Transcript Holt Algebra 2 3-1

3-1
Using Graphs and Tables
to Solve Linear Systems
Objectives
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 x- and y-values of the point satisfy both equations.
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
Use substitution to determine if the given ordered
pair is an element of the solution set for the system
of equations.
(1, 3);
x – 3y = –8
3x + 2y = 9
x – 3y = –8
3x + 2y = 9
(1) –3(3) –8
3(1) +2(3) 9
–8
–8 
Substitute 1 for x and 3
for y in each equation.
9
9
Because the point is a solution for both equations, it
is a solution of the system.
Holt Algebra 2
Using Graphs and Tables
to Solve Linear Systems
3-1
Use substitution to determine if the given ordered
pair is an element of the solution set for the system
of equations.
(–4,
);
x + 6 = 4y
2x + 8y = 1
x + 6 = 4y
(–4) + 6
2
2
2x + 8y = 1
Substitute –4 for x and
for y in each equation.
2(–4) +
1
–4
1x
Because the point is not a solution for both equations, it
is not a solution of the system.
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
Use a graph and a table to solve the system. Check
your answer.
2x – 3y = 3
y+2=x
On the graph, the lines
appear to intersect at
the ordered pair (3, 1)
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
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).
Holt Algebra 2
x
0
x
y
0
–2
1
1
–1
2
2
0
3
1
3
y
–1
1
3-1
Using Graphs and Tables
to Solve Linear Systems
Use a graph and a table to solve the system.
Check your answer.
x–y=2
2y – 3x = –1
Solve each equation for y.
Holt Algebra 2
y=x–2
y=
Using Graphs and Tables
to Solve Linear Systems
3-1
Use your graphing calculator to graph
the equations and make a table of
values. The lines appear to intersect
at (–3, –5). This is the confirmed by
the tables of values.
The solution to the system is (–3, –5).
Check Substitute (–3, –5) in the original
equations to verify the solution.
x–y = 2
(–3) – (–5)
2
Holt Algebra 2
2y – 3x = –1
2
2
2(–5) – 3(–3) –1

–1
–1 
3-1
Using Graphs and Tables
to Solve Linear Systems
Use a graph and a table to solve the system. Check your answer.
2y + 6 = x
4x = 3 + y
On the graph, the lines
appear to intersect at the
ordered pair (0, –3)
Check your answer
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
Use a graph and a table to solve the system. Check your answer.
x+y=8
2x – y = 4
On the graph, the lines
appear to intersect at the
ordered pair (4, 4).
Check your answer
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
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.
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
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.
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
Classify the system and determine the number of solutions.
4x + y = 1
y + 1 = –4x
y = –4x + 1
Solve each equation for y.
y = –4x – 1
The equations have
the same slope but
different y-intercepts
and are graphed as
parallel lines.
The system is inconsistent and has no solution.
Holt Algebra 2
3-1
Using Graphs and Tables
to Solve Linear Systems
Kayak Kottage charges $26 to rent a kayak plus $24 per
hour for lessons. Power Paddles charges $12 for rental
plus $32 per hour for lessons. For what number of hours
is the cost of equipment and lessons the same for each
company?
Holt Algebra 2