Transcript Solve Systems by Graphing

Warm Up
Evaluate each expression for x = 1 and y = –3.
1. x – 4y
2. –2x + y
13
–5
Write each expression in slope-intercept form.
y=x+1
3. y – x = 1
4. 2x + 3y = 6
y=
x+2
5. 0 = 5y + 5x
y = –x
Lesson 6.1
Solving Systems of Equations
by Graphing
Objective
California
Standards
9.0 Students solve a system of two linear equations in two variables
algebraically and are able to interpret the answer graphically. Students are
able to solve a system of two linear inequalities in two variables and to sketch
the solution sets. Also covered:
6.0
What is a system of equations?
A system of equations is when you have
two or more equations using the same
variables.
 The solution to the system is the point
that satisfies ALL of the equations. This
point will be an ordered pair (x, y).
 When graphing, you will encounter three
possibilities.

1) Intersecting Lines
The point where the lines
intersect is your solution.
 The solution of this graph
is (1, 2)

(1,2)
What is the solution of the system
graphed below?
1.
2.
3.
4.
(2, -2)
(-2, 2)
No solution
Infinitely many solutions
2) Parallel Lines

These lines never
intersect!
 Since the lines never
cross, there is
NO SOLUTION!
 Parallel lines have the
same slope with different
y-intercepts.
2
Slope = = 2
1
y-intercept = 2
y-intercept = -1
3) Coinciding Lines

These lines are the same!
 Since the lines are on top
of each other, there are
INFINITELY MANY
SOLUTIONS!
 Coinciding lines have the
same slope and
y-intercepts.
2
Slope = = 2
1
y-intercept = -1
What is the solution of this system?
y=3x – 8
y = 3x -8
1.
2.
3.
4.
(3, 1)
(4, 4)
No solution
Infinitely many solutions
Problem-Solving Application
Sally and Jenni are reading the same book. Wren
is on page 14 and reads 2 pages every night.
Jenni is on page 6 and reads 3 pages every night.
After how many nights will they have read the
same number of pages? How many pages will
that be?

(8, 30)
Nights
Lets try one: Find the solution to the
following system:
y = -2x+4
y = x -2
m=-2
b=4
m=1 b=-2
Step 1:Graph both equations
Step 2: Find
Where the lines intersect?
The point is (2, 0)
Check your answer!
To check your answer, substitute the
point back into both equations.
Y=-2x+4
2(2) + 4 = 0
y = x-2
(2) – 2 = 0
Nice job…let’s try another!
You try: Find the solution to the
following system
y = 2x – 1
y = –x + 5
 Graph
both equations
 Where do they
intersect?
The point
is (2, 3)
Try again: Find the solution to
the following system
 Graph
both equations
 Where do they
intersect?
no solution
Challenge: Find the solution to the
following system:
y = 2x – 3
-2x + y = 1
 Graph both equations.
 Put both equations in slope-intercept
y=mx+b
y = 2x – 3
y = 2x + 1
Graph using slope and y-intercept
Graph the equations.
y = 2x – 3
m = 2 and b = -3
y = 2x + 1
m = 2 and b = 1
Where do the lines intersect?
No solution!
Notice that the slopes are the same with different
y-intercepts. If you recognize this early, you don’t
have to graph them!
Solving a system of equations by graphing.
Let's summarize! There are 3 steps to
solving a system using a graph.
Step 1: Graph both equations.
Graph using slope and y – intercept
or x- and y-intercepts. Be sure to use
a ruler and graph paper!
Step 2: Do the graphs intersect?
This is the solution! LABEL the
solution!
Step 3: Check your solution.
Substitute the x and y values into
both equations to verify the point is a
solution to both equations.
Identifying Systems of Solutions
Tell whether the ordered pair is a solution of the given
system.
x + 3y = 4
(–2, 2);
–x + y = 2
x + 3y
=4
–2 + (3)2
4
–2 + 6
4
4 4 
Substitute –2
for x and 2
for y.
–x + y
=2
–(–2) + 2
2
4 2 
The ordered pair (–2, 2) makes one equation true, but
not the other. (–2, 2) is not a solution of the system.
Identifying Systems of Solutions
Tell whether the ordered pair is a solution of the given
system.
2x + y = 5
(1, 3);
–2x + y = 1
2x + y
=5
2(1) + 3
5
2+3 5

5 5
Substitute 1
for x and 3
for y.
–2x + y
=1
–2(1) + 3
–2 + 3
1
1

1 1
The ordered pair (1, 3) makes both equations true.
(1, 3) is the solution of the system.
Lesson Quiz: Part I
Tell whether the ordered pair is a solution of the given system.
1. (–3, 1);
no
2. (2, –4);
yes
Lesson Quiz: Part II
Solve the system by graphing.
3.
y + 2x = 9
y = 4x – 3
(2, 5)
4. Joy has 5 collectable stamps and will buy 2
more each month. Ronald has 25 collectable
stamps and will sell 3 each month. After how
many months will they have the same number
of stamps? How many will that be?
4 months
13 stamps