Name:
Date:
Period:
Topic: Solving
Systems of Equations by Graphing
Essential Question: How can graphing systems of equation help you find a solution?
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Warm-Up: Vocabulary Match-up (Review)
1. Equation
a) same variable and exponent
2. Like terms
b) intercept & slope is
opposite reciprocal
of each other
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3. Parallel Lines
c) uses an equal sign
4. Perpendicular Lines d) where a graph
crosses the y-axis
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5. y – intercept
e) never
intercept &
same slope
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.

What is the solution of
these lines?
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.
S lope =
2
=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.
S lo p e =
2
= 2
1
y-in tercep t = -1
What is the solution of the system
graphed below?
1.
2.
3.
4.
(2, -2)
(-2, 2)
No solution
Infinitely many solutions
1) Find the solution to the following system by graphing:
2x + y = 4
x-y=2
To find the solution we must graph both equations.
- I will first graph using x- and y-intercepts
(plug in zeros) and then I’ll graph using
the slope-intercept form. Choose the
one that you are more comfortable using.
Graph the equations.
2x + y = 4
x-y=2
Where do the lines
intersect?
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!
Answer to question # 1)
2x + y = 4
x-y=2
Using x- and y-intercepts (plug in zeros).
2x + y = 4
(0, 4) and (2, 0)
x–y=2
(0, -2) and (2, 0)
Answer: (2, 0)
2) Find the solution to the following system by graphing:
y = 2x – 3
-2x + y = 1
 Graph both equations.
 Put both equations in same form
(standard or slope-intercept form).

I’ll start with slope-intercept form on this one!
Graphing the Systems of equations:
y = 2x – 3
-2x + y = 1
Answer to question # 2)
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!
3) 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
Answer to question # 3)
3x – y = 8
2y = 6x -16
•Put both equations in slope-intercept form:
3x – y = 8
3x – y = 8 --- To isolate for ‘y,’ first move ‘mx’ (slope & ‘x’ coordinate to the other side
– y = – 3x + 8 --- When moving ‘mx’ to the other side of the equation change the symbol
y = 3x – 8 --- ‘y’ cannot be a negative value, so we divided everything by negative
2y = 6x -16
2y = 6x -16 --- To isolate for ‘y,’ divide everything by 2
y = 3x - 8
Page 363 - 364 (10, 22, 26)
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
2.
y
3
x 4
5
5 y  3x
3.
x y 3
2x  y  6
Challenge:
Writing a System of Equations:
A plant nursery is growing a tree that is 3ft tall and
grows at an average rate of 1 ft per year. Another
tree at the nursery is 4ft tall and grows at an
average rate of 0.5ft per year. After how
many years will the trees be the same height?
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– intercept form 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.
Activity: Name the Quadrant
Materials:
• Color Pencils
• Rulers
• Graphing Paper
• Lined Paper
Activity: Name the Quadrant
7. y = -x + 1
y = 2x + 7
1. y = x – 1
y = 2x
5.
1
y
x +1
2
2. y = 2x + 1
y = 3x
3
y =
8. -2x + y = 6
y = 4x + 3
x +7
2
3. y = 5x
y = -x – 6
9. 7 + y = 9x
-9x + y = – 7
6.
y  -
3
x
2
4.
y = 3x + 8
y = 3x + 4
y 
1
2
x + 3
10. -6 +y = 2x
y=x–3
Home-Learning Assignment #6
Page 363 (4)
 Page 364 (35, 36, 38)
 Page 365 (43)
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