Download Solving Systems of Equations by Graphing

Unit C
Solving Systems of Equations by
Graphing
ACT WARM-UP
• There are 30 antique cars in a parade. Six
of the cars are red, 14 are black, 5 are
blue, and 5 are white. If a circle graph is
used to represent this information, what
percent of the graph would accurately
represent the number of red cars?
• A) 5
B) 6 C) 16.7 D) 20 E) 24
• 6 out of 30 are red. This is 1/5 of the total,
therefore 1/5 = .2 or 20%. The answer is
D) 20.
Objectives
• Solve systems of linear equations by
graphing.
• Classify systems of equations, and
determine the number of solutions.
Essential Question:
How do you solve a system
of equations by graphing?
A system of equations is two or more equations
containing two or more variables. A linear system
is a system of equations containing only linear
equations. We will examine other systems later.
The solution of a system of equations is the set of
all points that satisfy each equation. Use
substitution to determine if an ordered pair is an
element of the solution set for a system of
equations.
On the graph of the system of two linear equations,
the solution is the set of points where the lines
intersect on the same coordinate plane.
Solve the system of equations by graphing.
Write each equation in slopeintercept form or standard form
using x- and y- intercepts and/or
slope and then graph.
The graphs appear to intersect
at (4, 2).
Check Substitute the coordinates into each equation.
Original equations
Replace x with 4
and y with 2.
Simplify.
Answer: The solution of the system is (4, 2).
Solve the system of equations by graphing.
Answer: (4, 1)
Systems of equations are used in
businesses to determine the break-even
point. The break-even point is the point at
which the income equals the cost. If a
business is operating at the break-even
point, it is neither making nor losing money.
Fund-raising A service club is selling copies of their
holiday cookbook to raise funds for a project. The
printer’s set-up charge is $200, and each book costs
$2 to print. The cookbooks will sell for $6 each. How
many cookbooks must the members sell before they
make a profit?
Let
Cost of books
is
cost per book
plus
set-up charge.
y
=
2x
+
200
Income from
books
y
Answer:
is
price per
book
=
6
The graphs intersect
at (50, 300). This is
the break-even point.
If the group sells less
than 50 books, they
will lose money. If the
group sells more than
50 books, they will
make a profit.
times
number of
books.
x
Classify Systems of Equations
Graphs of systems of linear equations may
be intersecting lines, parallel lines, or the
same line. A system of equations is
consistent if it has at least one solution and
inconsistent if it has no solutions. A
consistent system is independent if it has
exactly one solution or dependent if it has
an infinite number of solutions.
Systems of Equations
A system is consistent if it has one or more solutions.
Systems of Equations
A system is inconsistent if it has no solutions.
Systems of Equations
A consistent system is independent if it has exactly one solution.
Systems of Equations
A consistent system is dependent if it has an infinite number
of solutions. Two equations that represent the same line are said
to coincide.
Systems of Equations
Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
Since the equations are equivalent, their graphs
are the same line.
Any ordered pair representing a point on that line will satisfy both equations.
So, there are infinitely many solutions. This system is consistent and
dependent.
Graph the system of equations and describe it
as consistent and independent, consistent and
dependent, or inconsistent.
Answer:
inconsistent
Essential Question:
How do you solve a system
of equations by graphing?
Graph both equations on the same
coordinate plane. The point(s) of
intersection are the solutions of the system.
Math Fact
• The term pencil can be used to describe
the set of all lines that pass through a
given point. A pencil may be composed of
many consistent, independent linear
systems.