Download Math 142–Rodriguez Lehmann–3.1 system of linear equations

Math 142–Rodriguez
Lehmann–3.1
Using Graphs and Tables to Solve Systems
I. Systems of Linear Equations
A. A system of linear equations in two variables consists of two (or more) equations
of the form Ax + By = C, where A, B and C are real numbers.
⎧
⎪ x + 2y = 4
y = − 2x − 1
⎪
⎩
Example: ⎨
B. A ___________ to a system of equations consists of an ordered pair, ____, that
satisfies _____ the equations in the system.
Solution to the above system is (–2,3).
II. Solving systems by graphing
A.
The graph of a linear equation represents_________________________________.
Every point on the line is a _______________ of the equation.
B. If we were trying to solve systems by graphing, what might happen when we graph
two lines?
⎧⎪ equation 1
⎨
equation 2
⎩⎪
•
Case 1: The two lines intersect at one point.
• The two lines intersect at one point so that point of
intersection is the solution of the system.
• Solution:
• This is a one-solution system.
Case 2: The two lines are parallel
• The two lines never intersect so there is no ordered pair
that satisfies both equations. This means there is no
solution to the system.
• Solution:
• This is an inconsistent system.
Case 3: The two lines are the same line
• Since both lines are the same line, all ordered pairs
satisfy both equations so there are an infinite number
of solutions.
• Solution:
• This is a dependent system.
C. To solve by graphing:
1. Graph each equation on the same coordinate system (grid).
2. By inspection, state the solution set. (What do the lines do: intersect, parallel, or
same line?)
3. Check the solution in both original equations (when there is one solution).
Examples: Find the solution set of the system by graphing the equations by hand. If the
system is inconsistent or dependent, say so. If your result is one ordered pair, check that it
satisfies both equations.
⎧
⎪ x + 2y = 4
y = − 2x − 1
⎪
⎩
1) ⎨
⎧
⎪ 3x − 3y = 6
2y = 2x − 4
⎪
⎩
2) ⎨
⎧
⎪ 2x − 3y = −6
x + 3y = −3
⎪
⎩
3) ⎨
Lehmann – 3.1
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Examples: Use “intersect” on a graphing calculator to solve the system. Round coordinates
of solutions to the second decimal place. If the system is inconsistent or dependent, say so.
If your result is one ordered pair, check that it satisfies both equations approximately.
(Remember: check in the original equations.)
⎧⎪ y = 5.437x − 2.136
⎪⎩ y = −2.752x + 3.984
1) ⎨
⎧⎪
2) ⎨
y = 2x + 1
⎪⎩ 3y − 1 = 2 ( 3x + 1)
⎧ 1
y + x = −2
⎪
⎪
3) ⎨ 3
⎪ −1 x+ 1y=2
⎪
2
⎩ 4
⎧
⎪ y = −2x + 4
0.5y + x = −2
⎪
⎩
4) ⎨
III. Solving systems by using tables of solutions of equations
A table shows a list of ordered pairs that are solutions to an equation. By comparing the
tables we can see if the same ordered pair, or pairs, is listed for both equations. Is
there an x value that gives the same y value for both equations?
1)
Some solutions to
x
–3
y= –2x–1
5
y=–0.5x+2 3.5
Lehmann – 3.1
the
-2
3
3
equations
-1
0
1
1
–1 –3
2.5 2 1.5
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2)
Values of functions
x
0
1
2
3
f(x)
5
7
9 11
g(x) 85 82 79 76
4
13
73
IV. Applications of Systems
Lehmann – 3.1
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2.
3.
Lehmann – 3.1
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