A system of equations is a set of two or more equations containing
two or more variables. A linear system is a system of equations
containing only linear equations.
Recall that a line is an infinite set of points that are solutions to a
linear equation. The solution of a system of equations is the set of all
points that satisfy each equation.
On the graph of the system of two equations, the solution is the set of points
where the lines intersect. A point is a solution to a system of equation if the xand y-values of the point satisfy both equations.
Use substitution to determine if the given ordered pair is an
element of the solution set for the system of equations.
Ex 1:
(1, 3);
x – 3y = –8
3x + 2y = 9
3x + 2y = 9
x – 3y = –8
(1) –3(3) –8
–8
–8 
Substitute 1 for x and 3
3(1) +2(3)
for y in each equation.
9
9
9
Because the
point is a
solution for both
equations, it is a
solution of the
system.
Ex 2:
Use a graph and a table to solve the system. Check
your answer.
2x – 3y = 3
y+2=x
y=
Solve each equation for y.
On the graph, the lines appear
to intersect at the ordered pair
(3, 1)
x–1
y= x – 2
Make a table of values
for each equation.
Notice that when x = 3,
the y-value for both
equations is 1.
The solution to the
system is (3, 1).
y=
x
0
x–1
x
y
0
–2
1
1
–1
2
2
0
3
1
3
y
–1
y= x – 2
1
The systems of equations in Example 2 have exactly one solution.
However, linear systems may also have infinitely many or no
solutions. A consistent system is a set of equations or inequalities
that has at least one solution, and an inconsistent system will have
no solutions.
You can classify linear systems by comparing the slopes and y-intercepts of the
equations. An independent system has equations with different slopes. A
dependent system has equations with equal slopes and equal y-intercepts.
Ex 3: Classify the system and determine the number
of solutions.
x = 2y + 6
3x – 6y = 18
Solve each equation for y.
y=
x–3
y=
x–3
The equations have
the same slope and
y-intercept and are
graphed as the same
line.
The system is consistent and dependent with infinitely
many solutions.
Ex 4: City Park Golf Course charges $20 to rent golf clubs plus
$55 per hour for golf cart rental. Sea Vista Golf Course
charges $35 to rent clubs plus $45 per hour to rent a
cart. For what number of hours is the cost of renting
clubs and a cart the same for each course?
Step 1 Write an equation for the cost of renting clubs and a cart at
each golf course.
City Park Golf Course: y = 55x + 20
Sea Vista Golf Course: y = 45x + 35
Because the slopes are different, the system is independent and
has exactly one solution.
Step 2 Solve the system by using a table of values.
Use increments of
represent 30 min.
When x =
, the yvalues are both
102.5. The cost of
renting clubs and
renting a cart for
hours is $102.50 at
either company. So
the cost is the same
at each golf course
for hours.
to
y = 55x + 20
x
0
y
20
y = 45x + 35
x
0
57.5
47.5
1
75
1
120
80
102.5
102.5
2
y
35
2
125