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Math 154 — Rodriguez
Angel — 5.1
Solving Systems of Equations Graphically
I. Systems of Linear Equations
A. A system of linear equations in two variables consists of two equations of the form
Ax + By = C.
"$ x + 2y = 4
$% y = ! 2x ! 1
Example: #
B. A solution to a system of equations consists of an ordered pair, (x,y), that satisfies
both equations.
Example: Determine if (2,1) is a solution to the above system. What about (—2,3)?
II. Solving systems by graphing
A. Recall: the graph of a linear equation represents all of the solutions to that equation.
Since we are looking for a solution to both equations, we want to see if there is one
point that is on both lines.
B.
What can happens when you graph two lines?
Case 1: The two lines intersect at one point.
y
•
x
•
The point of intersection satisfies both equations so it is
the solution of the system.
•
Solution:
•
The system is classified as a _______________ and
__________________ system.
Case 2: The two lines are parallel
y
x
•
The two lines never intersect so there is no ordered pair
that satisfies both equations.
•
Solution:
•
This system is classified as an ________________ system.
Case 3: The two lines are the same line
y
x
•
The two lines are the same line. All ordered pairs on the
line satisfy both equations.
•
Solution:
•
This system is classified as a _________________ and
_________________ system.
C. To solve by graphing:
1. Graph each equation on the same coordinate system (grid).
2. Look at what the lines do—intersect, parallel, same line—and state the solution.
3. If there is a solution, check it in both equations.
Examples: Determine the solution to each system of equations graphically. If the
system is dependent or inconsistent, so state. (Note: I expect ‘no solution’ or ‘infinite
number of solutions’ for those types of systems.)
"$ x + y = !4
$% y = !3x ! 6
1) #
"$ 3x ! 3y = 6
$% 2y = 2x ! 4
2) #
Angel — 5.1
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"$ !x + 2y = 0
$% 2x ! y = !3
3) #
"$
4) #
y = 2x ! 1
$% 4x ! 2y = !5
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