Download 5.1 Solving Systems of Linear Equations by Graphing

4.1 Solving Systems of Linear
Equations by Graphing
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)
Ex. For the system:
x+y=5
x–y=1
(a) is (3, 2) a soln?
(b) is (-1, 6) a soln?
x+y=5
3+2=5
5=5
true
x+y=5
-1 + 6 = 5
5=5
true
x–y=1
3–2=1
1=1
true
x–y=1
-1 – 6 = 1
-7 = 1
false
Since (3, 2) satisfies BOTH
eqns, YES, it is a soln to
the system
Since (-1, 6) DOES NOT satisfy
BOTH eqns, NO, it is NOT a
soln to the system
Solving by Graphing
1. Graph first and second eqn. on same set of axes
2. Look for a point of intersection
a. The point of intersection is the soln.
b. If there is no point of intersection no solution
c. If lines intersect everywhereinfinitely many solns.
3. Check the soln. in BOTH eqns., if necessary
Worksheet
Notes:
• consistent system: a system with at least one
solution (#1 and 3 on worksheet)
• inconsistent system: a system with no soln.
(#2 on worksheet)
• dependent equations: eqns. that produce the
same line (#3 on worksheet)
• independent equations: equations that
produce distinct lines (#1 and 2 on worksheet)
Ex. Solve the system by graphing: y = -2x + 1
x = -1
y
3
2
1
-3
-2
-1
1
-1
-2
-3
2
3
x
1) Graph y = -2x + 1
y-int: 1, m = -2/1
rise = -2, run = 1
2) Graph x = -1 (vert. line
crossing x-axis at -1)
3) Point of intersection is
soln. (-1, 3)
4) Check (-1, 3) in both eqns.
y = -2x + 1
x = -1
3 = -2(-1) + 1 -1 = -1
3=2+1
true
3 = 3 true
Ex. Without graphing, determine the number of
solutions of the system.
3x – y = -3
y – 3x = 3
Find slope and y-intercept of each line by putting each equation
in slope intercept from.
3x – y = -3
y – 3x = 3
3x – y – 3x = -3 – 3x
y – 3x + 3x = 3 + 3x
-y = -3x – 3
y = 3x + 3
-1y = -3x – 3
m= 3, y-int: (0, 3)
-1 -1 -1
Since the slopes are the same and
y = 3x + 3
the y-intercepts are the same, the
m = 3, y-int: (0, 3)
graphs are the same. There are
infinitely many solutions.
Summary
One point of intersection
consistent system
independent eqns
Parallel Lines
inconsistent system
independent eqns
No solution
Identical Lines
consistent system
dependent eqns.
Infinitely many solutions
solution: (x, y)
no solution (empty set Ø)
solution: infinitely many
solutions
Lines (distinct lines)
slope  different
Lines (parallel)
slope  same
y-int  different
Lines (coincide - same line)
slope  same
y-int  same