Lesson 2.7
Solving Systems of
Linear Equations by Graphing
Concept: Solving systems of equations in two variables.
EQ: How can I manipulate equations to solve a system of
equations? (REI 5-6,10-11)
Vocabulary: System of equations, Inconsistent, Consistent
(independent/dependent)
‘In Common’ Ballad: http://youtu.be/Br7qn4yLf-I
‘All I do is solve’ Rap: http://youtu.be/1qHTmxlaZWQ
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2.2.2: Solving Systems of Linear Equations
Introduction
The solution to a system of equations is the point or
points that make both equations true. Systems of
equations can have one solution, no solutions, or an
infinite number of solutions.
There are various methods to solving a system of
equations. One is the graphing method.
On a graph, the solution to a system of equations can be
easily seen. The solution to the system is the point of
intersection, the point at which two lines cross or
meet.
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2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
Intersecting Lines
Parallel Lines
Same Line
One solution
No solutions
Infinitely many
solutions
Consistent
Independent
Inconsistent
Consistent
Dependent
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2.2.2: Solving Systems of Linear Equations
Guided Practice
Example 1
Graph the system of equations. Then determine whether
the system has one solution, no solution, or infinitely
many solutions. If the system has a solution, name it.
ì4x - 6y = 12
í
î y = -x + 3
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 1, continued
1. Solve each equation for y.
The first equation needs to be solved for y.
4x – 6y = 12
Original equation
–6y = 12 – 4x
2
y = -2 + x
3
2
y = x-2
3
Subtract 4x from both sides.
Divide both sides by –6.
Write the equation in slopeintercept form (y = mx + b).
The second equation (y = –x + 3) is already in slopeintercept form.
2.2.2: Solving Systems of Linear Equations
5
Guided Practice: Example 1, continued
2. Graph both equations using the slopeintercept method.
The y-intercept of y =
2
3
x - 2 is –2. The slope is
2
3
.
The y-intercept of y = –x + 3 is 3. The slope is –1.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 1, continued
3. Observe the
graph.
The lines intersect at
the point (3, 0).
This appears to be the
solution to this
system of
equations.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 1, continued
To check, substitute (3, 0) into both original
equations. The result should be a true statement.
4x – 6y = 12
First equation in the system
4(3) – 6(0) = 12
Substitute (3, 0) for x and y.
12 – 0 = 12
Simplify.
12 = 12
This is a true statement.
y = –x + 3
Second equation in the system
(0) = –(3) + 3
Substitute (3, 0) for x and y.
0 = –3 + 3
Simplify.
0=0
This is a true statement.
2.2.2: Solving Systems of Linear Equations
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Guided Practice: Example 1, continued
ì4x - 6y = 12
4. The system í
has one
î y = -x + 3
solution, (3, 0).
This system is consistent because it has
at least one solution and it is independent
because it only has 1 solution.
✔
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2.2.2: Solving Systems of Linear Equations
Guided Practice
Example 2
Graph the system of equations. Then determine whether
the system has one solution, no solution, or infinitely
many solutions. If the system has a solution, name it.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 2, continued
1. Solve each equation for y.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 2, continued
2. Graph both equations using the slopeintercept method.
The y-intercept of both equations is 1.
The slope of both equations is 2.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 2, continued
3. Observe the graph.
The graphs of y = 2x +1
and -8x + 4y = 4 are the
same line.
There are infinitely many
solutions to this system of
equations.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 2, continued
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 2, continued
4. The system
has infinitely
many solutions.
This system is consistent because it has
at least one solution and it is dependent
✔
because it has more than one solution.
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2.2.2: Solving Systems of Linear Equations
Guided Practice
Example 3
Graph the system of equations. Then determine whether
the system has one solution, no solution, or infinitely
many solutions. If the system has a solution, name it.
ì-6x + 2y = 8
í
î y = 3x - 5
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 3, continued
1. Solve each equation for y.
The first equation needs to be solved for y.
–6x + 2y = 8
Original equation
2y = 8 + 6x
Add 6x to both sides.
y = 4 + 3x
Divide both sides by 2.
Write the equation in slopey = 3x + 4
intercept form (y = mx + b).
The second equation (y = 3x – 5) is already in slopeintercept form.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 3, continued
2. Graph both equations using the slopeintercept method.
The y-intercept of y = 3x + 4 is 4. The slope is 3.
The y-intercept of y = 3x – 5 is –5. The slope is 3.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 3, continued
3. Observe the graph.
The graphs of –6x + 2y = 8
and y = 3x – 5 are parallel
lines and never cross.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 3, continued
ì-6x + 2y = 8
4. The systemí
has no solutions
î y = 3x - 5
because there are no values for x and y that
will make both equations true. This
system is inconsistent because it has no
solutions.
✔
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2.2.2: Solving Systems of Linear Equations