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. 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.
1
2.2.2: Solving Systems of Linear Equations
Key Concepts
• There are various methods to solving a system of
equations. One is the graphing method.
2
2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
Solving Systems of Equations by Graphing
• Graphing a system of equations on the same
coordinate plane allows you to visualize the solution
to the system.
• Use a table of values, the slope-intercept form of the
equations (y = mx + b), or a graphing calculator.
• Creating a table of values can be time consuming
depending on the equations, but will work for all
equations.
3
2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
• Equations not written in slope-intercept form will need
to be rewritten in order to determine the slope and
y-intercept.
• Once graphed, you can determine the number of
solutions the system has.
• Graphs of systems with one solution have two
intersecting lines. The point of intersection is the
solution to the system. These systems are considered
consistent, or having at least one solution, and are
also independent, meaning there is exactly one
solution.
4
2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
• Graphs of systems with no solution have parallel
lines. There are no points that satisfy both of the
equations. These systems are referred to as
inconsistent.
• Systems with an infinite number of solutions are
equations of the same line—the lines for the
equations in the system overlap. These are referred to
as dependent and also consistent, having at least
one solution.
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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
Key Concepts, continued
• Graphing the system of equations can sometimes be
inaccurate, but solving the system algebraically will
always give an exact answer.
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2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
Graphing Equations Using a TI-83/84:
Step 1: Press [Y=] and key in the equation using
[X, T, Θ, n] for x.
Step 2: Press [ENTER] and key in the second
equation.
Step 3: Press [WINDOW] to change the viewing
window, if necessary.
Step 4: Enter in appropriate values for Xmin, Xmax,
Xscl, Ymin, Ymax, and Yscl, using the arrow
keys to navigate.
Step 5: Press [GRAPH].
2.2.2: Solving Systems of Linear Equations
8
Key Concepts, continued
Step 6: Press [2ND] and [TRACE] to access the
Calculate Menu.
Step 7: Choose 5: intersect.
Step 8: Press [ENTER] 3 times for the point of
intersection.
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2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
Graphing Equations Using a TI-Nspire:
Step 1: Press the home key.
Step 2: Arrow over to the graphing icon (the picture of
the parabola or the U-shaped curve) and
press [enter].
Step 3: At the blinking cursor at the bottom of the
screen, enter in the equation and press
[enter].
Step 4: To change the viewing window: press [menu],
arrow down to number 4: Window/Zoom, and
click the center button of the navigation pad.
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2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
Step 5: Choose 1: Window settings by pressing the
center button.
Step 6: Enter in the appropriate XMin, XMax, YMin,
and YMax fields.
Step 7: Leave the XScale and YScale set to auto.
Step 8: Use [tab] to navigate among the fields.
Step 9: Press [tab] to “OK” when done and press
[enter].
Step 10: Press [tab] to enter the second equation, then
press [enter].
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2.2.2: Solving Systems of Linear Equations
Key Concepts, continued
Step 11: Enter the second equation and press [enter].
Step 12: Press [menu] and choose 6: Analyze Graph.
Step 13: Choose 4: Intersection.
Step 14: Select the lower bound.
Step 15: Select the upper bound.
12
2.2.2: Solving Systems of Linear Equations
Common Errors/Misconceptions
• incorrectly graphing each equation
• misidentifying the point of intersection
13
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
15
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
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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.
To check, substitute (3, 0) into both original
equations. The result should be a true statement.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 1, continued
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.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 1, continued
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.
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 1, continued
ì4x - 6y = 12
4. The system í
has one
î y = -x + 3
solution, (3, 0).
✔
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2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 1, continued
22
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.
y = 3x + 4
Write the equation in slopeintercept form (y = mx + b).
The second equation (y = 3x – 5) is already in slopeintercept form.
2.2.2: Solving Systems of Linear Equations
24
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
26
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.
There are no values for x and y that will make both
equations true.
27
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
✔
28
2.2.2: Solving Systems of Linear Equations
Guided Practice: Example 3, continued
29
2.2.2: Solving Systems of Linear Equations