259 Lecture 12 Spring 2017
Solving Systems of Equations
A System of Equations
 Consider the system of two equations
in two unknowns:
 x + 2y = 1
(1)
3x + 4y = -1
(2)
 Recall from algebra class that two
ways to solve a problem like this are
substitution and elimination.
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Example 1 (Substitution)
 Solve (1) or (2) for one variable in
terms of the other and substitute for
that variable in the other equation.
 x + 2y = 1
(1)
3x + 4y = -1
(2)
 Solving (1) for x yields
 x = 1 – 2y
(3)
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Example 1 (cont.)
 Using (3) we can substitute 1 – 2y for
x in (2):
 3(1 – 2y) + 4y = -1
3 – 6y + 4y = -1
-2y = -4
y=2
(4)
 (4) in (3) implies x = 1 – 2(2) = -3.
 Check (x, y) = (-3, 2) solves (1), (2).
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Example 2 (Elimination)
 x + 2y = 1
(1)
3x + 4y = -1
(2)
 Multiply both sides of (1) or (2) by a
non-zero constant to get coefficients
in front of one variable that can be
added to get zero.
 Then add to eliminate that variable
and solve for the remaining variable.
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Example 2 (cont.)
 Multiplying (1) by -2 on both sides,
we get an equivalent system;
 -2x - 4y = -2
(3)
3x + 4y = -1
(2)
 Adding (3) to (2) gives an equation
involving only x which can be solved
for x.
 x = -3
(4)
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Example 2 (cont.)
 Substituting x = -3 from (4) back into
either (1) or (2) (why these?), we can solve
for y.
 In this case, we choose (1):
 x + 2y = 1
-3 + 2y = 1
2y = 4
y=2
 Again, we find (x, y) = (-3, 2) and would
need to check it works in (1),(2)!
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Solving a System of Equations with
Technology
 How could we use technology to help
us solve system (1), (2).
 One way might be to use a calculator
with this capability.
 How about Mathematica or Excel?
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Example 3: Solving a System of
Equations in Excel





We can use the Solver to
solve a system of equations!
Let’s try with (1), (2).
To do so, we need to choose
one target cell (i.e. objective
cell) and specify appropriate
constraints.
Think of the system as
a*x + b*y = e
c*x + d*y = f,
with a=1, b=2, c=3, d=4,
e=1, and f=-1.
Then the objective can be to
set ax+by = 1, subject to the
constraints e = 1 and f = -1.
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Example 3: Solving a System of
Equations in Excel
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Example 4: Solving a System of
Equations in Mathematica
 For Mathematica,
we use the Solve
command:
Solve[{x+2y==1,
3x+4y==-1},{x,y}]
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Example 4: Solving a System of
Equations in Mathematica
 We can also solve
a system like this
in general:
 To solve the
system
a*x + b*y = e
c*x + d*y = f,
use the command:
Solve[{a*x+b*y==e,
c*x+d*y==f},{x,y}]
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Example 4: Solving a System of
Equations in Mathematica
 We find that for the
system
a*x + b*y = e
c*x + d*y = f,
the solution is:
x = (bf – de)/(ad – bc)
y = (af – ce)/(ad – bc)
 Try with coefficients and
right-hand side from (1),
(2) to see if we get the
same solution!
 For what choices of a, b,
c, d, e and f, do we get a
solution to this system?
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Systems of Equations in General
 We can generalize the
problem discussed
above to a system of n
equations in n
unknowns!
 For example, here is a
system of three
equations in three
unknowns:
 ax + by + cz = j
dx + ey + fz = k
gx + hy + iz = l
 How could we solve
this?
 By hand?
 Calculator?
 With Mathematica?
 With Excel?
 Other?
 When are we
guaranteed a solution?
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Example 5: Mathematica Solution
for 3 x 3 Case
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Example 5 (cont.)
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References
 The idea of how to use the Solver
comes from a web supplement for
Finite Mathematics (7th ed) by
Margaret Lial et al. found at this link:
 http://web.archive.org/web/2014101
7004542/http://wps.aw.com/aw_lial_
finitemath_7/0,1769,12520-,00.html
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