Prerequisite Skills Review
1.) Simplify: 8r + (-64r)
2.) Solve: 3x + 7(x – 1) = 23
3.) Decide whether the ordered pair (3, -7)
is a solution of the equation 5x + y = 8
Section 7.1
Solving Systems by Graphing
What is a system????
Working with 2 equations at one time:
Example:
2x – 3y = 6
X + 5y = -12
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.
 When graphing, you will encounter three
possibilities.

Intersecting Lines
The point where the lines
intersect is your solution.
 The solution of this graph
is (1, 2)

(1,2)
IDENTIFYING THE NUMBER OF SOLUTIONS
NUMBER OF SOLUTIONS OF A LINEAR SYSTEM
y
y
y
x
x
x
Lines intersect
Lines are parallel
Lines coincide
one solution
no solution
infinitely many solutions
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
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 the system
graphed below?
1.
2.
3.
4.
(2, -2)
(-2, 2)
No solution
Infinitely many solutions
Name the Solution
Name the Solution
Name the Solution
How to Use Graphs to Solve Linear
Systems
y
Consider the following system:
x – y = –1
x + 2y = 5
We must ALWAYS verify that
your coordinates actually satisfy
both equations.
(1 , 2)
To do this, we substitute the
coordinate (1 , 2) into both
equations.
x – y = –1
(1) – (2) = –1 
x + 2y = 5
(1) + 2(2) =
1+4=5
Since (1 , 2) makes both
equations true, then (1 , 2) is the
solution to the system of linear
equations.
x
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.
1) Find the solution to the following
system:
2x + y = 4
x-y=2
Graph both equations. I will graph using
x- and y-intercepts (plug in zeros).
2x + y = 4
(0, 4) and (2, 0)
x–y=2
(0, -2) and (2, 0)
Graph the ordered pairs.
Graph the equations.
2x + y = 4
(0, 4) and (2, 0)
x-y=2
(0, -2) and (2, 0)
Where do the lines intersect?
(2, 0)
Check your answer!
To check your answer, plug
the point back into both
equations.
2x + y = 4
2(2) + (0) = 4
x-y=2
(2) – (0) = 2
Nice job…let’s try another!
2) Find the solution to the following
system:
y = 2x – 3
-2x + y = 1
Graph both equations. Put both equations
in slope-intercept or standard form. I’ll do
slope-intercept form on this one!
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!
y = 2x + 0 & y = -1x + 3
Slope = -1/1
y-intercept=
0
y-intercept= +3
Slope = 2/1
Up 2
Down
and
(1,2)
1 and
right
The solution is the point they cross at (1,2)
right11
y = x - 3 & y = -3x + 1
Slope = -3/1
y-intercept= 3
y-intercept= +1
Slope = 1/1
The solution is the point they cross at (1,-2)
y =-2x + 4 & y = 2x + 0
Slope = 2/1
y-intercept=
4
y-intercept= 0
Slope = -2/1
The solution is the point they cross at (1,2)
Graphing to Solve a Linear System
Solve the following system by
graphing:
3x + 6y = 15
y
–2x + 3y = –3
Using the slope intercept form of
these equations, we can graph
them carefully on graph paper.
y = - 12 x +
y = 23 x - 1
(3 , 1)
x
5
2
Label the
Start at the y - intercept, then use the
solution!
slope.
Lastly, we need to verify our solution is correct, by substituting (3 , 1).
Since3(3)+ 6(1)= 15
and
- 2(3)+ 3(1)= - 3
, then our solution is correct!
Practice – Solving by Graphing



y – x = 1  (0,1) and (-1,0)
y + x = 3  (0,3) and (3,0)
(1,2)
Solution is probably (1,2) …
Check it:
2 – 1 = 1 true
2 + 1 = 3 true
therefore, (1,2) is the solution
Practice – Solving by Graphing
Inconsistent: no solutions



y = -3x + 5  (0,5) and (3,-4)
y = -3x – 2  (0,-2) and (-2,4)
They look parallel: No solution
Check it:
m1 = m2 = -3
Slopes are equal
therefore it’s an inconsistent
system
Practice – Solving by Graphing
Consistent: infinite sol’s



