x + 1 x = -1
Graph y = -2x + 1
y-int: 1, m = -2/1
rise = -2, run = 1
Graph x = -1 (vert. line crossing x-axis at -1)
Point of intersection is soln. (-1, 3)
Check (-1, 3) in both eqns.
y = -2x + 1 x = -1
3 = -2(-1) = -1
3 = true
3 = 3 true y 3 2 1 x -3 -2 -1 1 2 3 -1 -2 -3

6. Worksheet
Notes:
Inconsistent system: a system with no soln. (#2 on worksheet)
Dependent eqns: eqns. that produce the same line (#3 on worksheet)

7. Summary One point of intersection solution: {(x, y)}
Lines (distinct lines)
slope  different
No point of intersection
inconsistent system
No solution
empty set Ø
Lines (parallel)
slope  same
y-int  different
Lines intersect everywhere
dependent eqns.
Infinite number of solutions
solution: {(x, y)|eqn.}
Lines (coincide - same line)
slope  same
y-int  same