3y – 2x = 6  (0,2) and (-3,0)
-12y + 8x = -24  (0,2) and (-3,0)
(1,2)
Looks like a dependant system …
Check it:
divide all terms in the 2nd equation by -4
and it becomes identical to the 1st
equation
therefore, consistent, dependant system
Ex: Check whether the ordered pairs
are solns. of the system.
x-3y= -5
-2x+3y=10
(1,4)
1-3(4)= -5
1-12= -5
-11 = -5
*doesn’t work in the 1st
eqn, no need to check
the 2nd.
Not a solution.
A.
(-5,0)
-5-3(0)= -5
-5 = -5
B.
-2(-5)+3(0)=10
10=10
Solution
Ex: Solve the system graphically.
2x-2y= -8
2x+2y=4
(-1,3)
Ex: Solve the system graphically.
2x+4y=12
x+2y=6
1st eqn:
x-int (6,0)
y-int (0,3)
 2ND eqn:
x-int (6,0)
y-int (0,3)
 What does this mean?
the 2 eqns are for the
same line!
 ¸ many solutions

Ex: Solve graphically: x-y=5
2x-2y=9
 1st eqn:




x-int (5,0)
y-int (0,-5)
2nd eqn:
x-int (9/2,0)
y-int (0,-9/2)
What do you notice
about the lines?
They are parallel! Go
ahead, check the slopes!
No solution!
What is the solution of this system?
3x – y = 8
2y = 6x -16
1.
2.
3.
4.
(3, 1)
(4, 4)
No solution
Infinitely many solutions
Graph the system of equations. Determine whether the system has
one solution, no solution, or infinitely many solutions. If the system
has one solution, determine the solution.
1.
x  3y  3
3x  9 y  9
3
2. y  x  4
5
5 y  3x
3.
x y3
2x  y  6
y
The two equations in slopeintercept form are:
x
1
y x1
3
3
9
1
y x
or y   x  1
9
9
3
Plot points for each line.
Draw in the lines.
These two equations represent the same line.
Therefore, this system of equations has infinitely many solutions .
y
The two equations in slopeintercept form are:
3
y x4
5
3
y x
5
x
Plot points for each line.
Draw in the lines.
This system of equations represents two parallel lines.
This system of equations has
no points in common.
no solution
because these two lines have
y
The two equations in slopeintercept form are:
y  x  3
x
y  2x  6
Plot points for each line.
Draw in the lines.
This system of equations represents two intersecting lines.
The solution to this system of equations is a single point (3,0) .
Key Skills
Solve a system of two linear equations
in two variables graphically.
y
y = 2x  1
y= 1 x+4
2
6
4
–6 –4 –2
–2
solution: (2, 3)
–4
–6
2
2 4 6
x
Key Skills
Solve a system of two linear equations
in two variables graphically.
y
y + 2x = 2
6
4
y+x=1
2
solution:≈ (1, 0)
–6 –4 –2
–2
–4
–6
2 4 6
x
Key Skills
Solve a system of two linear equations
in two variables graphically.
y
y = 2x + 2
y = 2x + 4
No solution, why?
Because the 2 lines have
the same slope.
6
4
–6 –4 –2
–2
–4
–6
2
2 4 6
x
Key Skills
TRY THIS
Solve a system of two linear equations
in two variables graphically.
y
y = 3x + 2
6
4
y= 1 x-2
2
3
solution:≈ (-3, -1)
–6 –4 –2
–2
–4
–6
2 4 6
x
Key Skills
TRY THIS
Solve a system of two linear equations
in two variables graphically.
y
6
4
2x + 3y = -12
4x – 4y = 4
solution:≈ (-1.5, -3)
–6 –4 –2
–2
–4
–6
2
2 4 6
x
Consider the System
x y 5
x y 3
x
0
2
5
y
5
3
0
x
0
2
3
4, 1
y
-3
-1
0
BACK
Graph each system to find the
solution:
1.) x + y = -2
2x – 3y = -9
2.) x + y = 4
2x + y = 5
(-3, 1)
(1, 3)
3.) x – y = 5
2x + 3y = 0
(????)
4.) y = x + 2
y = -x – 4
(????)
5.) x = -2
y=5
(-2, 5)
Check whether the ordered pair
is a solution of the system:
1.) 3x + 2y = 4
-x + 3y = -5
(2, -1)
2.) 2x + y = 3
x – 2y = -1
(1, 1) or (0, 3)
3.) x – y = 3
3x – y = 11
(-5, -2) or (4, 1)